numerical methods for time-dependent differential equationsdurrand/amath_586_review.pdf · finite...

Numerical Methods for Time-Dependent DifferentialEquations

Dale DurranUniversity of Washington

22 August 2013

Dale Durran (Atmospheric Sci.) Amath 586 Review University of Washington 1 / 44

Introduction

Broad Categories

ODE’s

PDE’s (emphasizing low-viscosity fluid motion)Spatially discrete

Finite differenceFinite volume

Series expansions

Orthogonal expansion functionsFinite elements

Hybrids

Spectral element and discontinuous Galerkin


Introduction

Broad Categories

ODE’sPDE’s (emphasizing low-viscosity fluid motion)

Spatially discreteFinite differenceFinite volume

Series expansionsOrthogonal expansion functionsFinite elements

Hybrids

Spectral element and discontinuous Galerkin


Introduction

Broad Categories

ODE’sPDE’s (emphasizing low-viscosity fluid motion)

Spatially discreteFinite differenceFinite volume

Series expansionsOrthogonal expansion functionsFinite elements

HybridsSpectral element and discontinuous Galerkin


Introduction

Illustrative Equation

Transport in 2D nondivergent flow.

Advective form:∂ψ

∂t+ u

∂ψ

∂x+ v

∂ψ

∂y= 0.

Flux form, an equivalent expression obtained using continuity:

∂ψ

∂t+∂uψ∂x

+∂vψ∂y

= 0.

Flux form facilitates the construction of schemes with local(cell-wise) and global conservation.Advective form helps preserve uniform ψ in non-trivial velocityfields


Introduction




∂t+ u

∂ψ

∂x+ v

∂ψ

∂y= 0.


∂ψ

∂t+∂uψ∂x

+∂vψ∂y

= 0.

Flux form facilitates the construction of schemes with local(cell-wise) and global conservation.

Advective form helps preserve uniform ψ in non-trivial velocityfields


Introduction




∂t+ u

∂ψ

∂x+ v

∂ψ

∂y= 0.


∂ψ

∂t+∂uψ∂x

+∂vψ∂y

= 0.

Flux form facilitates the construction of schemes with local(cell-wise) and global conservation.Advective form helps preserve uniform ψ in non-trivial velocityfields


ODE’s

Basic Theory – ODE’s

Order of accuracy, consistency

Stability + Consistency→ ConvergenceFlavors of stability, A-stable, L-stable


ODE’s


Order of accuracy, consistencyStability + Consistency→ Convergence

Flavors of stability, A-stable, L-stable


ODE’s


Order of accuracy, consistencyStability + Consistency→ ConvergenceFlavors of stability, A-stable, L-stable


ODE’s

Basic Method’s – ODE’s

Single-step, single-stage (Forward, Backward Euler, Trapezoidal)

Multistage methods (Runge-Kutta)Multistep methods (Leapfrog, Adams-Bashforth, Backward)


ODE’s


Single-step, single-stage (Forward, Backward Euler, Trapezoidal)Multistage methods (Runge-Kutta)

Multistep methods (Leapfrog, Adams-Bashforth, Backward)


ODE’s


Single-step, single-stage (Forward, Backward Euler, Trapezoidal)Multistage methods (Runge-Kutta)Multistep methods (Leapfrog, Adams-Bashforth, Backward)


Finite Difference Methods

Theory

General:Order of accuracy, consistency

Stability + Consistency→ ConvergenceStability: Von Neumann, Energy method, CFL condition

Discrete dispersion relationModified equation



Theory

General:Order of accuracy, consistencyStability + Consistency→ Convergence

Stability: Von Neumann, Energy method, CFL condition




Theory

General:Order of accuracy, consistencyStability + Consistency→ ConvergenceStability: Von Neumann, Energy method, CFL condition




Theory


For wavelike flows:Discrete dispersion relation

Modified equation



Theory


For wavelike flows:Discrete dispersion relationModified equation



Finite difference notation

δnx f (x) =f (x + n∆x/2)− f (x − n∆x/2)

n∆x

Example:

δx fj =fj+ 1

2− fj− 1

2

∆xFlux form with local conservation:

dφi,j

dt+ δx (ui,jφi,j) + δy (vi,jφi,j) = 0




δnx f (x) =f (x + n∆x/2)− f (x − n∆x/2)

n∆xExample:

δx fj =fj+ 1

2− fj− 1

2

∆x

Flux form with local conservation:

dφi,j





δnx f (x) =f (x + n∆x/2)− f (x − n∆x/2)

n∆xExample:

δx fj =fj+ 1

2− fj− 1

2

∆xFlux form with local conservation:

dφi,j




Finite-Difference Methods

Unknowns are

values at points on a grid: φnj ≈ ψ(j∆x ,n∆t)

