Local discontinuous Galerkin method for the Keller-Segel chemotaxis model

Xingjie Helen Li1, Chi-Wang Shu2 and Yang Yang3

Abstract

In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller–

Segel (KS) chemotaxis model. We improve the results upon (Y. Epshteyn and A. Kurganov,

SIAM Journal on Numerical Analysis, 47 (2008), 368-408) and give optimal rate of con-

vergence under special finite element spaces. Moreover, to construct physically relevant

numerical approximations, we develop a positivity-preserving limiter to the scheme, extend-

ing the idea in (Y. Zhang, X. Zhang and C.-W. Shu, Journal of Computational Physics,

229 (2010), 8918-8934). With this limiter, we can prove the L1-stability of the numerical

scheme. Numerical experiments are performed to demonstrate the good performance of

the positivity-preserving LDG scheme. Moreover, it is known that the chemotaxis model

will yield blow-up solutions under certain initial conditions. We numerically demonstrate

how to find the numerical blow-up time by using the L2-norm of the L1-stable numerical

approximations.

Keywords: Local discontinuous Galerkin method, Keller-Segel chemotaxis model, positivity

preserving, error estimate, Neumann boundary condition, blow-up, L1 stability

1Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC

28223. E-mail: xli47@uncc.edu2Division of Applied Mathematics, Brown University, Providence, RI 02912. E-mail:

shu@dam.brown.edu. Research supported by ARO grant W911NF-15-1-0226 and NSF grant DMS-

1418750.3Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931. E-mail:

yyang7@mtu.edu

1

1 Introduction

In this paper, we study the Keller-Segel (KS) chemotaxis model in two space dimensions

[31, 27] and focus on the following common formulation [3],

ut − div(∇u − χu∇v) = 0, x ∈ Ω, t > 0,vt −v = u − v, x ∈ Ω, t > 0,

(1.1)

where Ω is assumed to be a convex, bounded and open set in R2. Chemotaxis is the highly

nonlinear term which indicates movements by cells in reaction to a chemical substance, where

cells approach chemically favorable environments and avoid unpleasant ones. In (1.1), u and

v denote the densities of cells and the chemical concentration, respectively. The chemotactic

sensitivity function χ is assumed to be a positive constant. For simplicity, we take χ ≡ 1.

The boundary conditions are set to be

∇u · n = ∇v · n = 0, (1.2)

where n is the outer normal of the boundary ∂Ω. With this boundary condition,∫Ω

u is a

constant during the time evolution and the system is thus isolated.

The exact solutions of the KS chemotaxis model are always positive. Moreover, the

model exhibits blow-up patterns with certain initial conditions [18, 17]. When the blow-up

patterns occur, the density u of cells will strengthen in the neighborhood of isolated points,

and these regions become sharper and eventually result in finite time point-wise blow-up.

Mathematical proofs for spherically symmetric solutions in a ball have been investigated in

[23]. The blow-up point in this case is the center of the ball, and it is proved to be the

only possible singularity. For nonsymmetric cases, blow-up results have also been studied

in [24, 25], and the blow-up points will be located on the boundary. More theoretical works

can be found in [28, 18].

It is difficult to construct numerical schemes for (1.1), and most of the previous works

are for the following simplified system

ut − div(∇u − χu∇v) = 0, x ∈ Ω, t > 0,−v = u − v, x ∈ Ω, t > 0,

2

(See, for example, [38, 29, 16, 22] and the references therein). Recently, there are some signif-

icant works designed to solve (1.1) directly [15, 37]. In [37], the authors constructed implicit

second-order positivity preserving finite-volume schemes in three-dimensional space, and

their technique requires solving a large linear system of equations coupling together all grid

points at each stage of the two stage TR-BDF2 method when updating the diffusion terms

at each time step. In [15], the authors applied the interior penalty discontinuous Galerkin

(IPDG) method on rectangular meshes to obtain suboptimal rate of convergence, and the

finite element space is assumed to be piecewise polynomials of degree k ≥ 2. Other related

works in this direction include [14, 12, 13] and the positivity-preserving property was demon-

strated by numerical experiments only. Besides the above, in [2], the authors constructed a

second order positivity-preserving scheme to a revised system by differentiating (1.1) with

respect to x and y. Hence, the scheme was not designed to solve (1.1) directly, similar work

can also be found in [34]. Recently, Zhang and Shu introduced positivity-preserving limiter

to hyperbolic equations in [45, 46, 47]. Subsequently, the idea was extended to parabolic

problems in [48]. In this paper, we will apply the positivity-preserving limiter introduced

in [48] to construct second-order positivity-preserving local discontinuous Galerkin (LDG)

schemes to obtain physically relevant numerical approximations. The method we plan to

use preserves the positivity of the numerical solutions, and can be applied to unstructured

meshes.

The DG method was first introduced in 1973 by Reed and Hill [33] in the framework

of neutron linear transport. Subsequently, Cockburn et al. developed Runge-Kutta discon-

tinuous Galerkin (RKDG) methods for hyperbolic conservation laws in a series of papers

[9, 6, 8, 10]. In [11], Cockburn and Shu introduced the LDG method to solve the convection-

diffusion equations. Their idea was motivated by Bassi and Rebay [1], where the compressible

Navier-Stokes equations were successfully solved. Recently, the DG methods were applied

to linear hyperbolic equations with δ-singularities [41] to obtain high-order approximations

under suitable negative-order norms. Subsequently, the methods have also been applied to

3

nonlinear hyperbolic equations with δ-singularities [42, 49]. Combined with special limiters,

the schemes were proved to be L1 stable [42, 32]. Recently, the idea has been extended to

parabolic equations with blow-up solutions by using the LDG method [21]. In this paper, we

follow the same direction and employ the LDG method to capture the blow-up phenomenon.

In the LDG method, we introduce auxiliary variables p = ux and q = uy. Numerical ex-

periments will be given to demonstrate the optimal rate of convergence. However, in this

problem the approximations of p and q are discontinuous across the cell interfaces and we

can hardly obtain error estimates if we analyze the convection and diffusion terms separately.

To explain this point, let us consider the following hyperbolic equation

ut + (a(x)u)x = 0,

where a(x) is discontinuous at x = x0. In [19, 26], the authors studied such a problem and

defined

Q =a(x0 + b) − a(x0)

b.

If Q is bounded from below for all b, then the solution exists, but may not be unique. If

Q is bounded from above for all b, we can guarantee the uniqueness, but the solution may

not exist. To bridge this gap, most of the previous works for similar problems are based on

mixed finite element methods (see e.g. [29]). In this paper, we consider a new technique for

error analysis following the idea in [39, 40]. Moreover, we also apply the positivity-preserving

limiter to guarantee positivity of the numerical approximation. With this special technique,

the L1 norm of the numerical approximation is a constant during the time evolution due

to the mass conservation. However, the L2 norm might still be unbounded when blow-up

patterns occur. We thus introduce a new idea to capture the numerical blow-up time based

on the L2 norm of the numerical approximations. To our best knowledge, this is the first

paper studying the numerical blow-up time with a concrete theoretical background.

The organization of this paper is as follows. In Section 2, we construct the LDG scheme.

In Section 3, we give the error estimates based on two different finite element spaces. In

4

Section 4, we discuss the positivity-preserving technique, the foundation of limiters and high

order time discretizations. Numerical experiments are given in Section 5. Finally, we will

end in Section 6 with concluding remarks and remarks for future work.

