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J Sci ComputDOI 10.1007/s10915-014-9946-6
One-Sided Position-Dependent Smoothness-IncreasingAccuracy-Conserving (SIAC) Filtering Over Uniformand Non-uniform Meshes
Jennifer K. Ryan · Xiaozhou Li ·Robert M. Kirby · Kees Vuik
Received: 14 January 2014 / Revised: 17 October 2014 / Accepted: 2 November 2014© Springer Science+Business Media New York 2014
Abstract In this paper, we introduce a new position-dependent smoothness-increasingaccuracy-conserving (SIAC) filter that retains the benefits of position dependence as pro-posed in van Slingerland et al. (SIAM J Sci Comput 33:802–825, 2011) while amelioratingsome of its shortcomings. As in the previous position-dependent filter, our new filter canbe applied near domain boundaries, near a discontinuity in the solution, or at the interfaceof different mesh sizes; and as before, in general, it numerically enhances the accuracy andincreases the smoothness of approximations obtained using the discontinuous Galerkin (dG)method. However, the previously proposed position-dependent one-sided filter had two sig-nificant disadvantages: (1) increased computational cost (in terms of function evaluations),brought about by the use of 4k + 1 central B-splines near a boundary (leading to increasedkernel support) and (2) increased numerical conditioning issues that necessitated the use ofquadruple precision for polynomial degrees of k ≥ 3 for the reported accuracy benefits to berealizable numerically. Our new filter addresses both of these issues—maintaining the samesupport size and with similar function evaluation characteristics as the symmetric filter in
Jennifer K. Ryan, Xiaozhou Li: Supported by the Air Force Office of Scientific Research (AFOSR), AirForce Material Command, USAF, under Grant No. FA8655-13-1-3017.Robert M. Kirby: Supported by the Air Force Office of Scientific Research (AFOSR), ComputationalMathematics Program (Program Manager: Dr. Fariba Fahroo), under Grant No. FA9550-08-1-0156.
J. K. Ryan (B)School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UKe-mail: [email protected]
X. Li · K. VuikDelft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlandse-mail: [email protected]
R. M. KirbySchool of Computing, University of Utah, Salt Lake City, UT, USAe-mail: [email protected]
K. Vuike-mail: [email protected]
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a way that has better numerical conditioning—making it, unlike its predecessor, amenablefor GPU computing. Our new filter was conceived by revisiting the original error analysisfor superconvergence of SIAC filters and by examining the role of the B-splines and theirweights in the SIAC filtering kernel. We demonstrate, in the uniform mesh case, that ournew filter is globally superconvergent for k = 1 and superconvergent in the interior (e.g.,region excluding the boundary) for k ≥ 2. Furthermore, we present the first theoreticalproof of superconvergence for postprocessing over smoothly varying meshes, and explainthe accuracy-order conserving nature of this new filter when applied to certain non-uniformmeshes cases. We provide numerical examples supporting our theoretical results and demon-strating that our new filter, in general, enhances the smoothness and accuracy of the solution.Numerical results are presented for solutions of both linear and nonlinear equations solvedon both uniform and non-uniform one- and two-dimensional meshes.
Keywords Discontinuous Galerkin method · Post-processing · SIAC filtering ·Superconvergence · Uniform meshes · Smoothly varying meshes · Non-uniform meshes
1 Introduction
Computational considerations are always a concern when dealing with the implementationof numerical methods that claim to have practical (engineering) value. The focus of thispaper is the formerly introduced smoothness-increasing accuracy-conserving (SIAC) class offilters, a class of filters that exhibit superconvergence behavior when applied to discontinuousGalerkin (dG) solutions. Although the previously proposed position-dependent filter (whichwe will, henceforth, call the SRV filter) introduced in van Slingerland et al. [20] met its statedgoals of demonstrating superconvergence, it contained two deficiencies which often made itimpractical for implementation and usage within engineering scenarios. The first deficiencyof the SRV filter was its reliance on 4k+1 central B-splines, which increased both the width ofthe stencil generated and increased the computational cost (in terms of functions evaluations)a disproportionate amount compared to the symmetric SIAC filter. The second deficiencyis one of numerical conditioning: the SRV filter requires the use of quadruple precision toobtain consistent and meaningful results, which makes it unsuitable for practical CPU-basedcomputations and for GPU computing. In this paper, we introduce a position-dependent SIACfilter that, like the SRV filter, allows for one-sided post-processing to be used near boundariesand solution discontinuities and which exhibits superconvergent behavior; however, our newfilter addresses the two stated deficiencies: it has a smaller spatial support with a reducednumber of function evaluations, and it does not require extended precision for error reductionto be realized.
To give context to what we will propose, let us review how we arrived at the currentlyavailable and used one-sided filter given in van Slingerland et al. [20]. The SIAC filter hasits roots in the finite element superconvergence extraction technique for elliptic equationsproposed by Bramble and Schatz [1], Mock and Lax [12] and Thomée [19]. The linearhyperbolic system counterpart for discontinuous Galerkin (dG) methods was introduced byCockburn et al. [5] and extended to more general applications in [9–11,13,14,18]. The post-processing technique can enhance the accuracy order of dG approximations from k + 1 to2k + 1 in the L2-norm. This symmetric post-processor uses 2k + 1 central B-splines oforder k + 1. However, a limitation of this symmetric post-processor was that it required asymmetric amount of information around the location being post-processed. To overcomethis problem, Ryan and Shu [15] used the same ideas in Cockburn et al. [5] to develop a
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one-sided post-processor that could be applied near boundaries and discontinuities in theexact solution. However, their results were not very satisfactory as the errors had a stair-stepping-type structure, and the errors themselves were not reduced when the post-processorwas applied to some dG solutions over coarse meshes. Later, van Slingerland et al. [20] recastthis formulation as a position-dependent SIAC filter by introducing a smooth shift functionλ(x̄) that aided in redefining the filter nodes and helped to ease the errors from the stair-stepping-type structure. In an attempt to reduce the errors, the authors doubled to 4k + 1 thenumber of central B-splines used in the filter when near a boundary. Further, they introduceda convex function that allowed for a smooth transition between boundary and symmetricregions.
The results obtained with this strategy were good for linear hyperbolic equations overuniform meshes, but new challenges arose. Issues were manifest when the position-dependentfilter was applied to equations whose solution lacked the (high) degree of regularity requiredfor valid post-processing. In some cases, this filter applied to certain dG solutions gave worseresults than when the original one-sided filter which used only 2k+ 1 central B-splines [15].Furthermore, it was observed that in order for the superconvergence properties expressed invan Slingerland et al. [20] to be fully realized, extended precision (beyond traditional doubleprecision) had to be used. Lastly, the addition of more B-splines did not come without cost.Figure 1 shows the difference between the symmetric filter, which was introduced in [1,5]and is applied in the domain interior, and the SRV filter when applied to the left boundary. Thesolution being filtered is at x = 0, and the filter extends into the domain. Upon examination,one sees that the position-dependent filter has a larger filter support and, by necessity, is notsymmetric near the boundary. The vast discrepancy in spatial extent is due to the numberof B-splines used: 2k + 1 in the symmetric case versus 4k + 1 in the one-sided case. Thepractical implications of this discrepancy is two-fold: (1) filtering at the boundary with the4k + 1 filter is noticeably more costly (in terms of function evaluations) than filtering in theinterior with the symmetric filter; and (2) the spatial extent of the one-sided filter forces oneto use the one-sided filter over a larger area/volume before being able to transition over tothe symmetric filter.
Fig. 1 Comparison of (a) the symmetric filter centered around x = 0 when the kernel is applied in the domaininterior and (b) the position-dependent filter at the boundary (represented by x = 0) before convolution witha quadratic approximation. Notice that the boundary filter requires a larger spatial support, the amplitude issignificantly larger in magnitude and the filter does not emphasize the point x = 0, which is the point beingpost-processed
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Recall that the SRV filter added two new features when attempting to improve the orig-inal Ryan and Shu one-sided filter: the authors added position-dependence to the filter andincreased the number of B-splines used. In attempting to overcome the deficiencies of thefilter in van Slingerland et al. [20], we reverted to the 2k+1 B-spline approach as in Ryan andShu [15] but added position-dependence. Although going back to 2k+1 B-splines does makethe filter less costly in the function evaluation sense, unfortunately, this approach did not leadto a filter that reduced the errors in the way we had hoped (i.e., at least error-order conserving,if not also superconvergent). We were forced to reconsider altogether how superconvergenceis obtained in the SIAC filter context; we outline a sketch of our thinking chronologically inwhat follows.
To conceive of a new one-sided filter containing all the benefits mentioned above withnone of the deficiencies, we had to harken back to the fundamentals of SIAC filtering. Wenot only examined the filter itself (e.g., its construction, etc.), but also the error analysis in[5,8]. From the analysis in [5,8] we can conclude that the main source of the aforementionedconditioning issue (expressed in terms of the necessity for increased precision) is the constantterm found in the expression for the error, which relies on the B-spline coefficients c(2k+1,�)
γ
in the SIAC filtering kernel∑
γ
∣∣∣c(2k+1,�)γ
∣∣∣ .
This quantity increases when the number of B-splines is increased. Further, the conditionnumber of the system used to calculate c(2k+1,�)
γ becomes quite large, on the order of 1024
for P4, increasing the possibility for round-off errors and thus requiring higher levels of
precision. As mentioned before, we attempted to still use 2k + 1 position-dependent centralB-splines, but this approach did not lead to error reduction. Indeed, the constant in the errorterm remains quite large at the boundaries. To reduce the error term, we concluded that oneneeded to add to the set of central B-splines the following: one non-central B-spline nearthe boundary. This general B-spline allows the filter to maintain the same spatial supportthroughout the domain, including boundary regions; it provides only a slight increase incomputational cost as there are now 2k + 2 B-splines to evaluate as part of the filter; andpossibly most importantly, it allows for error reduction. We note that our modifications to theprevious filter (e.g., going beyond central B-splines) do come with a compromise: we mustrelax the assumption of obtaining superconvergence. Instead, we merely require a reductionin error and a smoother solution. This new filter remains globally superconvergent for k = 1.
The new contributions of this paper are:
• A new one-sided position-dependent SIAC filter that allows filtering up to boundaries andthat ameliorates the two principle deficiencies identified in the previous 4k + 1 one-sidedposition-dependent filter;• Examination and documentation of the reasoning concerning the constant term in the error
analysis that led to the proposed work;• Demonstration that for the linear polynomial case the filtered approximation is always
superconvergent for a uniform mesh; and• Application of the scaled new filter to both smoothly varying and non-uniform (random)
meshes. In the smoothly varying case, we prove and demonstrate that we obtain supercon-vergence. For the general non-unform case, we still observe significant improvement inthe smoothness and an error reduction over the original dG solution, although full super-convergence is not always achieved. We show however that we remain accuracy-orderconserving.