Can approximate flux or advective formBasic methods are quite simple:(

dψdx

)j

= δ2xφj+O[(∆x)2],

(dψdx

)j

=43δ2xφj−

13δ4xφj+O[(∆x)4]

More advanced approach: a 4th-order compact scheme (LC)

124

[5(

dψdx

)j+1

+ 14(

dψdx

)j

+ 5(

dψdx

)j−1

]=

112(11δ2xφj + δ4xφj

)




Unknowns are values at points on a grid: φnj ≈ ψ(j∆x ,n∆t)


dψdx

)j

= δ2xφj+O[(∆x)2],

(dψdx

)j

=43δ2xφj−

13δ4xφj+O[(∆x)4]


124

[5(

dψdx

)j+1

+ 14(

dψdx

)j

+ 5(

dψdx

)j−1

]=


)





Can approximate flux or advective form

Basic methods are quite simple:(dψdx

)j

= δ2xφj+O[(∆x)2],

(dψdx

)j

=43δ2xφj−

13δ4xφj+O[(∆x)4]


124

[5(

dψdx

)j+1

+ 14(

dψdx

)j

+ 5(

dψdx

)j−1

]=


)






dψdx

)j

= δ2xφj+O[(∆x)2],

(dψdx

)j

=43δ2xφj−

13δ4xφj+O[(∆x)4]


124

[5(

dψdx

)j+1

+ 14(

dψdx

)j

+ 5(

dψdx

)j−1

]=


)



Phase Speed Error in 1D Advection

Semi-discrete approximation to∂ψ

∂t+ c

∂ψ

∂x= 0.



Utility of High Order

High order schemes converge more rapidly as the grid is refined whenthe solution is already almost correct.

They are essential for efficiency when really trying to converge.Convergence is never achieved in high-Reynolds-number flow.(Why not?)

Exception: all those complicated linear solutions for nontrivial basicstates!

In high-Reynolds-number flow, errors are generally dominated bythe most poorly resolved scales.High-order schemes may treat the marginally resolved scalesbetter.





They are essential for efficiency when really trying to converge.

Convergence is never achieved in high-Reynolds-number flow.(Why not?)









In high-Reynolds-number flow, errors are generally dominated bythe most poorly resolved scales.

High-order schemes may treat the marginally resolved scalesbetter.



Utility of High Order II

Is it best to approximate all terms with differences having the sameorder of accuracy?

Yes – if you are trying to achieve convergence.Not particularly if you are trying to improve the representation ofpoorly resolved scales.





Yes – if you are trying to achieve convergence.

Not particularly if you are trying to improve the representation ofpoorly resolved scales.





Yes – if you are trying to achieve convergence.Not particularly if you are trying to improve the representation ofpoorly resolved scales.



Example: Staggered Meshes

Arakawa C-grid

um+ 1 , n 2

wm,n+ 1

2

∆z

∆x

Pm,n bm,n

um– 1 , n 2

wm,n– 1

2



Example: Staggered Meshes II

Shallow-water phase speeds using staggered (S) or unstaggered (U) 2nd- or 4th-order differences

0 π/4∆ π/2∆ 3π/4∆ π/∆WAVE NUMBER

50∆ 10∆ 6∆ 4∆ 3∆ 2∆WAVELENGTH

0

c

PHAS

E S

PEED

E

2S

2U4U



Trouble with 2∆x-Waves

Finite-difference methods do not propagate 2∆x-waves on anunstaggered mesh. Why not?



2∆x-Waves and Negative Group Velocities

Animation of 2∆x-wide spike simulated by upstream and by explicitcentered 2nd and 4th-order finite differences.


http://www.atmos.washington.edu/numerical_methods/Ioops/spike_124.mpeg

http://www.atmos.washington.edu/numerical_methods/Ioops/spike_124.mpeg


2∆x-Waves and Negative Group Velocities

Group velocity is ∂(frequency)∂(wavenumber)



Killing Off 2∆x-Waves

2∆x waves are in serious errorVery short waves can generate aliasing error

Remove the very short waves via:A scale-selective global filter

(Nonlinear) local filters that become active only in regions withlarge-amplitude short-waves

e.g., WENO finite difference methodsMajor focus of finite-volume methods





Remove the very short waves via:A scale-selective global filter(Nonlinear) local filters that become active only in regions withlarge-amplitude short-waves







e.g., WENO finite difference methods

Major focus of finite-volume methods


Finite Volume Methods


Unknowns are cell averages:

φnj ≈

1∆x

∫ xj +∆x/2

xj−∆x/2ψ(x ,n∆t) dx

Essential for the simulation of shocksWeak solutions satisfy integral form of conservation law