2 LDG scheme

In this section, we define the finite element spaces and proceed to construct the LDG scheme.

In this paper, we consider the computational domain to be Ω = [0, 1] × [0, 1] and use

rectangular meshes for simplicity. The analyses for triangular meshes can be extended with

some minor changes, so we will skip this part. Let Kij = (xi− 1

2

, xi+ 1

2

) × (yj− 1

2

, yj+ 1

2

), i =

1, . . . , Nx, j = 1, . . . , Ny, be a partition of the computational domain Ω, and the mesh sizes in

the x and y directions are given as ∆xi = xi+ 1

2

−xi− 1

2

and ∆yj = yj+ 1

2

−yj− 1

2

, respectively. For

simplicity, in this paper, we assume uniform meshes and denote h = ∆xi = ∆yj . However,

this assumption is not essential in the proof. Moreover, we also denote Ii = [xi− 1

2

, xi+ 1

2

] and

Jj = [yj− 1

2

, yj+ 1

2

] to be the cells in x and y directions.

We define the finite element spaces V kh as

V kh =

z : z

∣∣Kij

∈ P k(Kij)

,

where P k(Kij) denotes the set of polynomials of degree up to k in cell Kij .

To construct the LDG scheme, we introduce the axillary variables p = ux, q = uy, r = vx

and s = vy, then (1.1) can be written as

ut = −(ru)x − (su)y + px + qy,

p = ux,

q = uy,

vt = rx + sy + u − v,

r = vx,

s = vy.

5

For simplicity, we use notations o to represent the exact solutions of the six variables

u, p, q, v, r, s and oh for the corresponding numerical approximations. Then the LDG scheme

is to find uh, ph, qh ∈ V k1

h and vh, rh, sh ∈ V k2

h , such that for any test functions wu, wp, wq ∈

V k1

h and wv, wr, ws ∈ V k2

h

(uht, wu)ij = (rhuh, w

ux)ij +

∫

Jj

rhuh(wu)+

i− 1

2,jdy −

∫

Jj

rhuh(wu)−

i+ 1

2,jdy

+ (shuh, wuy )ij +

∫

Ii

shuh(wu)+

i,j− 1

2

dx −

∫

Ii

shuh(wu)−

i,j+ 1

2

dx

− (ph, wux)ij −

∫

Jj

ph(wu)+

i− 1

2,jdy +

∫

Jj

ph(wu)−

i+ 1

2,jdy

− (qh, wuy )ij −

∫

Ii

qh(wu)+

i,j− 1

2

dx +

∫

Ii

qh(wu)−

i,j+ 1

2

dx, (2.1)

(ph, wp)ij = −(uh, w

px)ij −

∫

Jj

uh(wp)+

i− 1

2,jdy +

∫

Jj

uh(wp)−

i+ 1

2,jdy, (2.2)

(qh, wq)ij = −(uh, w

qy)ij −

∫

Ii

uh(wq)+

i,j− 1

2

dx +

∫

Ii

uh(wq)−

i,j+ 1

2

dx, (2.3)

(vht, wv)ij = −(rh, w

vx)ij −

∫

Jj

rh(wv)+

i− 1

2,jdy +

∫

Jj

rh(wv)−

i+ 1

2,jdy

− (sh, wvy)ij −

∫

Ii

sh(wv)+

i,j− 1

2

dx +

∫

Ii

sh(wv)−

i,j+ 1

2

dx

+ (uh − vh, wv)ij, (2.4)

(rh, wr)ij = −(vh, w

rx)ij −

∫

Jj

vh(wr)+

i− 1

2,jdy +

∫

Jj

vh(wr)−

i+ 1

2,jdy, (2.5)

(sh, ws)ij = −(vh, w

sy)ij −

∫

Ii

vh(ws)+

i,j− 1

2

dx +

∫

Ii

vh(ws)−

i,j+ 1

2

dx, (2.6)

where (u, v)ij =∫

Kijuvdxdy, and w+

i− 1

2,j

is the trace of w at x = xi− 1

2

in cell Kij. Likewise

for w−i+ 1

2,j, w+

i,j− 1

2

and w−i,j+ 1

2

. We also denote rhuh, shuh and oh to be the numerical fluxes.

In this paper, we use the alternating fluxes for the diffusion terms and Lax-Friedrichs fluxes

for the convection terms. For simplicity, if we consider the generic trace at the interior cell

interfaces, the subscript will be omitted. Due to the Neumann boundary condition (1.2), we

take

6

1. For j = 1, 2, · · · , Ny,

(uhi+ 1

2,j, phi+ 1

2,j

)=

(uh

+i+ 1

2,j, 0)

, i = 0,(uh

+i+ 1

2,j, ph

−i+ 1

2,j

), i = 1, · · · , Nx − 1,(

uh−i+ 1

2,j, 0)

, i = Nx,

(2.7)

(vhi+ 1

2,j, rhi+ 1

2,j

)=

(vh

+i+ 1

2,j, 0)

, i = 0,(vh

+i+ 1

2,j, rh

−i+ 1

2,j

), i = 1, · · · , Nx − 1,(

vh−i+ 1

2,j, 0)

, i = Nx.

(2.8)

2. For i = 1, 2, · · · , Nx

(uhi,j+ 1

2

, qhi,j+ 1

2

)=

(uh

+i+ 1

2,j, 0)

, j = 0,(uh

+i,j+ 1

2

, qh−i,j+ 1

2

), j = 1, · · · , Ny − 1,(

uh−i,j+ 1

2

, 0)

, j = Ny,

(2.9)

(vhi,j+ 1

2

, shi,j+ 1

2

)=

(vh

+i+ 1

2,j, 0)

, j = 0,(vh

+i,j+ 1

2

, sh−i,j+ 1

2

), j = 1, · · · , Ny − 1,(

vh−i,j+ 1

2

, 0)

, j = Ny.

(2.10)

For the convection term, we consider Lax-Friedrichs flux

rhuh =1

2

(r+h u+

h + r−h u−h − α(u+

h − u−h )), (2.11)

where α is chosen by the positivity-preserving technique. Due to the mass conservation, we

also take

rhuh 1

2,j = rhuhNx+ 1

2,j = 0, j = 1, · · · , Ny. (2.12)

We also define shuh is a similar way.