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These important results are presented as follows: first, as we present the SIAC filter in thecontext of discontinuous Galerkin approximations, we review the important properties of thedG method and the position-dependent SIAC filter in Sect. 2. In Sect. 3, we introduce thenewly proposed filter and further establish some theoretical error estimates for the uniformand non-uniform (smoothly varying) cases. We present the numerical results over uniform andnon-uniform one-dimensional mesh structures in Sect. 4 and two-dimensional quadrilateralmesh structures in Sect. 5. Finally, conclusions are given in Sect. 6.
2 Background
In this section, we present the relevant background for understanding how to improve theSIAC filter, which includes the important properties of the discontinuous Galerkin (dG)method that make the application of SIAC filtering attractive as well as the building blocksof SIAC filtering—B-splines and the symmetric filter.
2.1 Important Properties of Discontinuous Galerkin Methods
We frame the discussion of the properties of dG methods in the context of a one-dimensionalproblem as the ideas easily extend to multiple dimensions. Further details about the discon-tinuous Galerkin method can be found in [2–4].
Consider a one-dimensional hyperbolic equation such as
ut + a1ux + a0u = 0, x ∈ � = [xL , xR] (2.1)
u(x, 0) = u0(x). (2.2)
To obtain a dG approximation, we first decompose � as � = ⋃Nj=1 I j where I j =
[x j− 12, x j+ 1
2] = [x j − 1
2�x j , x j + 12�x j ]. Then Eq. (2.1) is multiplied by a test func-
tion and integrated by parts. The test function is chosen from the same function space as thetrial functions, a piecewise polynomial basis. The approximation can then be written as
uh(x, t) =k∑
�=0
u(�)j (t)ϕ(�)j (x), for x ∈ I j .
Herein, we choose the basis functions ϕ(�)j (x) = P(�)(2(x − x j )/�x j ), where P(�) is theLegendre polynomial of degree � over [−1, 1]. For simplicity, throughout this paper werepresent polynomials of degree less than or equal to � by P
�.
In order to investigate the superconvergence property of the dG solution, it is importantto look at the usual convergence rate of the dG method. By estimating the error of the dGsolution, we obtain u − uh ∼ O(hk+1) in the L2-norm for sufficiently smooth initial datau0 [2]:
‖u − uh‖0 ≤ C hk+1‖u0‖Hk+2 ,
where h is the measure of the elements, h = �x for a uniform mesh and h = max j �x j fornon-uniform meshes. Another useful property is the superconvergence of the dG solution inthe negative-order norm [5], where we have
∥∥∂αh (u − uh)∥∥−�,� ≤ C h2k+1
∥∥∂αh u0∥∥
k+1,�
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for linear hyperbolic equations. This expression represents why accuracy enhancementthrough post-processing is possible. Unfortunately, this superconvergence property does nothold for non-uniform meshes whenα ≥ 1, which makes extracting the superconvergence overnon-uniform meshes challenging. However, we prove that, for certain non-uniform meshescontaining smoothness in their construction (i.e., smoothly varying meshes), the accuracyenhancement through the SIAC filtering is still possible.
2.2 A Review of B-splines
As the SIAC filter relies heavily on B-splines, here we review the definition of B-splinesgiven by de Boor in [7] as well as central B-splines. The reader is also directed to [17] formore on Splines.
Definition 2.1 (B-spline) Let t := (t j ) be a nondecreasing sequence of real numbers thatcreate a so-called knot sequence. The j th B-spline of order � for the knot sequence t isdenoted by B j,�,t and is defined, for � = 1, by the rule
B j,1,t(x) ={
1, t j ≤ x < t j+1;0, otherwise.
In particular, t j = t j+1 leads to B j,1,t = 0. For � > 1,
B j,�,t(x) = ω j,l,t B j,�−1,t + (1− ω j+1,�,t)B j+1,�−1,t, (2.3)
with
ω j,�,t(x) = x − t j
t j+�−1 − t j.
This notation will be used to create a new kernel near the boundaries.The original symmetric filter [5,16] relied on central B-splines of order � whose knot
sequence was uniformly spaced and symmetrically distributed t = − �2 ,− �−22 , . . . , �−2
2 , �2 ,yielding the following recurrence relation for central B-splines:
ψ(1)(x) = χ[−1/2,1/2](x),ψ(�+1)(x) = (ψ(1) ψ(�))(x)
=(�+1
2 + x)ψ(�)(x + 1
2
)+ ( �+12 − x
)ψ(�)(x − 1
2
)
�, � ≥ 1. (2.4)
For the purposes of this paper, it is convenient to relate the recurrence relation for central B-splines to the definition of general B-splines given in Definition 2.1. Relating the recurrencerelation to the definition can be done by defining t = t0, . . . , t� to be a knot sequence, anddenoting ψ(�)t (x) to be the 0th B-spline of order � for the knot sequence t,
ψ(�)t (x) = B0,�,t(x).
Note that the knot sequence t also represents the so-called breaks of the B-spline. The B-spline in the region [ti , ti+1), i = 0, . . . , � − 1 is a polynomial of degree � − 1, but inthe entire support [t0, t�], the B-spline is a piecewise polynomial. When the knots (t j ) aresampled in a symmetric and equidistant fashion, the B-spline is called a central B-spline.Notice that Eq. (2.4) for a central B-spline is a subset of the general B-spline definition wherethe knots are equally spaced. This new notation provides more flexibility than the previouscentral B-spline notation.
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2.3 Position-Dependent SIAC Filtering
The original position-dependent SIAC filter is a convolution of the dG approximation with acentral B-spline kernel
u (x̄) =(
K (2k+1,�)h uh
)(x̄). (2.5)
The convolution kernel is given by
K (2k+1,�)(x) =2k∑
γ=0
c(2k+1,�)γ ψ(�)(x − xγ ), (2.6)
where 2k + 1 represents the number of central B-splines, � the order of the B-splines andKh = 1
h K ( xh ). The coefficients c(2k+1,�)
γ are obtained from the property that the kernelreproduces polynomials of degree ≤ 2k. For the symmetric central B-spline filter [5,16],� = k + 1 and xγ = −k + γ , where k is the highest degree of the polynomial used in the dGapproximation. More explicitly, the symmetric kernel is given by
K (2k+1,�)(x) =2k∑
γ=0
c(2k+1,�)γ ψ(�)(x − (−k + γ )). (2.7)
Note that this kernel is by construction symmetric and uses an equal amount of informationfrom the neighborhood around the point being post-processed. While being symmetric issuitable in the interior domain when the function is smooth, it is not suitable for applicationnear a boundary, or when the solution contains a discontinuity.
The one-sided position-dependent SRV filter defined in van Slingerland et al. [20] is called“position-dependent” because of its change of support according to the location of the pointbeing post-processed. For example, near a boundary or discontinuity, a translation of thefilter is done so that the support of the kernel remains inside the domain. Furthermore, inthese regions, a greater number of central B-splines is required. Using more B-splines aids inimproving the magnitude of the errors near the boundary, while allowing superconvergence.In addition, the authors in van Slingerland et al. [20] increased the number of B-splinesused in the construction of the kernel to be 4k + 1. The position-dependent (SRV) filter forelements near the boundaries can then be written as1
K (4k+1,�)(x) =4k∑
γ=0
c(4k+1,�)γ ψ(�)(x − xγ ), (2.8)
where xγ depends on the location of the evaluation point x̄ used in Eq. (2.5) and at theboundaries is given by
xγ = −4k + γ + λ(x̄),with
λ(x̄) =⎧⎨
⎩min{
0,− 4k+�2 + x̄−xL
h
}, x̄ ∈ [xL ,
xL+xR2 ),
max{
0, 4k+�2 + x̄−xR
h
}, x̄ ∈ [ xL+xR
2 , xR].
(2.9)
Here xL and xR are the left and right boundaries, respectively.
1 Note that the notation used in the current manuscript is slightly different from the notation used in vanSlingerland et al. [20]. Instead of using r2 = 4k to denote the SRV filter we chose to use the number ofB-splines directly for the clarity of the discussion.
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The authors chose 4k + 1 central B-splines because, in their experience, using fewer(central) B-splines was insufficient for enhancing the error. Furthermore, in order to providea smooth transition from the boundary kernel to the interior kernel, a convex combination ofthe two kernels was used:
u h(x) = θ(x)(
K (2k+1,l)h uh
)(x)+ (1− θ(x))
(K (4k+1,l)
h uh
)(x), (2.10)
where θ(x) ∈ Ck−1 such that θ = 1 in the interior and θ = 0 in the boundary regions. Thisposition-dependent filter demonstrated better behavior in terms of error than the originalone-sided filter given by Ryan and Shu in [15]. Throughout the article, we will refer to theposition-dependent filter using 4k + 1 central B-splines as the SRV filter.
3 Proposed One-Sided Position-Dependent SIAC Filter
In this section, we propose a new one-sided position-dependent filter for application nearboundaries. We first discuss the deficiencies in the current position-dependent SIAC filter.We then propose a new position dependent filter that ameliorates the deficiencies of the SRVfilter; however, our new filter must make some compromises with regards to superconvergence(which will be discussed). Lastly, we prove that our new filter is globally superconvergentfor k = 1 and superconvergent in the interior of the domain for k ≥ 2.
3.1 Deficiencies of the Previous Position-Dependent SIAC Filter
The SRV filter was reported to reduce the errors when filtering near a boundary. However,applying this filter to higher-order dG solutions (e.g., P
4-or even P3-polynomials in some
cases) required using a multi-precision package (or at least quadruple precision) to reduceround-off error, leading to significantly increased computational time. Figure 2 shows the sig-nificant round-off error near the boundaries when using double precision for post-processingthe initial condition. The multi-precision requirement also makes the position-dependentkernel [20] near the boundaries, unsuitable for GPU computing.