Approximations most naturally in flux formFluxes at cell boundaries computed from sub-cell reconstructionsLeads directly to two-level forward-in-time schemes





φnj ≈

1∆x

∫ xj +∆x/2



Approximations most naturally in flux form

Fluxes at cell boundaries computed from sub-cell reconstructionsLeads directly to two-level forward-in-time schemes





φnj ≈

1∆x

∫ xj +∆x/2



Approximations most naturally in flux formFluxes at cell boundaries computed from sub-cell reconstructions

Leads directly to two-level forward-in-time schemes





φnj ≈

1∆x

∫ xj +∆x/2



Approximations most naturally in flux formFluxes at cell boundaries computed from sub-cell reconstructionsLeads directly to two-level forward-in-time schemes



Sub-Cell Reconstructions

f

x0 2π

f

x0 2π

(a)

(b)



Avoiding overshoots and undershoots

When it it important?

When simulating shocks. Consistent monotonemethods in flux form converge to the entropy consistent solution.

φn+1j = H(φn

j−p, . . . , φnj+q+1),

is monotone if∂ H(φj−p, . . . , φj+q+1)

∂φi≥ 0

for each integer i in the interval [j − p, j + q+1].

Monotone schemesDo not produce spurious ripples.Are first-order accurate.




When it it important? When simulating shocks. Consistent monotonemethods in flux form converge to the entropy consistent solution.

φn+1j = H(φn

j−p, . . . , φnj+q+1),

is monotone if∂ H(φj−p, . . . , φj+q+1)

∂φi≥ 0

for each integer i in the interval [j − p, j + q+1].

Monotone schemesDo not produce spurious ripples.Are first-order accurate.



Advection of a Step Function

0.5 1 0.5 1x x

(a) (b)

Left: 2nd-order centered with global 4th-derivative smoother

Right: Upstream and Lax-Friedrichs (red) monotone methods

Strategy: Scheme becomes monotone near discontinuities and ishigh-order elsewhere.




When is it important in problems without shocks?

Chemically reactingflow

∂ψ1

∂t+ c

∂ψ1

∂x= −rψ1ψ2,

∂ψ2

∂t+ c

∂ψ2

∂x= rψ1ψ2.

ψ1

ψ2




When is it important in problems without shocks? Chemically reactingflow

∂ψ1

∂t+ c

∂ψ1

∂x= −rψ1ψ2,

∂ψ2

∂t+ c

∂ψ2

∂x= rψ1ψ2.

ψ1

ψ2



Limiters in Action

Minmod (long-dashed), MC (red), Superbee (short-dashed)

(a) (b)

2 3x

0 30∆x

Nice job at jump, but messes up smooth extremum.



Order of Accuracy

Scheme Estimated Order of AccuracyUpstream 0.9

Minmod Limiter 1.6Superbee Limiter 1.6

Zalesak FCT 1.7MC Limiter 1.9

Lax–Wendroff 2.0

Table: Empirically determined order of accuracy for constant-wind-speedadvection of a sine wave

Avoid limiting smooth extrema!


Spectral Methods

Orthogonal Expansion Functions

Unknowns are coefficients of expansion functions ak (t)

φ(x , t) =N∑

k=1

ak (t)ϕk (x)

Evolution equations for ak obtained viaGalerkin requirement that residual be orthogonal to all ϕk (x)

“Spectral"Global conservation of φ and φ2

Collocation requirement that sets the residual to zero at a set ofgrid points

“Pseudo-spectral"50% faster than spectralLose conservation


Spectral Methods

Orthogonal Expansion Functions – II

Using orthogonal expansion functionsDecouples the evolution equations for the ak (good for explicit timedifferencing).

Computation of the forcing for all N of the ak involves O(N2)operations (Fourier methods can be O(N log N)).

Jan 2010: ECMWF refines to T1279

“Spectral accuracy" when approximating smooth functions — errorgoes to zero faster than any finite power of the effective gridspacing.Not conducive to preserving positivity or treating steep gradients


Spectral Methods


Using orthogonal expansion functionsDecouples the evolution equations for the ak (good for explicit timedifferencing).Computation of the forcing for all N of the ak involves O(N2)operations (Fourier methods can be O(N log N)).




Spectral Methods




“Spectral accuracy" when approximating smooth functions — errorgoes to zero faster than any finite power of the effective gridspacing.