Also let [w] = w+ − w− denote the jump of w at the interface of each cell and

(u, v) =

Nx∑

i=1

Ny∑

j=1

(u, v)ij,

then (2.1)-(2.6) can be written as

(uht, wu) = Lc

x(rh, uh, wu) + Lc

y(sh, uh, wu) − Ld

x(ph, wu) − Ld

y(qh, wu), (2.13)

7

(ph, wp) = −Qx(uh, w

p), (2.14)

(qh, wq) = −Qy(uh, w

q), (2.15)

(vht, wv) = −Ld

x(rh, wv) − Ld

y(sh, wv) + (uh − vh, w

v), (2.16)

(rh, wr) = −Qx(vh, w

r), (2.17)

(sh, ws) = −Qy(vh, w

s), (2.18)

where

Lcx(r, u, w) = (ru, wx) +

1

2

Nx−1∑

i=1

Ny∑

j=1

∫ yj+ 1

2

yj− 1

2

(r+u+ + r−u− − α[u]

)[w]i+ 1

2,jdy, (2.19)

Lcy(s, u, w) = (su, wy) +

1

2

Nx∑

i=1

Ny−1∑

j=1

∫ xi+ 1

2

xi− 1

2

(s+u+ + s−u− − α[u]

)[w]i,j+ 1

2

dx, (2.20)

Ldx(p, w) = (p, wx) +

Nx−1∑

i=1

Ny∑

j=1

∫ yj+ 1

2

yj− 1

2

p−[w]i+ 1

2,jdy, (2.21)

Ldy(q, w) = (q, wy) +

Nx∑

i=1

Ny−1∑

j=1

∫ xi+ 1

2

xi− 1

2

q−[w]i,j+ 1

2

dx, (2.22)

Qx(u, w) = (u, wx) +

Nx−1∑

i=0

Ny∑

j=1

∫ yj+ 1

2

yj− 1

2

u+[w]i+ 1

2,jdy +

Ny∑

j=1

∫ yj+ 1

2

yj− 1

2

u−[w]Nx+ 1

2,jdy (2.23)

= −(ux, w) −

Nx−1∑

i=1

Ny∑

j=1

∫ yj+ 1

2

yj− 1

2

[u]w−i+ 1

2,jdy, (2.24)

Qy(u, w) = (u, wy) +Nx∑

i=1

Ny−1∑

j=0

∫ xi+ 1

2

xi− 1

2

u+[w]i,j+ 1

2

dx +Nx∑

i=1

∫ xi+1

2

xi−1

2

u−[w]i,Ny+ 1

2

dx (2.25)

= −(uy, w) −Nx∑

i=1

Ny−1∑

j=1

∫ xi+1

2

xi−1

2

[u]w−i,j+ 1

2

dx. (2.26)

Here, for simplicity, we take w−1

2,j

= w+Nx+ 1

2,j

= w−i, 1

2

= w+i,Ny+ 1

2

= 0. It is easy to check the

following identities by integration by parts on each cell: for any functions u and w,

Ldx(u, w) + Qx(w, u) = 0, (2.27)

Ldy(u, w) + Qy(w, u) = 0. (2.28)

8

3 Error estimates

In this section, we analyze the error between the numerical and exact solutions. We first

introduce the norms, construct the error equations and state the main theorem. Then we

proceed to the proof. The whole procedure can be divided into several steps. We first

construct the special projections that will be used in this section. Then we give an a priori

error estimate and its verification. Finally, we will consider another finite element space and

the corresponding error estimate. Let us define the norms first.

3.1 Norms

In this subsection ,we define several norms that will be used throughout the paper.

Denote ‖u‖0,Kijto be the standard L2 norm of u in cell Kij . For any natural number ℓ,

we consider the norm of the Sobolev space Hℓ(Kij), defined by

‖u‖ℓ,Kij=

∑

0≤α+β≤ℓ

∥∥∥∥∂α+βu

∂αx∂βy

∥∥∥∥2

0,Kij

1

2

.

Moreover, we define the norms on the whole computational domain as

‖u‖ℓ =

(Nx∑

i=1

Ny∑

j=1

‖u‖2ℓ,Kij

) 1

2

.

For convenience, if we consider the standard L2 norm, then the corresponding subscript will

be omitted.

Let Γij be the edges of Kij, and we define

‖u‖2Γij

=

∫

Jj

((u+

i− 1

2,j)2 + (u−

i+ 1

2,j)2)dy +

∫

Ii

((u+

i,j− 1

2

)2 + (u−i,j+ 1

2

)2)dx.

Moreover, we also define

‖u‖2Γh

=Nx∑

i=1

Ny∑

j=1

‖u‖2Γij

.

Finally, we define the standard L∞ norm of u in Kij as ‖u‖∞,Kij, and define the L∞ norm

on the whole computational domain as

‖u‖∞ = maxi,j

‖u‖∞,Kij.

9

3.2 Error equations and the main theorem

In this subsection, we proceed to construct the error equations. Denote the error between

the exact and numerical solutions to be eo = o− oh, then we have the equations satisfied by

the errors as

(eut, wu) =Lc

x(r, u, wu) − Lcx(rh, uh, w

u) + Lcy(s, u, wu) −Lc

y(sh, uh, wu)

− Ldx(ep, w

u) − Ldy(eq, w

u), (3.1)

(ep, wp) = −Qx(eu, w

p), (3.2)

(eq, wq) = −Qy(eu, w

q), (3.3)

(evt, wv) = − Ld

x(er, wv) −Ld

y(es, wv) + (eu − ev, w

v), (3.4)

(er, wr) = −Qx(ev, w

r), (3.5)

(es, ws) = −Qy(ev, w

s). (3.6)

Now, we can state our main theorem.

Theorem 3.1. Suppose u and v are smooth functions, r and s are uniformly bounded for

t ≤ T . The numerical approximations uh, ph, qh ∈ V k1

h and vh, sh, rh ∈ V k2

h . The initial

discretization is given as the standard L2-projection. If we take k1 ≥ 1 and k2 ≥ 2, then we

have

‖u − uh‖ + ‖v − vh‖ ≤ Chmink1+1,k2,

where the positive constant C does not depend on h.

3.3 Preliminaries and projections

In this subsection, we study the basic properties of the finite element space. Let us start

with the classical inverse properties.

Lemma 3.1. Assuming u ∈ V kh , there exists a positive constant C independent of h and u

such that

h‖u‖∞,Kij+ h1/2‖u‖Γij

≤ C‖u‖Kij.

10

In this paper, we consider several special projections. We define the elliptic projections

Pu,Pp,Pq ∈ V k1

h such that for any wu, wp, wq ∈ V k1

h

Ldx(p, w

u) + Ldy(q, w

u) =Ldx(Pp, wu) + Ld

y(Pq, wu), (3.7)

(Pp, wp) + (Pq, wq) = −Qx(Pu, wp) −Qy(Pu, wq). (3.8)

Similarly, we also define Pv,Pr,Ps ∈ V k2

h such that for any wv, wr, ws ∈ V k2

h

Ldx(r, w

v) + Ldy(s, w

v) =Ldx(Pr, wv) + Ld

y(Ps, wv), (3.9)

(Pr, wr) + (Ps, ws) = −Qx(Pv, wr) −Qy(Pv, ws). (3.10)

The existence of the elliptic projections will be given in the following lemma [5],

Lemma 3.2. Suppose∫ΩPudxdy =

∫Ω

udxdy and∫ΩPvdxdy =

∫Ω

vdxdy, then Po are

uniquely determined by (3.7)-(3.10). Moreover, we have the following estimates

‖p −Pp‖ + ‖q − Pq‖ ≤Chk1 , (3.11)

‖u −Pu‖ ≤Chk1+1, (3.12)

‖r −Pr‖ + ‖s − Ps‖ ≤Chk2 , (3.13)

‖v −Pv‖ ≤Chk2+1, (3.14)

where C is independent of h.

In addition, we also define the standard L2 projection P by

(Pu, v)ij = (u, v)ij, ∀v ∈ P k(Kij).

The L2 projection satisfies the following property [4].

Lemma 3.3. Suppose u ∈ Ck+1(Ω). Then there exists a positive constant C independent of

h and u such that

‖u − Pu‖ + h‖u − Pu‖∞ + h1/2‖u − Pu‖Γh≤ Chk+1‖u‖k+1.

11

Let us finish this section by proving the following lemma.