To discover why this challenge arises requires revisiting the foundations of the filter—inparticular, the existing error estimates. The L2-error estimate given in Cockburn et al. [5],
Fig. 2 Comparison of the pointwise errors in log scale of the (a) original L2 projection solution, (b) the SRVfilter in van Slingerland et al. [20] for the 2D L2 projection using basis polynomials of degree k = 4, mesh80× 80. Double precision was used in these computations
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∥∥u − u h∥∥
0,� ≤ Ch2k+1, (3.1)
provides us insight into the cause of the issue by examining the constant C in more detail.The constant C depends on:
κ(r+1,�) =r∑
γ=0
∣∣∣c(r+1,�)γ
∣∣∣ , (3.2)
where cγ denotes the kernel coefficients and the value of r depends on the number of B-splines used to construct the kernel. The kernel coefficients are obtained by ensuring that thekernel reproduces polynomials of degree r by the convolution:
K (r+1,�)(x) (x)p = x p, p = 0, 1, . . . , r. (3.3)
We note that for the error estimates to hold, it is enough to ensure that the kernel reproducespolynomials up to degree 2k (for r ≥ 2k), although near the boundaries it was required thatthe kernel reproduces polynomials of degree 4k (r = 4k) in [8,20]. For filtering near theboundary, the value κ defined in Eq. (3.2) is on the order of 105 for k = 3 and 107 for k = 4,as can be seen in Fig. 4. This indicates one possible avenue (i.e., lowering κ) by which wemight generate an improved filter. A second avenue is by investigating the round-off errorstemming from the large condition number of the linear system generated to satisfy Eq. (3.3)and solved to find the kernel coefficients. The condition number of the generated matrix is onthe order of 1024 for k = 4. This leads to significant round-off error (e.g., the rule of thumbin this particular case being that 24 digits of accuracy are lost due to the conditioning ofthis system), hence requiring the use of high-precision/extended precision libraries for SIACfiltering to remain accurate (in both its construction and usage).
The requirement of using extended precision in our computations increases the computa-tional cost. In addition, the aforementioned discrepancy in the spatial extent of the filters dueto the number of B-splines used—2k+1 in the symmetric case versus 4k+1 in the one-sidedcase—leads to the boundary filter costing even more due to extra function evaluations. Theseextra function evaluations have led us to reconsider the position-dependent filter and proposea better conditioned and less computationally intensive alternative.
In order to apply SIAC filters near boundaries, we first no longer restrict ourselves tousing only central B-splines. Secondly, we seek to maintain a constant support size for boththe interior of the domain and the boundaries. The idea we propose is to add one generalB-spline for boundary regions, which is located within the already defined support size.Using general B-splines provides greater flexibility and improves the numerical evaluation(eliminating the explicit need for precision beyond double precision). To introduce our newposition-dependent one-sided SIAC filter, we discuss the one-dimensional case and how tomodify the current definition of the SIAC filter. Multi-dimensional SIAC filters are a tensorproduct of the one-dimensional case.
3.2 The New Position-Dependent One-Sided Kernel
Before we provide the definition of the new position-dependent one-sided kernel, we firstintroduce a new concept, that of a knot matrix. The definition of a knot matrix helps us tointroduce the new position-dependent one-sided kernel in a concise and compact form. Italso aids in demonstrating the differences between the new position-dependent kernel, thesymmetric kernel and the SRV filter. Informally, the idea behind introducing a knot matrix is toexploit the definition of B-splines in terms of their corresponding knot sequence t := (t j ), in
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the definition of the SIAC filter. In order to introduce a knot matrix, we will use the followingnotation: ψ(�)t (x) = B0,�,t(x) for the zeroth B-Spline of order � with knot sequence t.
Definition 3.1 (Knot matrix) A knot matrix, T, is an n × m matrix such that the γ th row,T(γ ), of the matrix T is a knot sequence with �+ 1 elements (i.e., m = �+ 1) that are usedto create the B-spline ψ(�)T(γ )(x). The number of rows n is specified based on the number ofB-splines used to construct the kernel.
To provide some context for the necessity of the definition of a knot matrix, we firstredefine some of the previous SIAC kernels discussed in terms of their knot matrices. Recallthat the general definition of the SIAC kernel relies on r + 1 central B-splines of order �.Therefore, we can use Definition 3.1 to rewrite the symmetric kernel given in Eq. (2.7) interms of a knot matrix as follows
K (2k+1,�)Tsym
(x) =2k∑
γ=0
c(2k+1,�)γ ψ
(�)Tsym(γ )
(x), (3.4)
where Tsym in this relation is a (2k+1)×(�+1)matrix. Each row in Tsym corresponds to theknot sequence of one of the constituent B-splines in the symmetric kernel. More specifically,the elements of Tsym are defined as
Tsym(i, j) = − �2+ j + i − k, i = 0, . . . , 2k; and j = 0, . . . , �.
For instance, for the first order symmetric SIAC kernel we have: � = 2 and k = 1. Therefore,the corresponding knot matrix can be defined as
Tsym =⎛
⎝−2 −1 0−1 0 10 1 2
⎞
⎠. (3.5)
The definition of a SIAC kernel in terms of a knot matrix can also be used to rewrite theoriginal boundary filter [15], which uses only 2k + 1 central B-splines at the left boundary.The knot matrix Tone for this case is given by
Tone =⎛
⎝−4 −3 −2−3 −2 −1−2 −1 0
⎞
⎠. (3.6)
Now we can define our new position-dependent one-sided kernel by generating a knotmatrix. The new position-dependent one-sided kernel consists of r + 1 = 2k + 1 centralB-splines and one general B-spline, and hence the knot matrix is of size (2k + 2)× (�+ 1).At a high-level, using the scaling of the kernel, the new position-dependent one-sided kernelcan be written as
K (r+1,�)hT (x) =
r+1∑
γ=0
c(r+1,�)γ ψ
(�)hT(γ )(x), (3.7)
where T(γ ) represents the γ th row of the knot matrix T, which is the knot sequenceT (γ, 0), . . . , T (γ, �). For the central B-spline, γ = 0, . . . , 2k and
ψ(�)hT(γ )(x) =
1
hψ(�)T(γ )
( x
h
).
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The added B-spline is a monomial defined as
ψ(�)hT(r+1)(x) =
1
hx�−1
T(r+1)
( x
h
),
where
x�−1T(r+1) =
{(x − T (r + 1, 0))�−1 , T (r + 1, 0) ≤ x ≤ T (r + 1, �);0, otherwise.
Therefore near the left boundary, the kernel in Eq. (3.7) can be rewritten as
K (r+1,�)hT (x) =
r∑
γ=0
c(r+1,�)γ ψ
(�)hT(γ )(x)
︸ ︷︷ ︸Position-dependent kernel with r+1=2k+1 central B-splines
+ c(r+1,�)r+1 ψ
(�)hT(r+1)(x)︸ ︷︷ ︸
General B-spline
.
(3.8)The kernel coefficients, c(r+1,�)
γ , γ = 0, . . . , r + 1 are obtained through reproducing poly-nomials of degree up to r+1.We have imposed further restrictions on the knot matrix for thedefinition of the new position-dependent one-sided kernel. First, for convenience we require
T (γ, 0) ≤ T (γ, 1) ≤ · · · ≤ T (γ, �), for γ = 0, . . . , r,+1
and
T (γ + 1, 0) ≤ T (γ, �), for γ = 0, . . . , r.
Second, the knot matrix, T, should satisfy
T (0, 0) ≥ x̄ − xR
hand T (r, �) ≤ x̄ − xL
h,
where h is the element size in a uniform mesh. This requirement is derived from the supportof the B-spline as well as the support needing to remain inside the domain. Recall thatthe support of the B-spline ψ(�)T (γ ) is [T (γ, 0), T (γ, �)], and the support of the kernel is[T (0, 0), T (r, �)]. For any x̄ ∈ [xL , xR], the post-processed solution at point x̄ can then bewritten as
u (x̄) = (K (r+1,�)hT uh
)(x̄) =
∫ ∞
−∞K (r+1,�)
hT (x̄ − ξ)uh(ξ)dξ
=∫ x̄−hT (0,0)
x̄−hT (r,�)K (r+1,�)
hT (x̄ − ξ)uh(ξ)dξ, (3.9)
where hT represents the scaled knot matrix. For the boundary regions, we force the interval[x̄ − hT (r, �), x̄ − hT (0, 0)] to be inside the domain � = [xL , xR]. This implies that
xL ≤ x̄ − hT (r, �), x̄ − hT (0, 0) ≤ xR,
and hence the requirement of T (0, 0) ≥ x̄−xRh and T (r, �) ≤ x̄−xL
h . Finally, we require thatthe kernel remain as symmetric as possible. This means the knots should be chosen as
Left : T ← T −(
T (r, �)− x̄ − xL
h
), for
x̄ − xL
h<
3k + 1
2, (3.10)
Right : T ← T −(
T (0, 0)− x̄ − xR
h
), for
xR − x̄
h<
3k + 1
2, (3.11)
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J Sci Comput
This shifting will increase the error and it is therefore still necessary to increase the numberof B-splines used in the kernel.
Because the symmetric filter yields superconvergence results, we wish to retain the originalform of the kernel as much as possible. Near the boundary, where the symmetric filter cannotbe applied, we keep the 2k + 1 shifted central B-splines and add only one general B-spline.To avoid increasing the spatial support of the filter, we will choose the knots of this generalB-spline dependent upon the knots of the 2k + 1 central B-splines in the following way:near the left boundary, we let the first 2k + 1 B-splines be central B-splines whereas the lastB-spline will be a general spline. The elements of the knot matrix are then given by
T (i, j) =
⎧⎪⎨
⎪⎩
−�− r + j + i + x̄−xLh , 0 ≤ i ≤ r, 0 ≤ j ≤ �;
x̄−xLh − 1, i = r + 1, j = 0;
x̄−xLh , i = r + 1, j = 1, . . . , �.
(3.12)
Similarly, we can design the new kernel near the right boundary, where the general B-splineis given by
ψ(�)T(0)(x) = x�−1
T(0) ={(T (0, �)− x)�−1 , T (0, 0) ≤ x ≤ T (r + 1, �);0, otherwise.
The elements of the knot matrix for the right boundary kernel are defined as
T (i, j) =
⎧⎪⎨
⎪⎩
x̄−xRh , i = 0, j = 0, . . . , �− 1;
x̄−xRh + 1, i = 0, j = �;
j + i − 1+ x̄−xRh , 1 ≤ i ≤ r + 1, 0 ≤ j ≤ �,
(3.13)
and the form of the kernel is then
K (r+1,�)hT (x) = c(r+1,�)
0 ψ(�)hT(0)(x)+
r+1∑
γ=1
c(r+1,�)γ ψ
(�)hT(γ )(x). (3.14)
We note that this “extra” B-spline is only used when x̄−xLh < 3k+1
2 or xR−x̄h < 3k+1
2 , otherwisethe symmetric central B-spline kernel is used.
We present a concrete example for the P1 case with � = 2. In this case, the knot matrices
for our newly proposed filter at the left and right boundaries are
TLeft =
⎛
⎜⎜⎝
−4 −3 −2−3 −2 −1−2 −1 0−1 0 0
⎞
⎟⎟⎠ , TRight =
⎛
⎜⎜⎝
0 0 10 1 21 2 32 3 4
⎞
⎟⎟⎠ . (3.15)
These new knot matrices are 4×3 matrices where, in the case of the filter for the left boundary,the first three rows express the knots of the three central B-splines and the last row expressesthe knots of the general B-spline. For the filter applied to the right boundary, the first rowexpresses the knots of the general B-spline and the last three rows express the knots of thecentral B-splines.