Lemma 3.4. Let u ∈ Ck+1(Ω) and Πu ∈ V kh . Suppose ‖u − Πu‖ ≤ Chκ for some positive

constant C and κ ≤ k + 1. Then

h‖u − Πu‖∞ + h1/2‖u − Πu‖Γh≤ Chκ,

where the positive constant C does not depend on h.

Proof. Actually,

h‖u − Πu‖∞ + h1/2‖u − Πu‖Γh

≤ h‖Pu − Πu‖∞ + h‖u − Pu‖∞ + h1/2‖Pu − Πu‖Γh+ h1/2‖u − Pu‖Γh

≤ C‖Pu − Πu‖ + Chk+1 + C‖Pu − Πu‖ + Chk+1

≤ C‖u − Πu‖ + C‖u − Pu‖ + Chk+1

≤ Chκ,

where the first and third steps follow from triangle inequality, the second step requires

Lemma 3.1 and Lemma 3.3, and the last step we use Lemma 3.3.

3.4 A priori error estimate

In this subsection, we would like to make an a priori error estimate that

‖u − uh‖ ≤ h, (3.15)

which further implies ‖u − uh‖∞ ≤ C by Lemma 3.4. Moreover, if u is bounded, then

‖uh‖∞ ≤ C. This assumption, which will be verified in Section 3.6, is used for the estimate

of the convection terms. This idea has been used to obtain the error estimates for nonlinear

hyperbolic equations [43, 44, 30]. With this assumption, we can proceed to the main proof

of the theorem.

12

3.5 Proof of Theorem 3.1

In this subsection, we give the proof of Theorem 3.1. As the general treatment of the finite

element methods, we divide the error as eo = ηo − ξo, where ηo = o − Po and ξo = oh − Po.

Clearly, ξo is chosen from the desired finite element space, and following [39, 40] we have

Lemma 3.5.

‖(ξu)x‖2 + ‖(ξu)y‖

2 + h−1‖[ξu]‖2Γh

≤ C(‖ξp‖2 + ‖ξq‖

2).

Proof. By (2.14), (2.15) and (3.8), we have

(ξp, wp) + (ξq, w

q) = −Qx(ξu, wp) −Qy(ξu, w

q) (3.16)

We take wp(x, y) =x

i+ 12

−x

∆xi(ξu)x, (x, y) ∈ Kij and wq = 0, then by (2.24) and (2.14) it is

easy to show

|||(ξu)x|||2Kij

=

∣∣∣∣∣

(ξp,

xi+ 1

2

− x

∆xi(ξu)x

)

ij

∣∣∣∣∣ ≤ ‖(ξu)x‖Kij‖ξp‖Kij

.

where

|||u|||2Kij=

∫

Kij

xi+ 1

2

− x

∆xi

u2dxdy.

Clearly, ||| · |||Kijis a weighted L2-norm in P k1(Kij), and is equivalent to the standard

L2-norm, i.e. for any u ∈ P k1(Kij),

‖u‖Kij≤ C|||u|||Kij

,

where the constant C does not depend on Kij . Therefore,

‖(ξu)x‖2Kij

≤ C|||(ξu)x|||2Kij

≤ C‖(ξu)x‖Kij‖ξp‖Kij

,

which further yields

‖(ξu)x‖Kij≤ C‖ξp‖Kij

.

Similarly, we can also prove

‖(ξu)y‖ ≤ C‖ξq‖

13

following the same line.

Next, we will take care of interfacial terms. We take wp(x, y) = ξu(x, y) − ξ+u (xi+ 1

2

, y),

(x, y) ∈ Kij and wq = 0 in (3.16) to obtain in cell Kij ,

∫

Jj

[ξu(xi+ 1

2

, y)]2dy = (ξp − (ξu)x, wp)ij ≤ C‖wp‖Kij

‖ξp‖Kij.

It is easy to see that

wp(x, y) = −[ξu(xi+ 1

2

, y)] −

∫ x+ 1

2

x

(ξu)xdx.

Therefore, by Minkowski’s inequality,

∫

Jj

[ξu(xi+ 1

2

, y)]2dy ≤ C‖ξp‖Kij

(∥∥∥[ξu(xi+ 1

2

, y)]∥∥∥

Kij

+ ∆xi‖(ξu)x‖Kij

)

≤ C‖ξp‖Kij

∆x

1/2i

(∫

Jj

[ξu(xi+ 1

2

, y)]2dy

)1/2

+ ∆xi‖ξp‖Kij

≤1

2

∫

Jj

[ξu(xi+ 1

2

, y)]2dy + C∆xi‖ξp‖2Kij

which further yields ∫

Jj

[ξu(xi+ 1

2

, y)]2dy ≤ C∆xi‖ξp‖2Kij

. (3.17)

Similarly, we have ∫

Ii

[ξu(x, yj+ 1

2

)]2dx ≤ C∆yj‖ξq‖2Kij

. (3.18)

Hence,

h−1‖[ξu]‖2Γh

≤ C(‖ξp‖2 + ‖ξq‖

2).

With the above lemma, we can proceed to the proof of the main theorem. We take

wo = ξo in (3.1)-(3.6) to obtain

1

2

d‖ξu‖2

dt+ ‖ξp‖

2 + ‖ξq‖2 = T1 − T2 − T3, (3.19)

1

2

d‖ξv‖2

dt+ ‖ξr‖

2 + ‖ξs‖2 = T4, (3.20)

14

where

T1 = ((ηu)t, ξu),

T2 = (ru − rhuh, (ξu)x) + 〈ru − rhuh, [ξu]〉x,

T3 = (su − shuh, (ξu)y) + 〈su − shuh, [ξu]〉y,

T4 = ((ηv)t, ξv) − (ηu − ξu − ηv + ξv, ξv),

with

〈u, v〉x =

Nx−1∑

i=1

Ny∑

j=1

∫

Jj

uvi+ 1

2,jdy and 〈u, v〉y =

Nx∑

i=1

Ny−1∑

j=1

∫

Ii

uvi,j+ 1

2

dx.

Furthermore, we also define

‖u‖Γxh

= 〈u, u〉x and ‖u‖Γy

h= 〈u, u〉y.

Then by (3.18) and (3.17), we have

‖[ξu]‖Γxh≤ Ch1/2‖ξp‖ and ‖[ξu]‖Γy

h≤ Ch1/2‖ξq‖. (3.21)

Now we give the estimates of T ′is. By Cauchy-Schwarz inequality and Lemma 3.2

T1 ≤ ‖(ηu)t‖‖ξu‖ ≤ Chk1+1‖ξu‖. (3.22)

The estimates of T2 and T3 are similar, and we consider T2 only.