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Fig. 3 Comparison of the two boundary filters before convolution. (a) The SRV kernel and (b) the new kernelat the boundary with k = 2. The boundary is represented by x = 0. The new filter has reduced support andmagnitude
If we use the same form of the knot matrix to express the SRV kernel introduced in vanSlingerland et al. [20] at the left boundary for k = 1, we would have
TSRV =
⎛
⎜⎜⎜⎜⎝
−6 −5 −4−5 −4 −3−4 −3 −2−3 −2 −1−2 −1 0
⎞
⎟⎟⎟⎟⎠. (3.16)
Comparing the new knot matrix with the one used to obtain the SRV filter, we can see thatthey have the same number of columns, which indicates that they use the same order ofB-splines. There are fewer rows in the new matrix (2k + 2) than the number of rows fromthe original position-dependent filter (4k + 1). This indicates that the new filter uses fewerB-splines than the SRV filter.
To compare the new filter and the SRV filter, we plot the kernels used at the left boundary fork = 2. Figure 3 illustrates that the new position-dependent SIAC kernel places more weighton the evaluation point than the SRV kernel, and the SRV kernel has a significantly largermagnitude and support which we observed to cause problems, especially for higher-orderpolynomials (such as P
3 or P4). For this example, using the filter for quadratic approximations,
the scaling of the original position-dependent SIAC filter has a range from−150 to 150 versus−6 to 6 for the newly proposed filter.
3.3 Theoretical Results in the Uniform Case
The previous section introduced a new filter to reduce the errors of dG approximationswhile attempting to ameliorate the issues concerning the old filter. In this section, we discussthe theoretical results for the newly defined boundary kernel. Specifically, for k = 1 it isglobally superconvergent of order three. For higher degree polynomials, it is possible toobtain superconvergence only in the interior of the domain.
Recall from Eq. (3.7) that the new one-sided scaled kernel has the form
K (r+1,�)hT (x) =
r+1∑
γ=0
c(r+1,�)γ ψ
(�)hT(γ )(x). (3.17)
123
J Sci Comput
In the interior of the domain the symmetric SIAC kernel is used which consists of 2k + 1central B-splines
K (2k+1,�)hT (x) =
2k∑
γ=0
c(2k+1,�)γ ψ
(�)hT(γ )(x), (3.18)
and, near the left boundary the new one-sided kernel can be written as
K (r+1,�)hT (x) =
⎛
⎝r∑
γ=0
c(r+1,�)γ ψ
(�)hT(γ )(x)
⎞
⎠+ c(r+1,�)r+1 ψ
(�)hT(r+1)(x), r = 2k
where 2k+1 central B-splines are used together with one general B-spline. The scaled kernelK (r+1,�)
hT has the property that the convolution K (r+1,�)hT uh only uses information inside the
domain �.
Theorem 3.1 Let u be the exact solution to the linear hyperbolic equation
ut +d∑
i=1
Ai uxi + A0u = 0, x ∈ �× (0, T ],
u(x, 0) = u0(x), x ∈ �, (3.19)
where the initial condition u0(x) is a sufficiently smooth function. Here, � ⊂ Rd . Let uh be
the numerical solution to Eq. (3.19), obtained using a discontinuous Galerkin scheme withan upwind flux over a uniform mesh with mesh spacing h. Let u h(x̄) = (K (r+1,�)
hT uh)(x̄)be the solution obtained by applying our newly proposed filter which uses r + 1 = 2k + 1central B-splines of order � = k+ 1 and one general B-spline in boundary regions. Then theSIAC-filtered dG solution has the following properties:
(i) ‖(u−u h)(x̄)‖0,� ≤ C h3 for k = 1. That is, u h(x̄) is globally superconvergent of orderthree for linear approximations.
(ii) ‖(u− u h)(x̄)‖0,�\supp{Ks } ≤ C hr+1 when r + 1 ≤ 2k+ 1 central B-splines are used inthe kernel. Here supp{Ks} represents the support of the symmetric kernel. Thus, u h(x̄)is superconvergent in the interior of the domain.
(iii) ‖(u − u h)(x̄)‖0,� ≤ C hk+1 globally.
Proof We neglect the proof of properties (i) and (ii) as they are similar to the proofs inCockburn et al. [5] and Ji et al. [8]. Instead we concentrate on ‖(u − u h)(x̄)‖ ≤ Chk+1,
which is rather straight-forward.Consider the one-dimensional case (d = 1). Then the error can be written as
∥∥∥u − K (r+1,�)hT uh
∥∥∥0,�≤ �h,1 +�h,2,
where
�h,1 = ‖u − K (r+1,�)hT u‖0,� and �h,2 = ‖K (r+1,�)
hT (u − uh)‖0,�.The proof of higher order convergence for the first term,�H,1, is the same as in Cockburn
et al. [5] as the requirement on KhT does not change (reproduction polynomials up to degreer + 1). This means that
�h,1 ≤ hr+1
(r + 1)!C1|u|r+1,�.
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J Sci Comput
Now consider the second term,�h,2. Without loss of generality, we consider the filter forthe left boundary in order to estimate �h,2. The proofs for the filter in the interior and rightboundary are similar. We use the form of the kernel given in Eq. (3.8), which decomposesthe new filter into two parts: 2k + 1 central B-splines and one general B-spline. That is, wewrite
K (r+1,�)hT (x) =
⎛
⎝r∑
γ=0
c(r+1,�)γ ψ
(�)hT(γ )(x)
⎞
⎠
︸ ︷︷ ︸central B-splines
+ c(r+1,�)r+1 ψ
(�)hT(r+1)(x)︸ ︷︷ ︸
general B-spline
.
Setting e(x) = u(x)− uh(x), then
�h,2=∥∥∥K (r+1,�)
hT e∥∥∥
0,�1≤∥∥∥K (r+1,�)
hT
∥∥∥L1‖e‖0 ≤ sup
x∈�
⎛
⎝r∑
γ=0
|c(r+1,�)γ | + |c
(r+1,�)2k+1 |�
⎞
⎠ ‖e‖0.
Hence
�h,2 ≤ C supx∈�
⎛
⎝r∑
γ=0
|c(r+1,�)γ | + |c
(r+1,�)2k+1 |�
⎞
⎠ hk+1.
�Remark 3.1 Note that in this analysis we steered away from the negative-order norm argu-ment. Technically, the terms involving the central B-splines have a convergence rate ofr + 1 ≤ 2k + 1 as given in [5,8]. It is the new addition, the term involving the generalB-spline that presents the limitation and reduces the convergence rate to that of the dGapproximation itself (i.e., it is accuracy-order conserving).
To extend this to the multidimensional case (d > 1), given an arbitrary x = (x1, . . . , xd) ∈R
d , we set
ψ(�)T(γ )(x) =
d∏
i=1
ψ(�)T(γ )(xi ).
The filter for the multidimensional space considered is of the form
K (r+1,�)hT (x) =
r+1∑
γ=0
c(r+1,�)γ ψ
(�)hT(γ )(x),
where the coefficients c(�)γ are tensor products of the one-dimensional coefficients. To empha-size the general B-spline used near the boundary, we assume, without loss of generality, thatin the xk1 , . . . , xkd0
directions we need the general B-spline, where 0 ≤ d0 ≤ d . Then
ψ(�)hT(2k+1) =
d0∏
i=1
ψ(�)hT(2k+1)(xki ).
By applying the same idea we used for the one-dimensional case, the theorem is also true formulti-dimensional case.
We note that the constant in the final estimation is a product of two other constants, one of
them is determined by the filter (∑rγ=0 |c(r+1,�)
γ | + |c(r+1,�)r+1 |�
) and the other one is determined
123
J Sci Comput
Fig. 4 Plots demonstrating the effect of the coefficients on the error estimate for (a) P3- and (b) P
4-
polynomials. Shown is∑rγ=0 |c(r+1,�)
γ | using 2k + 1 central B-splines (blue dashed), using 4k + 1 central
B-splines (green dash dot-dot) and∑rγ=0 |c(r+1,�)
γ | + |c(r+1,�)r+1 |�
for the new filter (red line)
by the dG approximation. To further illustrate the necessity of examining the constant in theerror term which contributed by the filter, we provide Fig. 4. This figure demonstrates thedifference between
∑rγ=0 |c(r+1,�)
γ | for the previously introduced filters and our new filter in
which the constant gets modified to∑rγ=0 |c(r+1,�)
γ |+ |c(r+1,�)r+1 |�
. In Fig. 4, one can clearly seethat by adding a general spline to the r + 1 central B-splines, we are able, in the boundaryregions, to reduce the constant in the error term significantly.
3.4 Theoretical Results in the Non-uniform (Smoothly Varying) Case
In this section, we give a theoretical interpretation to the computational results presented inCurtis et al. [6]. This is done by using the newly proposed filter for non-uniform meshes andshowing that the new position-dependent filter maintains the superconvergence property inthe interior of the domain for smoothly varying meshes and is accuracy order conserving nearthe boundaries for non-uniform meshes. We begin by defining what we mean by a smoothlyvarying mesh.
Definition 3.2 (Smoothly varying mesh) Let ξ be a variable defined over a uniform mesh ondomain� ⊂ R, then a smoothly varying mesh defined over� is a non-uniform mesh whosevariable x satisfies
x = ξ + f (ξ), (3.20)
where f is a sufficiently smooth function and satisfies
f ′(ξ) > −1, ξ ∈ ∂�⇐⇒ ξ + f (ξ) ∈ ∂�.For example, we can choose f (ξ) = 0.5 sin(ξ) over [0, 2π]. The multi-dimensional defini-tion can be defined in the same way.
Lemma 3.2 Let u be the exact solution of a linear hyperbolic equation
ut +d∑
n=1
Anuxn + A0u = 0, x ∈ �× (0, T ], (3.21)
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J Sci Comput
with a smooth enough initial function and � ⊂ Rd . Let ξ be the variable for the uniform
mesh defined on � with size h, and x be the variable of a smoothly varying mesh defined in(3.20). Let uh(ξ) be the numerical solution to Eq. (3.21) over uniform mesh ξ , and uh(x)be the approximation over smoothly varying mesh x, both of them obtained by using thediscontinuous Galerkin scheme. Then the post-processed solution obtained by applying SIACfilter Kh(ξ) for uh(ξ) and K H (x) for uh(x) with a proper scaling H, are related by
‖u(x)− K H uh(x)‖0,� ≤ C‖u(ξ)− Kh uh(ξ)‖0,�.Here, the filter K can be any filter we mentioned in the previous section (symmetric filter,the SRV filter, and newly proposed position-dependent filter). Note that this means that weobtain the full 2k+1 superconvergence rate behavior for both the SRV and symmetric filters.
Proof The proof is straightforward. If the scaling H is properly chosen, a simple mappingcan be done from the smoothly varying mesh to the corresponding uniform mesh. The resultholds if the Jacobian is bounded (from the definition of smoothly varying mesh).
‖u(x)− K H uh(x)‖20,� =∫
�
(u(x)− K H uh(x))2 dx
x→ξ=∫
�̃
(u(ξ)− u h(ξ)
)2(1+ f ′(ξ))dξ ≤ ‖u(ξ)− Kh uh(ξ)‖20,�̃ ·max |1+ f ′(ξ)|.
According to the definition of smoothly varying mesh, � = �̃, we have
‖u(x)− K H uh(x)‖0,� ≤ C‖u(ξ)− Kh uh(ξ)‖0,�,
where C = (max� |1+ f ′|) 12 . �
Remark 3.2 The proof seems obvious, but it is important to choose a proper scaling for Hin the computations. Due to the smoothness and computational cost requirements, we needto keep H constant when treating points within the same element. Under this condition, thenatural choice is H = �x j when post-processing the element I j . It is now easy to see thatthere exists a position c in the element I j , such that
H = �x j = h(1+ f ′(c)).
Remark 3.3 Note that Theorem 3.1 (iii) still holds for generalized non-uniform meshes. Thisis due to the proof not relying on the properties (i.e., structure) of the mesh.
We have now shown that superconvergence can be achieved for interior solutions oversmoothly varying meshes. In the subsequent sections, we present numerical results thatconfirm our results on uniform and non-uniform (smoothly varying) meshes.
4 Numerical Results for One Dimension
The previous section introduced a new SIAC kernel by adding a general B-spline to a modifiedcentral B-spline kernel. The addition of a general B-spline helps to maintain a consistentsupport size for the kernel throughout the domain and eliminates the need for a multi-precisionpackage. This section illustrates the performance of the new position-dependent SIAC filteron one-dimensional uniform and non-uniform (smoothly varying and random) meshes. Wecompare our results to the SRV filter [20]. In order to provide a fair comparison between
123
J Sci Comput
the SRV and new filters, we mainly show the results using quadruple precision for a fewone-dimensional cases. Furthermore, in order to reduce the computational cost of the filterthat uses 4k+1 central B-splines, we neglect to implement the convex combination describedin Eq. (2.10). This is not necessary for the new filter, and was implemented in the SRV filterto ensure the transition from the one-sided filter to the symmetric filter was smooth.
This is the first time that the position-dependent filters have been tested on non-uniformmeshes. Although tests were performed using scalings of H = �x j and H = max�x j ,
we only present the results using a scaling of H = �x j . This scaling provides better resultsin boundary regions, which is one of the motivations of this paper. We note that the errorsproduced using a scaling of H = max�x j are quite similar and often produce smoothererrors in the interior of the domain for smoothly varying meshes.
Remark 4.1 The SRV filter requires using quadruple precision in the computations to elim-inate round-off error, which typically involves more computational effort than natively sup-ported double precision. The new filter only requires double precision. In order to give a faircomparison between the SRV filter and the new filter, for the one-dimensional examples wehave used quadruple precision to maintain a consistent computational environment.
4.1 Linear Transport Equation Over a Uniform Mesh
The first equation that we consider is a linear hyperbolic equation with periodic boundaryconditions,
ut + ux = 0, (x, t) ∈ [0, 1] × (0, T ] (4.1)
u(x, 0) = sin(2πx), x ∈ [0, 1]. (4.2)
The exact solution is a periodic translation of the sine function,
u(x, t) = sin(2π(x − t)).
For T = 0, this is simply the L2-projection of the initial condition. Here, we consider a finaltime of T = 1 and note that we expect similar results at later times.
The discontinuous Galerkin approximation error and the position-dependent SIAC fil-tered error results are shown in Tables 1 and 2 for both quadruple precision and doubleprecision. Using quadruple precision, both filters reduce the errors in the post-processedsolution, although the new filter has only a minor reduction in the quality of the error. How-ever, using double precision only the new filter can maintain this error reduction for P
3-and P
4-polynomials. We note that we concentrate on the results for P3- and P
4-polynomialsas there is no noticeable difference between double and quadruple precision for P
1- andP
2-polynomials.The pointwise error plots are given in Figs. 5 and 6. When using quadruple precision as
in Fig. 5, the SRV filter can reduce the error of the dG solution better than the new filterfor fine meshes. However, it uses 2k − 1 more B-splines than the newly generated filter.This difference is noticeable when using double precision, which is almost ten times fasterthan using quadruple precision for P
3 and P4. For such examples the new filter performs
better both computationally and numerically (in terms of error). Tables 1 and 2 show that theSRV filter can only reduce the error for fine meshes when using P
4 piecewise polynomials.The new filter performs as good as when using quadruple precision and reduces the errormagnitude at a reduced computational cost.
123
J Sci Comput
Tabl
e1
L2-a
ndL∞
-err
ors
fort
hedG
appr
oxim
atio
nto
geth
erw
ithth
eSR
Van
dne
wfil
ters
fort
helin
eart
rans
port
equa
tion
usin
gpo
lyno
mia
lsof
degr
eek=
1,...,
4(q
uadr
uple
prec
isio
n)ov
erun
ifor
mm
eshe
s
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
Qua
drup
lepr
ecis
ion
P1 20
4.02
E−0
3–
1.45
E−0
2–
1.98
E−0
3–
2.80
E−0
3–
1.98
E−0
3–
2.80
E−0
3–
401.
02E−0
31.
973.
82E−0
31.
922.
44E−0
43.
023.
46E−0
43.
022.
44E−0
43.
023.
46E−0
43.
02
802.
58E−0
41.
999.
79E−0
41.
963.
02E−0
53.
014.
28E−0
53.
013.
03E−0
53.
014.
28E−0
53.
01
P2 20
1.07
E−0
4–
3.67
E−0
4–
3.73
E−0
6–
5.82
E−0
6–
1.21
E−0
5–
8.27
E−0
5–
401.
34E−0
53.
004.
62E−0
52.
999.
42E−0
85.
311.
34E−0
75.
445.
52E−0
74.
455.
31E−0
63.
96
801.
67E−0
63.
005.
78E−0
63.
002.
48E−0
95.
243.
52E−0
95.
264.
79E−0
83.
536.
19E−0
73.
10
P3 20
2.06
E−0
6–
6.04
E−0
6–
1.53
E−0
7–
1.02
E−0
6–
2.30
E−0
6–
8.71
E−0
6–
401.
29E−0
74.
003.
80E−0
73.
992.
70E−1
09.
154.
00E−1
011
.32
4.14
E−0
99.
122.
27E−0
88.
58
808.
07E−0
94.
002.
38E−0
84.
001.
22E−1
27.
791.
73E−1
27.
858.
18E−1
28.
981.
20E−1
07.
56
P4 20
3.19
E−0
8–
7.02
E−0
8–
7.53
E−0
3–
7.33
E−0
2–
5.31
E−0
7–
1.99
E−0
6–
401.
00E−0
94.
992.
25E−0
94.
971.
99E−1
231
.82
3.12
E−1
234
.45
2.97
E−1
010
.80
1.58
E−0
910
.30
803.
14E−1
15.
007.
14E−1
14.
982.
23E−1
59.
803.
19E−1
59.
931.
37E−1
311
.08
1.55
E−1
29.
99
123
J Sci Comput
Tabl
e2
L2-
and
L∞
-err
ors
for
the
dGap
prox
imat
ion
toge
ther
with
the
SRV
and
new
filte
rsfo
rth
elin
ear
tran
spor
teq
uatio
nus
ing
poly
nom
ials
ofde
gree
k=
3,4
(dou
ble
prec
isio
n)ov
erun
ifor
mm
eshe
s
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
Dou
ble
prec
isio
n
P3 20
2.06
E−0
6–
6.04
E−0
6–
1.53
E−0
7–
1.02
E−0
6–
2.30
E−0
6–
8.71
E−0
6–
401.
29E−0
74.
003.
80E−0
73.
992.
70E−1
09.
154.
00E−1
011
.32
4.14
E−0
99.
122.
27E−0
88.
58
808.
07E−0
94.
002.
38E−0
84.
001.
25E−1
27.
753.
85E−1
26.
708.
18E−1
28.
981.
20E−1
07.
56
P4 20
3.19
E−0
8–
7.02
E−0
8–
7.53
E−0
3–
7.33
E−0
2–
5.31
E−0
7–
1.99
E−0
6–
401.
00E−0
94.
992.
25E−0
94.
973.
97E−1
127
.50
6.14
E−1
026
.83
2.97
E−1
010
.80
1.58
E−0
910
.30
803.
14E−1
15.
007.
14E−1
14.
981.
48E−1
11.
423.
28E−1
00.
901.
37E−1
311
.08
1.55
E−1
29.
99
123
J Sci Comput
Fig
.5C
ompa
riso
nof
the
poin
twis
eer
rors
inlo
gsc
ale
ofth
eor
igin
aldG
solu
tion
(lef
tcol
umn)
,the
SRV
filte
r(m
iddl
eco
lum
n)an
dth
ene
wfil
ter
(rig
htco
lum
n)fo
rth
elin
ear
tran
spor
tequ
atio
nov
erun
ifor
mm
eshe
sus
ing
poly
nom
ials
ofde
gree
k=
3,4
(top
and
bott
omro
ws,
resp
ectiv
ely)
.Qua
drup
lepr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.6C
ompa
riso
nof
the
poin
twis
eer
rors
inlo
gsc
ale
ofth
eor
igin
aldG
solu
tion
(lef
tcol
umn)
,the
SRV
filte
r(m
iddl
eco
lum
n)an
dth
ene
wfil
ter
(rig
htco
lum
n)fo
rth
elin
ear
tran
spor
tequ
atio
nov
erun
ifor
mm
eshe
sus
ing
poly
nom
ials
ofde
gree
k=
3,4
(top
and
bott
omro
ws,
resp
ectiv
ely)
.Dou
ble
prec
isio
nw
asus
edin
the
com
puta
tions
123
J Sci Comput
Additionally, we point out that the accuracy of the SRV filter depends on (1) having higherregularity of C4k+1, (2) a well-resolved dG solution, and (3) a wide enough support (at least5k + 1 elements). The same phenomenon will also be observed in the following tests suchas for a nonlinear equation. For the new filter, the support size remains the same throughoutthe domain – 3k + 1 elements—and a higher degree of regularity is not necessary.