T2 = (r(u − uh) + (r − rh)uh, (ξu)x) + 〈ru − rhuh, [ξu]〉x,

= T21 + T22 + T23, (3.23)

where

T21 = (r(u − uh) + (r − rh)uh, (ξu)x)

T22 =1

2〈2ru − r+

h u+h − r−h u−

h , [ξu]〉x

T23 = −〈α[ηu − ξu], [ξu]〉x,

Using the fact that ‖r‖∞ ≤ C and the a priori error estimate ‖uh‖∞ ≤ C, we have

T21 ≤ C (‖u − uh‖ + ‖r − rh‖) ‖(ξu)x‖

15

≤ C(‖ηu‖ + ‖ξu‖ + ‖ηr‖ + ‖ξr‖) ‖ξp‖

≤ C(hk1+1 + ‖ξu‖ + hk2 + ‖ξr‖) ‖ξp‖, (3.24)

where the first step is based on Cauchy-Schwarz inequality, the second step follows from

Lemma 3.5 and triangle inequality, and in the last one we use Lemma 3.2. The estimate of

T22 also requires the boundedness of r and uh,

T22 =1

2〈r(u − u+

h ) + (r − r+h )u+

h + r(u − u−h ) + (r − r−h )u−

h , [ξu]〉x

≤ C (‖u − uh‖Γh+ ‖r − rh‖Γh

) ‖[ξu]‖Γxh

≤ Ch1/2 (‖ηu‖Γh+ ‖ξu‖Γh

+ ‖ηr‖Γh+ ‖ξr‖Γh

) ‖ξp‖

≤ C(hk1+1 + ‖ξu‖ + hk2 + ‖ξr‖) ‖ξp‖. (3.25)

Here in the second step we use Cauchy-Schwarz inequality, the third step follows from

triangle inequality and (3.21), the last one requires Lemma 3.4 and Lemma 3.2. Now we

proceed to the estimate of T23,

T23 ≤ C(‖[ηu]‖Γh+ ‖[ξu]‖Γx

h) ‖[ξu]‖Γx

h

≤ Ch1/2(‖[ηu]‖Γh+ ‖[ξu]‖Γh

) ‖ξp‖

≤ C(hk1+1 + ‖ξu‖) ‖ξp‖, (3.26)

where the first step follows from Cauchy-Schwarz inequality, the second step is based on

(3.21), the third one requires Lemma 3.4 and Lemma 3.2. Plug (3.24), (3.25) and (3.26) into

(3.23) to obtain

T2 ≤ (Chk1+1 + Chk2 + C‖ξu‖ + C‖ξr‖) ‖ξp‖. (3.27)

Following the same analysis of T2, we obtain a similar estimate of T3:

T3 ≤ (Chk1+1 + Chk2 + C‖ξu‖ + C‖ξs‖) ‖ξq‖. (3.28)

The estimate of T4 is simple, we only use Cauchy-Schwartz inequality and Lemma 3.2,

T4 ≤ (‖(ηv)t‖ + ‖ηu‖ + ‖ξu‖ + ‖ηv‖ + ‖ξv‖) ‖ξv‖

16

≤ C(hmin(k1,k2)+1 + ‖ξu‖ + ‖ξv‖

)‖ξv‖ (3.29)

Plug (3.22), (3.27) and (3.28) into (3.19) to obtain

1

2

d‖ξu‖2

dt+ ‖ξp‖

2 + ‖ξq‖2 ≤ Chk1+1‖ξu‖ + C(hmin(k1+1,k2) + ‖ξu‖)(‖ξp‖ + ‖ξq‖)

+C‖ξr‖‖ξp‖ + C‖ξs‖‖ξq‖

≤ C(h2 min(k1+1,k2) + ‖ξu‖

2 + ‖ξs‖2 + ‖ξr‖

2)

+ ‖ξp‖2 + ‖ξq‖

2,

which further implies

1

2

d‖ξu‖2

dt≤ C

(h2min(k1+1,k2) + ‖ξu‖

2 + ‖ξs‖2 + ‖ξr‖

2). (3.30)

Moreover, plug (3.29) into (3.20) to yield

1

2

d‖ξv‖2

dt+ ‖ξr‖

2 + ‖ξs‖2 ≤ C

(hmin(k1,k2)+1 + ‖ξu‖ + ‖ξv‖

)‖ξv‖. (3.31)

Combining (3.30) and (3.31), we can find a constant γ such that

1

2

d‖ξu‖2

dt+ γ

(1

2

d‖ξv‖2

dt+ ‖ξr‖

2 + ‖ξs‖2

)

≤ C(h2 min(k1+1,k2) + ‖ξu‖

2 + ‖ξs‖2 + ‖ξr‖

2)

+ γC(hmin(k1,k2)+1 + ‖ξu‖ + ‖ξv‖

)‖ξv‖

Take γ to be sufficiently large, we have

d‖ξu‖2

dt+ γ

d‖ξv‖2

dt≤ Ch2min(k1+1,k2) + C‖ξu‖

2 + C‖ξv‖2.

Therefore, by the Gronwall’s inequality and use L2-projection as the initial discretization,

‖ξu‖2 + ‖ξv‖

2 ≤ Ch2min(k1+1,k2),

which further yield

‖u − uh‖ + ‖v − vh‖ ≤ Chmin(k1+1,k2)

by Lemma 3.2.

17

3.6 Verification of the a priori error estimate

In this section, we proceed to verify the a priori error estimate (3.15). Actually, (3.15) is

true at t = 0. Denote t⋆ = inft|‖(u − uh)(t)‖ > h. If t⋆ < T , then at t = t⋆, we have

h = ‖u − uh‖ ≤ Chmin(k1+1,k2), which is a contradiction if k1 ≥ 1, k2 ≥ 2 and h is small.

Therefore, we have ‖u − uh‖ ≤ Chmin(k1+1,k2). Now we finish the whole proof.

3.7 Qk polynomials

In this section, we consider another finite element space and give the error estimates. If not

otherwise stated, we follow the same notations used before. We first introduce the finite

element space W kh defined as

W kh =

z : z

∣∣Kij

∈ Qk(Kij)

,

where Qk(Kij) denotes the set of tensor product polynomials of degree up to k in cell Kij . In

this section, we will take vh, rh, sh ∈ W kh and uh, ph, qh ∈ V k

h to obtain optimal error estimate.

We first define several special projections. We define P+ into W kh which is, for each cell Kij ,

(P+u − u, v)ij = 0, ∀v ∈ Qk−1(Kij),

∫

Jj

(P+u − u)(xi− 1

2

, y)v(y)dy = 0, ∀v ∈ P k−1(Jj),

(P+u − u)(xi− 1

2

, yj− 1

2

) = 0,

∫

Ii

(P+u − u)(x, yj− 1

2

)v(x)dx = 0, ∀v ∈ P k−1(Ii). (3.32)

Moreover, we also define Πxk and Πy

k into W kh which are, for each cell Kij ,

(Πxku − u, vx)ij = 0, ∀v ∈ Qk(Kij),

∫

Jj

(Πxku − u)(xi+ 1

2

, y)v(y)dy = 0, ∀v ∈ P k(Jj),

(Πyku − u, vy)ij = 0, ∀v ∈ Qk(Kij),

∫

Ii

(Πyku − u)(x, yj+ 1

2

)v(x)dx = 0, ∀v ∈ P k(Ii).

(3.33)

Following the same analysis before, we define

ηv = v − P+v, ηr = r − Πxkr, ηs = s − Πy

ks,

and

ξv = vh − P+v, ξr = rh − Πxkr, ξs = sh − Πy

ks.

Following the proof of Lemma 3.5, we obtain another similar one

18

Lemma 3.6. Suppose ξv, ξr and ξs are defined above, we have

‖(ξv)x‖ ≤ C‖ξr‖, ‖(ξv)y‖ ≤ C‖ξs‖, h−1‖[ξv]‖2Γx

h≤ C‖ξr‖

2, h−1‖[ξv]‖2Γy

h≤ C‖ξs‖

2.