4.2 Non-uniform Meshes
We begin by defining three non-uniform meshes that are used in the numerical examples.The meshes tested are:
Mesh 4.1 Smoothly varying mesh with periodicity The first mesh is a simple smoothlyvarying mesh. It is defined by x = ξ + b sin(ξ), where ξ is a uniform mesh variable andb = 0.5 as in Curtis et al. [6]. We note that the tests were also performed for different valuesof b; similar results were attained in all cases. This mesh has the nice feature that it is aperiodic mesh and that the elements near the boundaries have a larger element size.
Mesh 4.2 Smooth polynomial mesh The second mesh is also a smoothly varying mesh butdoes not have a periodic structure. It is defined by x = ξ − 0.05(ξ − 2π)ξ. For this mesh,the size of elements gradually decrease from left to right.
Mesh 4.3 Randomly varying mesh The third mesh is a mesh with randomly distributedelements. The element size varies between [0.8 h, 1.2 h], where h is the uniform mesh size.
We will now present numerical results demonstrating the usefulness of the position-dependent SIAC filter in van Slingerland et al. [20] and our new one-sided SIAC filter forthe aforementioned meshes.
4.3 Linear Transport Equation
The first example that we consider is a linear transport equation,
ut + ux = 0, (x, t) ∈ [0, 2π] × (0, T ]u(x, 0) = sin(x), (4.3)
with periodic boundary conditions and the errors calculated at T = 2π . We calculate thediscontinuous Galerkin approximations for this equation over the three different non-uniformmeshes (Meshes 4.1, 4.2, 4.3). The approximation is then post-processed at the final timein order to analyze the numerical errors. We note that although the boundary conditions forthe equation are periodic, in the boundary regions we implement the one-sided filter in vanSlingerland et al. [20] as the SRV filter and compare them with the new filter presented above.
The pointwise error plots for the periodically smoothly varying mesh are given in Fig. 7with the corresponding errors presented in Table 3. In the boundary regions, the SRV filterbehaves slightly better for coarse meshes than the new filter. However, we recall that thisfilter essentially doubles the support in the boundary regions. Additionally, we see that thenew filter has a higher convergence rate than k + 1 which is better than the theoreticallypredicted convergence rate.
For the smooth polynomial Mesh 4.2 (without a periodic property), the results of usingthe scaling of H = �x j are presented in Fig. 8 and Table 3. Unlike the previous example,without the periodic property, the SRV filter leads to significantly worse performance. TheSRV filter no longer enhances the accuracy order and has larger errors near the boundaries.
123
J Sci Comput
Fig
.7C
ompa
riso
nof
the
poin
twis
eer
rors
inlo
gsc
ale
ofth
eor
igin
aldG
solu
tion
(lef
tcol
umn)
,the
SRV
filte
r(m
iddl
eco
lum
n)an
dth
ene
wfil
ter
(rig
htco
lum
n)fo
rth
elin
ear
tran
spor
tEq.
(4.3
)ov
ersm
ooth
lyva
ryin
gm
esh
(Mes
h4.
1).T
heke
rnel
scal
ing
H=�
xj
and
quad
rupl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Tabl
e3
L2-
and
L∞
-err
ors
for
the
dGap
prox
imat
ion
toge
ther
with
the
SRV
and
the
new
filte
rsfo
rth
elin
ear
tran
spor
tEq.
(4.3
)ov
erth
eth
ree
Mes
hes
4.1,
4.2
and
4.3
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
Err
orO
rder
L2
erro
ror
der
L∞
erro
ror
der
L2
Err
orO
rder
L∞
erro
rO
rder
Mes
h1:
Smoo
thly
vary
ing
mes
h
P1 20
7.13
E−0
3–
1.97
E−0
2–
3.57
E−0
3–
5.83
E−0
3–
3.54
E−0
3–
5.29
E−0
3–
401.
65E−0
32.
115.
43E−0
31.
864.
35E−0
43.
046.
42E−0
43.
184.
34E−0
43.
036.
42E−0
43.
04
804.
04E−0
42.
031.
43E−0
31.
935.
37E−0
53.
027.
94E−0
53.
025.
37E−0
53.
027.
94E−0
53.
02
P2 20
2.43
E−0
4–
1.22
E−0
3–
2.41
E−0
4–
1.51
E−0
3–
1.56
E−0
4–
7.32
E−0
4–
403.
02E−0
53.
011.
55E−0
42.
989.
97E−0
77.
929.
73E−0
67.
282.
55E−0
65.
942.
29E−0
55.
00
803.
77E−0
63.
001.
95E−0
53.
008.
30E−0
96.
911.
34E−0
89.
501.
98E−0
73.
682.
13E−0
63.
42
P3 20
5.45
E−0
6–
1.87
E−0
5–
6.43
E−0
6–
5.51
E−0
5–
6.36
E−0
5–
2.02
E−0
4–
403.
39E−0
74.
011.
20E−0
63.
963.
11E−0
74.
373.
25E−0
64.
091.
72E−0
78.
537.
62E−0
78.
05
802.
12E−0
84.
007.
48E−0
84.
011.
45E−1
011
.06
1.81
E−0
910
.81
2.81
E−1
09.
262.
16E−0
98.
46
P4 20
1.56
E−0
7–
5.20
E−0
7–
5.15
E−0
7–
4.11
E−0
6–
1.72
E−0
5–
5.41
E−0
5–
404.
83E−0
95.
011.
66E−0
84.
976.
43E−0
96.
326.
56E−0
85.
972.
56E−0
89.
391.
12E−0
78.
91
801.
51E−1
05.
005.
22E−1
04.
991.
25E−1
19.
011.
80E−1
08.
511.
15E−1
111
.13
7.77
E−1
110
.50
Mes
h2:
Smoo
thpo
lyno
mia
lmes
h
P1 20
5.36
E−0
3–
1.68
E−0
2–
2.38
E−0
3–
3.48
E−0
3–
2.39
E−0
3–
3.48
E−0
3–
401.
26E−0
32.
104.
69E−0
31.
842.
93E−0
43.
024.
37E−0
42.
992.
94E−0
43.
034.
37E−0
42.
99
803.
08E−0
42.
031.
23E−0
31.
933.
63E−0
53.
015.
43E−0
53.
013.
64E−0
53.
015.
43E−0
53.
01
123
J Sci Comput
Tabl
e3
cont
inue
d
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
Err
orO
rder
L2
erro
ror
der
L∞
erro
ror
der
L2
Err
orO
rder
L∞
erro
rO
rder
P2 20
1.43
E−0
4–
8.00
E−0
4–
2.88
E−0
5–
2.45
E−0
4–
4.62
E−0
5–
2.18
E−0
4–
401.
80E−0
52.
991.
03E−0
42.
952.
24E−0
63.
682.
43E−0
53.
331.
22E−0
65.
249.
29E−0
64.
55
802.
25E−0
63.
001.
31E−0
52.
993.
91E−0
72.
526.
37E−0
61.
939.
79E−0
83.
641.
37E−0
62.
76
P3 20
3.15
E−0
6–
1.25
E−0
5–
1.52
E−0
5–
1.75
E−0
4–
1.53
E−0
5–
7.35
E−0
5–
401.
96E−0
74.
018.
05E−0
73.
969.
80E−0
87.
271.
50E−0
66.
863.
50E−0
88.
772.
34E−0
78.
29
801.
22E−0
84.
004.
97E−0
84.
022.
31E−0
95.
403.
77E−0
85.
325.
63E−1
19.
288.
26E−1
08.
15
P4 20
6.25
E−0
8–
2.67
E−0
7–
6.79
E−0
7–
5.38
E−0
6–
4.45
E−0
6–
2.13
E−0
5–
401.
96E−0
95.
008.
77E−0
94.
932.
13E−0
98.
322.
34E−0
87.
854.
12E−0
910
.08
2.76
E−0
89.
59
806.
14E−1
15.
002.
79E−1
04.
973.
03E−1
16.
135.
01E−1
05.
541.
74E−1
211
.21
1.59
E−1
110
.76
Mes
h3:
Ran
dom
lyva
ryin
gm
esh
P1 20
4.88
E−0
3–
1.49
E−0
2–
2.31
E−0
3–
6.73
E−0
3–
2.18
E−0
3–
3.83
E−0
3–
401.
15E−0
32.
094.
45E−0
31.
742.
84E−0
43.
028.
11E−0
43.
052.
76E−0
42.
986.
41E−0
42.
58
802.
87E−0
42.
001.
20E−0
31.
894.
85E−0
52.
551.
60E−0
42.
354.
71E−0
52.
551.
37E−0
42.
23
P2 20
1.17
E−0
4–
5.63
E−0
4–
3.78
E−0
4–
3.57
E−0
3–
2.23
E−0
5–
1.06
E−0
4–
401.
52E−0
52.
957.
90E−0
52.
833.
35E−0
53.
503.
45E−0
43.
379.
58E−0
74.
547.
44E−0
63.
83
801.
93E−0
62.
989.
85E−0
63.
008.
04E−0
62.
061.
17E−0
41.
561.
27E−0
72.
928.
51E−0
73.
13
P3 20
2.49
E−0
6–
1.06
E−0
5–
3.35
E−0
5–
3.61
E−0
4–
5.64
E−0
6–
2.90
E−0
5–
123
J Sci Comput
Tabl
e3
cont
inue
d
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
Err
orO
rder
L2
erro
ror
der
L∞
erro
ror
der
L2
Err
orO
rder
L∞
erro
rO
rder
401.
55E−0
74.
007.
46E−0
73.
837.
42E−0
75.
509.
11E−0
65.
316.
23E−0
99.
823.
80E−0
89.
58
801.
02E−0
83.
934.
67E−0
84.
002.
46E−0
84.
914.
57E−0
74.
321.
54E−1
05.
348.
71E−1
05.
45
P4 20
4.03
E−0
8–
1.49
E−0
7–
1.40
E−0
6–
1.47
E−0
5–
1.52
E−0
6–
7.85
E−0
6–
401.
37E−0
94.
885.
25E−0
94.
831.
42E−0
86.
631.
78E−0
76.
363.
20E−1
012
.21
1.98
E−0
911
.95
804.
40E−1
14.
961.
70E−1
04.
954.
03E−1
05.
137.
93E−0
94.
493.
68E−1
39.
773.
95E−1
28.
97
Asc
alin
gof
H=�
xj
alon
gw
ithqu
adru
ple
prec
isio
nw
asus
edin
the
com
puta
tions
123
J Sci Comput
Fig
.8C
ompa
riso
nof
the
poin
twis
eer
rors
inlo
gsc
ale
ofth
eor
igin
aldG
solu
tion
(lef
tcol
umn)
,the
SRV
filte
r(m
iddl
eco
lum
n)an
dth
ene
wfil
ter
(rig
htco
lum
n)fo
rth
elin
ear
tran
spor
tEq.