Moreover, we will use the following lemma [4]

Lemma 3.7. Suppose v is sufficiently smooth, then we have

‖ηv‖ + h‖∇ηv‖ + h1/2‖ηv‖Γh≤Chk+1, (3.34)

‖ηr‖ + h‖∇ηr‖ + h1/2‖ηr‖Γh≤Chk+1, (3.35)

‖ηs‖ + h‖∇ηs‖ + h1/2‖ηs‖Γh≤Chk+1. (3.36)

Finally, we also need the following superconvergence property [7].

Lemma 3.8. For any v ∈ W kh , we have

|Qx(ηu, v)| ≤ Chk+1‖u‖k+2‖v‖, |Qy(ηu, v)| ≤ Chk+1‖u‖k+2‖v‖.

Now we can state the main theorem.

Theorem 3.2. Suppose the exact solutions u, v are smooth and the derivatives r, s are

uniformly bounded. The LDG scheme is defined as (2.1)-(2.6) with uh, qh, qh ∈ V kh and

vh, rh, sh ∈ W kh for k ≥ 1. Moreover, the initial discretization is also given as the L2-

projection. Then we have

‖u − uh‖ + ‖v − vh‖ ≤ Chk+1.

Proof. In the proof, we skip the a posterior error estimate, its verification in Section 3.5 and

most of the derivation steps, since they can be obtained from Section 3.5 with some minor

changes. We take wo = ξo in (3.1)-(3.6) to obtain

1

2

d‖ξu‖2

dt+ ‖ξp‖

2 + ‖ξq‖2 = T1 − T2 − T3, (3.37)

1

2

d‖ξv‖2

dt+ ‖ξr‖

2 + ‖ξs‖2 = T4 + T5, (3.38)

19

where

T1 = ((ηu)t, ξu),

T2 = (ru − rhuh, (ξu)x) + 〈ru − rhuh, [ξu]〉x,

T3 = (su − shuh, (ξu)y) + 〈su − shuh, [ξu]〉y,

T4 = ((ηv)t, ξv) − (ηu − ξu − ηv + ξv, ξv),

T5 = Qx(ηu, ξv) + Qy(ηu, ξv)

Following the analyses in Section 3.5, we obtain the estimates of T ′is.

T1 ≤ Chk1+1‖ξu‖. (3.39)

T2 ≤ (Chk+1 + C‖ξu‖ + C‖ξr‖) ‖ξp‖ (3.40)

T3 ≤ (Chk+1 + C‖ξu‖ + C‖ξs‖) ‖ξq‖ (3.41)

T4 ≤ C(hk+1 + ‖ξu‖ + ‖ξv‖

)‖ξv‖ (3.42)

Finally, the estimate of T5 follows from Lemma 3.8 directly,

T5 ≤ Chk+1(‖ξr‖ + ‖ξs‖). (3.43)

Plug (3.39), (3.40) and (3.41) into (3.37) to obtain

1

2

d‖ξu‖2

dt+ ‖ξp‖

2 + ‖ξq‖2 ≤ Chk+1‖ξu‖ + C(hk+1 + ‖ξu‖)(‖ξp‖ + ‖ξq‖)

+C‖ξr‖‖ξp‖ + C‖ξs‖‖ξq‖

≤ C(hk+1 + ‖ξu‖

2 + ‖ξs‖2 + ‖ξr‖

2)

+ ‖ξp‖2 + ‖ξq‖

2,

which further implies

1

2

d‖ξu‖2

dt≤ C

(h2k+2 + ‖ξu‖

2 + ‖ξs‖2 + ‖ξr‖

2). (3.44)

Moreover, plug (3.42) and (3.43) into (3.38) to yield

1

2

d‖ξv‖2

dt+ ‖ξr‖

2 + ‖ξs‖2 ≤ C

(hk+1 + ‖ξu‖ + ‖ξv‖

)‖ξv‖ + Chk+1(‖ξr‖ + ‖ξs‖)

20

≤ C(hk+1 + ‖ξu‖ + ‖ξv‖

)‖ξv‖ + Ch2k+2 +

1

2‖ξr‖

2 +1

2‖ξs‖

2,

which further implies

d‖ξv‖2

dt+ ‖ξr‖

2 + ‖ξs‖2 ≤ C

(h2k+2 + ‖ξu‖

2 + ‖ξv‖2)

(3.45)

Combining (3.44) and (3.45), we can find a constant γ such that

d‖ξu‖2

dt+ γ

(d‖ξv‖

2

dt+ ‖ξr‖

2 + ‖ξs‖2

)

≤ C(h2k+2 + ‖ξu‖

2 + ‖ξs‖2 + ‖ξr‖

2)

+ γC(h2k+2 + ‖ξu‖

2 + ‖ξv‖2)

Take γ to be sufficiently large, then

d‖ξu‖2

dt+ γ

d‖ξv‖2

dt≤ Ch2k+2 + C‖ξu‖

2 + C‖ξv‖2.

Therefore, by the Gronwall’s inequality and L2 initial discretization

‖ξu‖2 + ‖ξv‖

2 ≤ Ch2k+2,

which further yield

‖u − uh‖ + ‖v − vh‖ ≤ Chk+1

by Lemma 3.2 and Lemma 3.7.

4 Positivity preserving property

In this subsection, we apply the positivity-preserving technique to construct positive nu-

merical approximations. Following the idea in [48], we restrict ourselves to the P 1-LDG

formulation. For simplicity, we consider Euler forward time discretization and assume the

numerical solutions at time level n are positive: unh > 0, vn

h > 0. Then we will construct

positive numerical approximations at time level n+1. To do so, we will firstly prove that the

cell average at time level n + 1 is positive. Next, we can use a simple limiter to modify the

DG polynomial and make it to be positive without changing its cell average. Throughout

this section, we use oij for the numerical approximation oh in Kij, and the cell average is oij .

21

Let us consider how to construct positive uh first, and the equation satisfied by the cell

averages is

un+1ij = Hc

ij(u, r, s) + Hdij(u, p, q),

where

Hcij(u, r, s) =

1

2un

ij −∆t

∆x∆y

(∫

Jj

(uri+ 1

2,j − uri− 1

2,j

)dy +

∫

Ii

(usi,j+ 1

2

− usi,j− 1

2

)dx

),

Hdij(u, p, q) =

1

2un

ij +∆t

∆x∆y

(∫

Jj

(pi+ 1

2,j − pi− 1

2,j

)dy +

∫

Ii

(qi,j+ 1

2

− qi,j− 1

2

)dx

).

For simplicity, if not otherwise stated, we will omit the subscript ij in Hcij and Hd

ij.

To analyze Hc, we approximate the integral by a 2-point Gauss quadrature. The Gauss

quadrature points on Ii and Jj are denoted by

pxi =

x

βi : β = 1, 2

and p

yj =

y

βj : β = 1, 2

,

respectively. Also, we denote wβ as the corresponding weights on the interval[−1

2, 1

2

].

Let λ1 = ∆t∆x

and λ2 = ∆t∆y

, then Hc becomes

Hc(u, r, s) =1

2un

ij + λ1

2∑

β=1

wβ

(uri− 1

2,β − uri+ 1

2,β

)+ λ2

2∑

β=1

wβ

(usβ,j− 1

2

− usβ,j+ 1

2

),

where uri− 1

2,β is the numerical flux at (xi− 1

2

, yβj ). Likewise for the other fluxes. As the general

treatment, we rewrite the cell average on the right hand side as

unij =

1

2

2∑

β=1

wβ

(u+

i− 1

2,β

+ u−i+ 1

2,β

)=

1

2

2∑

β=1

wβ

(u+

β,j− 1

2

+ u−β,j+ 1

2

),

where u−i− 1

2,β

= u−i− 1

2,j(yβ

j ) is a point value of the numerical approximation in the Gauss

quadrature. Likewise for the other point values. Define µ = λ1 + λ2, then

Hc(u, r, s) = λ1

2∑

β=1

wβ

[1

4µ

(u+

i− 1

2,β

+ u−i+ 1

2,β

)+(uri− 1

2,β − uri+ 1

2,β

)]

+ λ2

2∑

β=1

wβ

[1

4µ

(u+

β,j− 1

2

+ u−β,j+ 1

2

)+(usβ,j− 1

2

− usβ,j+ 1

2

)].

22

We need to suitably choose the parameter α in the Lax-Friedrichs flux, and the result is

given below.

Lemma 4.1. We can choose

α > max1 ≤ i ≤ Nx − 11 ≤ j ≤ Ny − 1

β = 1, 2

r+i+ 1

2,β, −r−

i+ 1

2,β

, s+β,j+ 1

2

, −s−β,j+ 1

2

,

then un+1j > 0 under a CFL condition

A

(∆t

∆x+

∆t

∆y

)≤

1

2, (4.1)

where

A = α − min1 ≤ i ≤ Nx − 11 ≤ j ≤ Ny − 1

β = 1, 2

r+i+ 1

2,β

, −r−i+ 1

2,β

, s+β,j+ 1

2

, −s−β,j+ 1

2

≥ 0,

Proof. It is easy to check, for i = 2, 3, · · · , Nx,

1

4µu+

i− 1

2,β

+ uri− 1

2,β

=1

4µu+

i− 1

2,β

+1

2

(u−

i− 1

2,βr−i− 1

2,β

+ u+i− 1

2,β

r+i− 1

2,β− α(u+

i− 1

2,β− u−

i− 1

2,β

))

=1

2

(α + r−

i− 1

2,β

)u−

i− 1

2,β

+1

2

(1

2µ+ r+

i− 1

2,β− α

)u+

i− 1

2,β.

Therefore, we can choose α+r−i− 1

2,β

> 0 and 12µ

+r+i− 1

2,β−α > 0 to obtain 1

4µu+

i− 1

2,β

+uri− 1

2,β >

0. If i = 1, then by (2.12),

1

4µu+

i− 1

2,β

+ uri− 1

2,β =

1

4µu+

i− 1

2,β

> 0.

Also, for i = 1, 2, · · · , Ny − 1,

1

4µu−

i+ 1

2,β− uri+ 1

2,β

=1

4µu−

i+ 1

2,β−

1

2

(u−

i+ 1

2,β

r−i+ 1

2,β

+ u+i+ 1

2,β

r+i+ 1

2,β− α(u+

i+ 1

2,β− u−

i+ 1

2,β

))

=1

2

(α − r+

i+ 1

2,β

)u+

i+ 1

2,β

+1

2

(1

2µ− r−

i+ 1

2,β− α

)u−

i+ 1

2,β

.

23

We can choose α − r+i+ 1

2,β

> 0 and 12µ

− r−i+ 1

2,β− α > 0 such that 1

4µu−

i+ 1

2,β− uri+ 1

2,β > 0. If

i = Ny, then by (2.12),

1

4µu−

i+ 1

2,β− uri+ 1

2,β =

1

4µu−

i+ 1

2,β

> 0.

Similarly, for j = 2, 3, · · · , Ny we require α + s−β,j− 1

2

> 0 and 12µ

+ s+β,j− 1

2

− α > 0 to obtain

14µ

u+β,j− 1

2

+usβ,j− 1

2

> 0. For j = 1, 2, · · · , Ny−1, we need α−s+β,j+ 1

2

> 0 and 12µ−s−

β,j+ 1

2

−α > 0

such that 14µ

u−β,j+ 1

2

− usβ,j+ 1

2

> 0.

Now, we proceed to analyze Hd(p, q). Following the same analysis in [48] with some

minor changes, we can show the following lemma.

Lemma 4.2. Suppose unij > 0 for all i and j, then

Hd(u, p, q) > 0

under a CFL condition

∆t

∆x2+

∆t

∆y2≤

1

20. (4.2)

Proof. Define Hd(u, p, q) = 1∆y

H1(u, p) + 1∆x

H2(u, q), where

Hd1 (u, p) =

∫

Jj

[Λ1

4Λ

(u+

i− 1

2,j

+ u−i+ 1

2,j

)− λ1

(pi− 1

2,j − pi+ 1

2,j

)]dy (4.3)

Hd2 (u, q) =

∫

Ii

[Λ2

4Λ

(u+

i,j− 1

2

+ u−i,j+ 1

2

)− λ2

(qi,j− 1

2

− qi,j+ 1

2

)]dx, (4.4)

with Λ1 = ∆t∆x2 , Λ2 = ∆t

∆y2 and Λ = Λ1 + Λ2. It is easy to check that

∫

Jj

p−i+ 1

2,jdy =

∫

Jj

1

∆x

(4u+

i+ 1

2,j− 3u−

i+ 1

2,j− u+

i− 1

2,j

)dy, i = 1, · · · , Nx − 1,

∫

Ii

q−i,j+ 1

2

dy =

∫

Ii

1

∆y

(4u+

i,j+ 1

2

− 3u−i,j+ 1

2

− u+i,j− 1

2

)dx, j = 1, · · · , Ny − 1.

Plug the flux (2.7) into (4.3) to obtain

1. i = 1

Hd1 (u, p)

24

=

∫

Jj

[Λ1

4Λu−

i+ 1

2,j

+Λ1

4Λu+

i− 1

2,j

+ λ1p−i+ 1

2,j

]dy

=

∫

Jj

[(Λ1

4Λ− Λ1

)u+

i− 1

2,j

+

(Λ1

4Λ− 3Λ1

)u−

i+ 1

2,j

+ 4Λ1u+i+ 1

2,j

]dy

> 0.

2. 2 ≤ i ≤ Nx − 1

Hd1 (u, p)

=

∫

Jj

[Λ1

4Λu−

i+ 1

2,j

+Λ1

4Λu+

i− 1

2,j

+ λ1

(p−

i+ 1

2,j− p−

i− 1

2,j

)]dy

=

∫

Jj

[Λ1u

+i− 3

2,j

+ 3Λ1u−i− 1

2,j

+

(Λ1

4Λ− 5Λ1

)u+

i− 1

2,j

+

(Λ1

4Λ− 3Λ1

)u−

i+ 1

2,j

+ 4Λ1u+i+ 1

2,j

]dy

> 0.

3. i = Nx

Hd1 (u, p)

=

∫

Jj

[Λ1

4Λu−

i+ 1

2,j

+Λ1

4Λu+

i− 1

2,j− λ1p

−i− 1

2,j

]dy

=

∫

Jj

[Λ1u

+i− 3

2,j

+ 3Λ1u−i− 1

2,j

+

(Λ1

4Λ− 4Λ1

)u+

i− 1

2,j

+Λ1

4Λu−

i+ 1

2,j

]dy

> 0.

We can analyze Hd2 in a similar way, so we skip the proof.

Based on the above two lemmas, we have the following theorem.