(4.3
)ov
ersm
ooth
poly
nom
ialm
esh
(Mes
h4.
2).T
heke
rnel
scal
ing
H=�
xj
and
quad
rupl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
On the other hand, the new filter still improves accuracy when the mesh is sufficiently refined(N = 40). Numerically the new filter obtains higher accuracy order than k + 1. For higherorder polynomials, P
3 and P4, we see that it achieves accuracy order of 2k + 1, but this is
not theoretically guaranteed.Lastly, the filters were applied to dG solutions over a randomly distributed mesh. For
this randomly varying mesh, the new filter again reduces the errors except for a very coarsemesh (see Table 3). The accuracy order is decreased compared to the smoothly varying meshexample, but it is still higher than k + 1. Unlike the smoothly varying mesh, there are moreoscillations in the errors (Fig. 9). However, the oscillations are still reduced compared to thedG solutions. We note that the results suggest that the SRV filter may be only suitable foruniform meshes.
4.4 Variable Coefficient Equation
In this example, we consider the variable coefficient equation:
ut + (au)x = f, x ∈ [0, 2π] × (0, T ]a(x, t) = 2+ sin(x + t),
u(x, 0) = sin(x), (4.4)
at T = 2π . Similar to the previous constant coefficient Eq. (4.3), we also test this variablecoefficient Eq. (4.4) over three different non-uniform meshes (Meshes 4.1, 4.2, 4.3). Sincethe results are similar to the previous linear transport Eq. (4.3), here we do not re-describe thedetails of the results. We only note that the results of variable coefficient equation have morewiggles than the constant coefficient equation. This may be an important issue in extendingthese ideas to nonlinear equations. To save space, we only show the P
3 and P4 results, P
1
and P2 are similar to the previous examples.
Figure 10 shows the pointwise error plots for the dG and post-processed approximationsover a smoothly varying mesh. The corresponding errors are given in Table 4. The results aresimilar to the linear transport equation. The two filters perform similarly, with the new filterusing fewer function evaluations.
For the smooth polynomial mesh 4.2, we show the pointwise error plots in Fig. 11. Thecorresponding errors are given in Table 4. In this example we see that the new filter behavesbetter at the boundaries than the SRV filter. This may be due to the more compact kernelsupport size.
Finally, we test the variable coefficient Eq. (4.4) over randomly varying mesh 4.3. Similarto the linear transport example, the pointwise errors plots (Fig. 12) show more oscillationsthan the smoothly varying mesh examples. We again see the new filter has better errors atthe boundaries than the SRV filter.
5 Numerical Results for Two Dimensions
5.1 Linear Transport Equation Over a Uniform Mesh
To demonstrate the performance of the new filter in two-dimensions, we consider the solutionto a linear transport equation,
ut + ux + uy = 0, (x, y) ∈ [0, 2π] × [0, 2π], (5.1)
123
J Sci Comput
Fig
.9C
ompa
riso
nof
the
poin
twis
eer
rors
inlo
gsc
ale
ofth
eor
igin
aldG
solu
tion
(lef
tcol
umn)
,the
SRV
filte
r(m
iddl
eco
lum
n)an
dth
ene
wfil
ter
(rig
htco
lum
n)fo
rth
elin
ear
tran
spor
tEq.
(4.3
)ov
erra
ndom
lyva
ryin
gM
esh
4.3.
The
kern
elsc
alin
gH=�
xj
and
quad
rupl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.10
Com
pari
son
ofth
epo
intw
ise
erro
rsin
log
scal
eof
the
orig
inal
dGso
lutio
n(l
eftc
olum
n),t
heSR
Vfil
ter(
mid
dle
colu
mn)
and
the
new
filte
r(ri
ghtc
olum
n)fo
rthe
vari
able
coef
ficie
ntE
q.(4
.4)
over
smoo
thly
vary
ing
mes
h(M
esh
4.1)
.The
kern
elsc
alin
gH=�
xj
and
quad
rupl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Tabl
e4
L2-
and
L∞
-err
ors
for
the
dGap
prox
imat
ion
toge
ther
with
the
SRV
and
the
new
filte
rsfo
rth
eva
riab
leco
effic
ient
Eq.
(4.4
)us
ing
adG
appr
oxim
atio
nof
poly
nom
ial
degr
eek=
3,4
over
the
thre
eM
eshe
s4.
1,4.
2an
d4.
3
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
Mes
h1:
Smoo
thly
vary
ing
mes
h
P3 20
5.54
E−0
6–
1.93
E−0
5–
4.40
E−0
6–
3.66
E−0
5–
6.36
E−0
5–
2.02
E−0
4–
403.
41E−0
74.
021.
21E−0
64.
003.
14E−0
73.
813.
25E−0
63.
491.
72E−0
78.
537.
61E−0
78.
05
802.
12E−0
84.
017.
50E−0
84.
011.
45E−1
011
.08
1.81
E−0
910
.81
2.78
E−1
09.
272.
05E−0
98.
53
P4 20
1.62
E−0
7–
5.69
E−0
7–
1.89
E−0
5–
1.44
E−0
4–
1.72
E−0
5–
5.41
E−0
5–
404.
95E−0
95.
031.
77E−0
85.
005.
74E−0
911
.68
5.82
E−0
811
.28
2.56
E−0
89.
391.
12E−0
78.
91
801.
53E−1
05.
015.
48E−1
05.
021.
26E−1
18.
831.
76E−1
08.
371.
16E−1
111
.11
7.26
E−1
110
.59
Mes
h2:
Smoo
thpo
lyno
mia
lmes
h
P3 20
3.15
E−0
6–
1.27
E−0
5–
2.70
E−0
5–
3.05
E−0
4–
1.53
E−0
5–
7.36
E−0
5–
401.
96E−0
74.
018.
06E−0
73.
981.
31E−0
77.
691.
54E−0
67.
623.
55E−0
88.
752.
38E−0
78.
28
801.
22E−0
84.
004.
98E−0
84.
027.
51E−0
94.
131.
27E−0
73.
606.
25E−1
19.
157.
84E−1
08.
24
P4 20
6.40
E−0
8–
2.82
E−0
7–
2.95
E−0
6–
2.42
E−0
5–
4.45
E−0
6–
2.13
E−0
5–
401.
98E−0
95.
018.
94E−0
94.
986.
84E−0
72.
111.
12E−0
51.
104.
12E−0
910
.08
2.76
E−0
89.
60
806.
18E−1
15.
002.
80E−1
05.
001.
51E−0
98.
833.
50E−0
88.
331.
59E−1
211
.34
1.55
E−1
110
.80
Mes
h3:
Ran
dom
lyva
ryin
gm
esh
P3 20
2.49
E−0
6–
9.61
E−0
6–
1.11
E−0
4–
8.98
E−0
4–
5.63
E−0
6–
2.90
E−0
5–
123
J Sci Comput
Tabl
e4
cont
inue
d
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
401.
56E−0
74.
007.
18E−0
73.
742.
12E−0
65.
712.
55E−0
55.
147.
96E−0
99.
474.
31E−0
89.
39
801.
02E−0
83.
934.
72E−0
83.
935.
91E−0
85.
171.
06E−0
64.
593.
15E−1
04.
661.
91E−0
94.
50
P4 20
4.07
E−0
8–
1.56
E−0
7–
2.45
E−0
5–
1.96
E−0
4–
1.52
E−0
6–
7.85
E−0
6–
401.
37E−0
94.
895.
31E−0
94.
874.
48E−0
75.
776.
18E−0
64.
992.
98E−1
012
.31
1.79
E−0
912
.09
804.
41E−1
14.
961.
73E−1
04.
941.
26E−0
98.
481.
91E−0
88.
332.
64E−1
26.
822.
19E−1
16.
35
The
filte
rsar
eus
ing
scal
ing
H=�
xj.
Qua
drup
lepr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.11
Com
pari
son
ofth
epo
intw
ise
erro
rsin
log
scal
eof
the
orig
inal
dGso
lutio
n(l
eftc
olum
n),t
heSR
Vfil
ter(
mid
dle
colu
mn)
and
the
new
filte
r(ri
ghtc
olum
n)fo
rthe
vari
able
coef
ficie
ntE
q.(4
.4)
over
smoo
thpo
lyno
mia
lmes
h(M
esh
4.2)
.The
kern
elsc
alin
gH=�
xj
and
quad
rupl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.12
Com
pari
son
ofth
epo
intw
ise
erro
rsin
log
scal
eof
the
orig
inal
dGso
lutio
n(l
eftc
olum
n),t
heSR
Vfil
ter(
mid
dle
colu
mn)
and
the
new
filte
r(ri
ghtc
olum
n)fo
rthe
vari
able
coef
ficie
ntE
q.(4
.4)
over
rand
omly
vary
ing
mes
h(M
esh
4.3)
.The
kern
elsc
alin
gH=�
xj
and
quad
rupl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Tabl
e5
L2-
and
L∞
-err
ors
for
the
dGap
prox
imat
ion
toge
ther
with
the
SRV
and
new
filte
rsfo
ra
2Dlin
ear
tran
spor
tequ
atio
nso
lved
over
aun
ifor
mm
esh
usin
gpo
lyno
mia
lsof
degr
eek=
3,4
Mes
hdG
SRV
filte
rN
ewfil
ter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
P3 20×
203.
30E−0
6–
1.21
E−0
5–
2.60
E−0
7–
1.12
E−0
4–
2.39
E−0
6–
1.80
E−0
5–
40×
402.
06E−0
74.
007.
60E−0
63.
994.
69E−1
09.
113.
11E−0
95.
177.
01E−0
98.
415.
11E−0
88.
46
80×
801.
29E−0
84.
004.
76E−0
84.
001.
74E−1
14.
755.
50E−0
9-0
.82
7.97
E−1
16.
461.
02E−0
95.
65
P4 20×
204.
71E−0
8–
1.41
E−0
7–
2.77
E−0
8–
1.35
E−0
6–
5.25
E−0
7–
3.77
E−0
6–
40×
401.
46E−0
95.
014.
50E−0
94.
972.
55E−0
80.
122.
84E−0
6−1
.07
3.83
E−1
010
.42
3.40
E−0
910
.11
80×
804.
44E−1
15.
041.
43E−1
04.
982.
73E−0
8−0
.10
7.86
E−0
6-1
.47
3.00
E−1
310
.31
3.12
E−1
210
.09
Dou
ble
prec
isio
nw
asus
edin
the
com
puta
tions
123
J Sci Comput
Fig
.13
Com
pari
son
ofth
epo
intw
ise
erro
rsin
log
scal
eof
the
orig
inal
dGso
lutio
n(l
eft)
,the
SRV
filte
r(m
iddl
e)an
dth
ene
wfil
ter
(rig
ht)
for
the
2Dlin
ear
tran
spor
tequ
atio
nus
ing
poly
nom
ials
ofde
gree
k=
4an
da
unif
orm
80×
80m
esh.
Dou
ble
prec
isio
nw
asus
edin
the
com
puta
tions
123
J Sci Comput
Tabl
e6
L2-
and
L∞
-err
ors
for
the
dGap
prox
imat
ion
toge
ther
with
the
SRV
and
new
filte
rsfo
rth
e2D
linea
rtr
ansp
ortE
q.(5
.1)
usin
gpo
lyno
mia
lsof
degr
eek=
3,4
over
the
thre
em
eshe
s:M
eshe
s4.
1,4.
2an
d4.
3
Mes
hdG
SRV
Filte
rN
ewFi
lter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
Mes
h1:
Smoo
thly
vary
ing
mes
h
P3 20×
208.
74E−0
6–
5.39
E−0
5–
6.94
E−0
6–
1.10
E−0
4–
6.71
E−0
5–
4.04
E−0
4–
40×
405.
45E−0
74.
003.
39E−0
64.
003.
68E−0
74.
246.
49E−0
64.
082.
09E−0
78.
331.
66E−0
67.
93
80×
803.
40E−0
84.
002.
06E−0
74.
031.
50E−1
011
.76
9.01
E−0
99.
497.
33E−1
08.
167.
76E−0
98.
92
P4 20×
201.
93E−0
7–
1.05
E−0
6–
9.25
E−0
7–
8.56
E−0
6–
3.26
E−0
5–
1.12
E−0
4–
40×
406.
00E−0
95.
013.
32E−0
84.
983.
38E−0
84.
774.
17E−0
61.
042.
67E−0
810
.25
2.31
E−0
78.
92
80×
801.
88E−1
05.
001.
04E−0
95.
002.
07E−0
80.
719.
13E−0
6-1
.13
1.90
E−1
110
.46
1.61
E−1
010
.49
Mes
h2:
Smoo
thpo
lyno
mia
lmes
h
P3 20×
204.
56E−0
6–
3.03
E−0
5–
1.55
E−0
5–
3.49
E−0
4–
1.59
E−0
5–
1.38
E−0
4–
40×
402.
85E−0
74.
001.
92E−0
63.
981.
23E−0
76.
983.
04E−0
66.
844.
67E−0
88.
415.
14E−0
78.
67
80×
801.
78E−0
84.
001.
20E−0
74.
003.
35E−0
95.
208.
01E−0
85.
252.
43E−1
07.
594.
61E−0
96.
80
P4 20×
208.
48E−0
8–
3.27
E−0
7–
1.38
E−0
6–
1.48
E−0
5–
5.92
E−0
6–
3.27
E−0
5–
40×
402.
65E−0
95.
001.
74E−0
84.
923.
21E−0
85.
434.
99E−0
61.
574.
65E−0
910
.30
5.79
E−0
89.
14
80×
808.
31E−1
15.
005.
58E−1
04.
962.
51E−0
80.
356.
88E−0
6-0
.46
3.29
E−1
210
.46
3.49
E−1
110
.27
Mes
h3:
Ran
dom
lyva
ryin
gm
esh
P3 20×
203.
47E−0
6–
2.16
E−0
5–
3.46
E−0
5–
6.43
E−0
4–
3.90
E−0
6–
3.83
E−0
5–
123
J Sci Comput
Tabl
e6
cont
inue
d
Mes
hdG
SRV
Filte
rN
ewFi
lter
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
L2
erro
rO
rder
L∞
erro
rO
rder
40×
402.
23E−0
73.
961.
52E−0
63.
831.
90E−0
64.
193.
59E−0
54.
161.
28E−0
88.
251.
25E−0
78.
26
80×
801.
41E−0
83.
989.
65E−0
83.
989.
94E−0
84.
262.
71E−0
63.
732.
97E−1
05.
433.
88E−0
95.
01
P4 20×
205.
83E−0
8–
2.84
E−0
7–
3.06
E−0
6–
5.19
E−0
5–
1.06
E−0
6–
8.87
E−0
5–
40×
401.
90E−0
94.
941.
04E−0
84.
772.
64E−0
86.
862.
64E−0
64.
307.
80E−1
010
.41
1.01
E−0
89.
78
80×
806.
06E−1
14.
973.
48E−1
04.
901.
46E−0
80.
857.
09E−0
6-1
.43
6.60
E−1
310
.20
7.85
E−1
210
.33
The
filte
rsus
eth
esc
alin
gof
Hx=�
xj
inx-
dire
ctio
nan
dH
y=�
yj
iny-
dire
ctio
n.D
oubl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.14
Com
pari
son
ofth
epo
intw
ise
erro
rsin
log
scal
eof
the
orig
inal
dGso
lutio
n(l
eft)
,the
SRV
filte
r(m
iddl
e)an
dth
ene
wfil
ter
(rig
ht)
for
the
2Dlin
ear
tran
spor
tEq.
(5.1
)us
ing
poly
nom
ials
ofde
gree
k=
3,4
over
asm
ooth
lyva
ryin
g80×
80m
esh
(Mes
h4.
2).T
hefil
ters
use
the
scal
ing
ofH
x=�
xj
inx-
dire
ctio
nan
dH
y=�
yj
iny-
dire
ctio
n.D
oubl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.15
Com
pari
son
ofth
epo
intw
ise
erro
rson
alo
gsc
ale
betw
een
the
SRV
and
new
filte
rsfo
rth
e2D
linea
rtr
ansp
ort
equa
tion
usin
gpo
lyno
mia
lsof
degr
eek=
3,4.
Asm
ooth
-var
ying
mes
hde
fined
byx=ξ−
b(ξ−
2π)ξ,
y=ξ−
b(ξ−
2π)ξ
with
b=
0.05
was
used
.Filt
ersc
alin
gw
asba
sed
upon
the
loca
lele
men
tsiz
e.D
oubl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
Fig
.16
Com
pari
son
ofth
epo
intw
ise
erro
rsin
log
scal
eof
the
orig
inal
dGso
lutio
n(a
),th
eSR
Vfil
ter
(b)
and
the
new
filte
r(c
)fo
rth
e2D
linea
rtr
ansp
ort
Eq.
(5.1
)us
ing
poly
nom
ials
ofde
gree
k=
3,4
over
ara
ndom
lyva
ryin
g80×
80m
esh.
The
filte
rsus
eth
esc
alin
gof
Hx=�
xj
inx-
dire
ctio
nan
dH
y=�
yj
iny-
dire
ctio
n.D
oubl
epr
ecis
ion
was
used
inth
eco
mpu
tatio
ns
123
J Sci Comput
u(x, y, 0) = sin(x + y) (5.2)
at T = 2π . Due to the computational cost to obtain the post-processed solution, we onlypresent the 2D results using double precision. Table 5 shows that the accuracy is affectedby the round-off error, especially for the previous one-sided position-dependent filter. Suchsignificant round-off error appears to destroy the accuracy. Although the error magnitudenear the boundaries is larger than the regions where a symmetric filter is used, the new filterreduces the error and improves smoothness of the dG solution (Fig. 13).
5.2 Linear Transport Equation Over a Non-uniform Mesh
For the 2D example, we consider the same linear transport equation as above, now over non-uniform meshes. The non-uniform meshes we consider are rectangular grids, in which thetessellations in the x- and y-directions are generated similar to Meshes 4.1, 4.2 and 4.3. Dou-ble precision was used for all two-dimensional computations. Unlike the one-dimensionalexample, the results of the SRV filter are significantly affected by the round-off error, espe-cially near the four corners of the grids. This round-off error completely destroys the accuracyand smoothness near the boundaries. Compared to the SRV filter, the new filter performs muchbetter. In the following examples, we can clearly see the improvement in the accuracy andsmoothness compared to the original dG approximations. From all the tests we performed, itis easy to see that the new filter is more suitable than the SRV filter over non-uniform meshes,and the practical performance of the new filter is better than the theoretical prediction.
For the P3 case, because of the periodicity, the SRV filter seems slightly better in L2
norm than the new filter. However, if we look at the L∞ norm, we can see the new filter stillbehaves better than the SRV filter (see Table 6). For the P
4 case, we can see that even theideal periodic property can not hide the fact that the SRV filter is not suitable for non-uniformmeshes—the SRV filter is worse than the new filter and even the original dG solution. InFig. 14, the round-off error of the SRV filter is noticeably demonstrated. The new filter hasbetter errors in L2 and L∞ norm when the mesh is sufficiently refined.
Unlike the smoothly varying mesh we used in the previous example, the smooth-polynomial mesh and the randomly varying mesh do not have the nice periodic propertywhich is exactly where a one-sided filter is needed. The deficiencies of the SRV filter becomesignificant. The results near the boundaries are worse than the original dG solution (Figs. 15,16). The previous filters work well when all of their preconditions are met; however, if anyone of its assumptions are violated, we may not obtain the full benefit of the filter. The newfilter appears to still perform well in such circumstances.
6 Conclusion
In this paper, we have proposed a new position-dependent SIAC filter applied to discontinuousGalerkin approximations over uniform and non-uniform meshes. The new filter was devisedas a consequence of analyzing the constant in the previous error estimates. This filter wascreated by introducing an extra general B-spline to a filter consisting of 2k + 1 central B-splines. This strategy allows us to overcome two shortcomings of the SRV filter: we cannow reliably use double-precision to both produce and use our filter, and our new filter has asmaller geometric footprint and hence costs less (in terms of operations) to evaluate. We have,for the first time, proved the accuracy-order conserving nature of the SIAC filter globallyand shown that this boundary filter does not affect the interior superconvergence properties.
123
J Sci Comput
Additionally, we are able, for the first time, to extend our proofs of superconvergence for oursymmetric and SVR SIAC filters used over smoothly varying meshes. We demonstrated theapplicability of the position-dependent filter for non-uniform meshes by choosing a properscaling, H , which is obtained by analyzing smoothly varying meshes. Numerical resultsindicate that this scaling idea works, even in the random mesh case (although no proof existsto assert this). Future work will concentrate on extending these concepts to derivative filtering.
Acknowledgments We would like to first thank Dr. Mahsa Mirzargar for helping us drive this draft tocompletion. Without her help, we would have possibly never finished this manuscript. We would like to thankDr. Hanieh Mirzaee for her discovery and discussion concerning the conditioning issues of the SRV filter.The first and second authors are sponsored by the Air Force Office of Scientific Research (AFOSR), AirForce Material Command, USAF, under Grant No. and FA8655-13-1-3017. The third author is sponsored inpart by the Air Force Office of Scientific Research (AFOSR), Computational Mathematics Program (ProgramManager: Dr. Fariba Fahroo), under Grant No. FA9550-12-1-0428. The U.S Government is authorized toreproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.
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