Theorem 4.1. Suppose unij > 0 for all i and j, then

un+1ij > 0

under the CFL conditions (4.1) and (4.2).

Now, let us proceed to analyze v, and the equation satisfied by the numerical cell averages

is

vn+1ij =

1

2vn

ij + Hd(v, r, s) + ∆t(un

ij − vnij

)

25

= Hd(v, r, s) +

(1

2− ∆t

)vn

ij + ∆tunij .

Applying Lemma 4.2, it is easy to prove the following theorem.

Theorem 4.2. Suppose vnij > 0 for all i and j, then

vn+1ij > 0

provided

∆t

∆x2+

∆t

∆y2≤

1

20,

and

∆t ≤1

2.

Based on Theorems 4.1 and 4.2, the numerical cell averages we obtained are positive.

However, the numerical solutions unij and vn

ij might still be negative. Hence, we have to

modify the numerical solutions while keeping the cell averages untouched. For simplicity, we

discuss the modification of unij only and the procedure is given in the following steps.

• Set up a small number ε = 10−13.

• If unij > ε, then proceed to the following steps. Otherwise, un

ij is identified as the

approximation to vacuum, and we will take unij = un

ij as the numerical solution and

skip the following steps.

• Modify the density first: Compute

bij = min(x,y)∈Kij

unij(x, y).

If bij < ε, then take unij as

unij = un

ij + θij

(un

ij − unij

),

with

θij =un

ij − ε

unij − bij

,

and use unij as the new numerical density un

ij.

26

Following [42], we can show the L1-stability of the numerical scheme with the positivity-

preserving limiter. Since unh is positive, we have

‖unh‖L1 =

∫

Ω

unh(x)dx =

∫

Ω

u0h(x)dx = ‖u0

h‖L1 ,

where ‖u‖L1 is the standard L1-norm of u on Ω. This implies the L1-stability of the scheme.

4.1 High order time discretizations

All the previous analyses are based on first-order Euler forward time discretization. We can

also use strong stability preserving (SSP) high-order time discretizations to solve the ODE

system wt = Lw. More details of these time discretizations can be found in [36, 35, 20]. In

this paper, we use the third order SSP Runge-Kutta method [36]

w(1) = wn + ∆tL(wn),

w(2) =3

4wn +

1

4

(w(1) + ∆tL(w(1))

), (4.5)

wn+1 =1

3wn +

2

3

(w(2) + ∆tL(w(2))

),

and the third order SSP multi-step method [35]

wn+1 =16

27(wn + 3∆tL(wn)) +

11

27

(wn−3 +

12

11∆tL(wn−3)

). (4.6)

Since an SSP time discretization is a convex combination of Euler forward, by using the

limiter mentioned in Section 4, the numerical solutions obtained from the full scheme are

also positive.

5 Numerical experiments

In this section, we present numerical examples in two space dimensions to verify the theoret-

ical analysis and the positivity-preserving property of the proposed method. If not otherwise

stated, we consider the third-order RKDG method (k = 2).

27

Example 5.1. We solve the following problem on the domain Ω = [0, 2π] × [0, 2π]

ut − div(∇u − u∇v) = fu(x, y, t), x ∈ Ω, t > 0vt −v = u − v + fv(x, y, t), x ∈ Ω, t > 0,

(5.1)

subject to boundary condition (1.2). We choose suitable fu and fv such that the exact

solutions are

u(x, y, t) = exp(−t)(cos(x) + cos(y)), v(x, y, t) = exp(−t)(cos(x) + cos(y)).

We can easily check that the source terms are

fu(x, y, t) = −2 exp(−2t)(cos2(x) + cos(x) cos(y) + cos2(y) − 1), fv(x, y, t) = 0.

In Table 5.1, we present the numerical results for the proposed method without the bound

preserving limiter. From the table, we can observe optimal rate of convergence with V kh finite

element spaces.

k=1 k=2N L2 error order L2 error order10 2.69e-1 – 1.32e-2 –20 7.16e-2 1.91 1.66e-3 2.9940 1.74e-2 2.04 2.07e-4 3.0080 4.25e-3 2.03 2.59e-5 3.00

Table 5.1: Example 5.1: accuracy test at T = 0.1 for the second- and third-order LDGmethods without the positivity-preserving limiter. k is the degree of polynomial and Nx =Ny = N .

Example 5.2. In this example, we solve (1.1) with the following initial condition.

u0 = 840 exp(−84(x2 + y2)), v0 = 420 exp(−42(x2 + y2)).

We use second-order positivity-preserving LDG methods to solve and the numerical approx-

imations at time t is given in figure 5.1. In [15], the authors demonstrated that the blow-up

time should be approximately t = 1.21×10−4. However, we can continue our numerical sim-

ulation to t = 2×10−4, and the numerical approximation is given in figure 5.2. We also solve

28

Figure 5.1: Example 5.2: Numerical approximations of u at t = 6 × 10−5 (left) and t =1.2 × 10−4 with positivity-preserving limiter for P 1 polynomials and N = 160.

.

Figure 5.2: Example 5.2: Numerical approximations of u at t = 2.0 × 10−4 with positivity-preserving limiter for P 1 polynomials and N = 160.

.

29

the problem without the positivity-preserving limiter. We find that at about t = 8 × 10−5

the numerical scheme yields negative u.

Moreover, we also test the numerical blow-up time. For simplicity, we take Nx = Ny = N ,

and compute the L2-norm numerical approximations at time t with N × N cells, defined as

S(N, t). Define

tb(N) = inft : S(2N, t) ∗ 1.05% ≥ S(N, t) (5.2)

as the numerical blow-up time. We anticipate that as we refine the mesh, the numerical

blow-up time will converge to the exact value. However, due to the computational cost, we

have to apply adaptive methods and refine the mesh in the vicinity of the blow-up point,

and this work will be considered in the future. To verify our anticipation, we consider the

Figure 5.3: L2-norm of the numerical approximation for Example 5.2 with different N’s..

following function

f(x, t) =1

1 − texp

(−x2

2(t − 1)2

). (5.3)

It is easy to see that

limt→1−

f(x, t) → δ(x)

30

in the sense of distribution.

We consider the interval [-1,1] and divided in into N uniform cells. Denote uh to be the

L2-projection of f(x, t) in each cell and S(N, t) to be the L2-norm of uh over [-1,1] with

different t′s. We also compute the numerical blow-up time tb(N) by (5.2) and the results

are given in Table 5.2. We can clearly see that as we refine the meshes, tb(N) converges to

1, the exact blow-up time, in first order accuracy.

N 10 20 40 80 160 320Blow-up time - 0.671 0.836 0.918 0.959 0.980

Error - 0.329 0.164 0.092 0.041 0.020

Table 5.2: Numerical blow-up time with different mesh sizes for (5.3).

6 Conclusion

In this paper, we develop LDG methods to the KS chemotaxis model. We improve the result

given by [15], and give the optimal error estimate under special finite element spaces. A

special positivity-preserving limiter is constructed to obtain physically relevant numerical

approximations. Moreover, we also prove the L1-stability of the LDG scheme with the

limiter. Numerical experiments are given to demonstrate the good performance of the LDG

scheme and the estimate of the numerical blow-up time. In the future, we plan to apply

adaptive methods to calculate the blow-up time more accurately.

7 Acknowledgments

Xingjie Helen Li would like to thank the Shanghai Centre for Mathematics Science (SCMS),

Fudan University, for support during her visit.

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