Not conducive to preserving positivity or treating steep gradients


Spectral Methods






Spectral Methods

Global Overshoots and Undershoots with PoorResolution

(b)(a)


Spectral Methods

Finite Element Methods

Piecewise linear elements (chapeau functions)

x0 2π

0

16

0

17

0

15

0

18

0

13

0

14


Spectral Methods

The Galerkin Requirement

General PDE∂ψ

∂t+ F (ψ) = 0

The residual isR(φ) =

∂φ

∂t+ F (φ)

If the residual is orthogonal to every expansion function ϕk (x),

N∑n=1

Mn,kdan

dt= −

∫S

[F( N∑

n=1

anϕn

)ϕk

]dx for all k = 1, . . . ,N

The mass matrix isMn,k =

∫Sϕnϕk dx




Expansion functions have compact support

N∑n=1

Mn,kdan

dt= −

∫S

[F( N∑

n=1

anϕn

)ϕk

]dx

Evaluation of the RHS for all N of the ak involves only O(N)operations.The evolution equations for the ak are coupled

Implicit solve every time stepVery inefficient for wave-propagation problems

Implicit coupling in quadratic or cubic finite element methodsmakes them completely impractical for the simulation of waves.Higher-order finite element methods are the nevertheless thestandard approach for solving may elliptic problems.





N∑n=1

Mn,kdan

dt= −

∫S

[F( N∑

n=1

anϕn

)ϕk

]dx

Evaluation of the RHS for all N of the ak involves only O(N)operations.

The evolution equations for the ak are coupledImplicit solve every time stepVery inefficient for wave-propagation problems






N∑n=1

Mn,kdan

dt= −

∫S

[F( N∑

n=1

anϕn

)ϕk

]dx








N∑n=1

Mn,kdan

dt= −

∫S

[F( N∑

n=1

anϕn

)ϕk

]dx



Implicit coupling in quadratic or cubic finite element methodsmakes them completely impractical for the simulation of waves.

Higher-order finite element methods are the nevertheless thestandard approach for solving may elliptic problems.





N∑n=1

Mn,kdan

dt= −

∫S

[F( N∑

n=1

anϕn

)ϕk

]dx






Finite Element versus Compact Differencing in 1D

1D advection equation∂ψ

∂t+ c

∂ψ

∂x= 0

Piecewise linear finite-elements:

ddt

(aj+1 + 4aj + aj−1

6

)+ c

(aj+1 − aj−1

2∆x

)= 0

4th-order compact scheme:

dφj

dt+ c

(dψdx

)j

= 0

16

[(dψdx

)j+1

+ 4(

dψdx

)j

+

(dψdx

)j−1

]=φj+1 − φj−1

2∆x




Schemes are identical!




4th-order compact scheme:

dφi,j

dt+ u

(dψdx

)i,j

+ v(

dψdy

)i,j

= 0

16

[(dψdx

)i+1,j

+ 4(

dψdx

)i,j

+

(dψdx

)i−1,j

]=φi+1,j − φi−1,j

2∆x

16

[(dψdy

)i,j+1

+ 4(

dψdy

)i,j

+

(dψdy

)i,j−1

]=φi,j+1 − φi,j−1

2∆y

Finite-elements: huge implicit mess (element couples with itself and 8neighbors).


Hybrid Methods

The Challenge

How do weExtend finite-volume methods to high order (beyond piecewiseparabolic or cubic)?

Extend finite-element methods to high order while avoiding implicitcoupling (and other bad behaviors)?Avoid the global coupling and O(N2) operation counts to update aset of expansion coefficients in spectral or pseudo-spectralmethods?


Hybrid Methods

The Challenge

How do weExtend finite-volume methods to high order (beyond piecewiseparabolic or cubic)?Extend finite-element methods to high order while avoiding implicitcoupling (and other bad behaviors)?

Avoid the global coupling and O(N2) operation counts to update aset of expansion coefficients in spectral or pseudo-spectralmethods?


Hybrid Methods

The Challenge

How do weExtend finite-volume methods to high order (beyond piecewiseparabolic or cubic)?Extend finite-element methods to high order while avoiding implicitcoupling (and other bad behaviors)?Avoid the global coupling and O(N2) operation counts to update aset of expansion coefficients in spectral or pseudo-spectralmethods?


Hybrid Methods

One Solution

Use h-p methods – break domain into h elements and represent thesolution within each element by orthogonal polynomials of maximumorder p.

Spectral element methods – solution is continuous at cellboundaries (good for approximating 2nd-order derivatives).Discontinuous Galerkin methods – solution is discontinuousacross cell boundaries (localizes communication between cells).


Hybrid Methods

One Solution


Spectral element methods – solution is continuous at cellboundaries (good for approximating 2nd-order derivatives).

Discontinuous Galerkin methods – solution is discontinuousacross cell boundaries (localizes communication between cells).


Hybrid Methods

One Solution


Spectral element methods – solution is continuous at cellboundaries (good for approximating 2nd-order derivatives).Discontinuous Galerkin methods – solution is discontinuousacross cell boundaries (localizes communication between cells).


Hybrid Methods

Discontinuous Galerkin Method

Enforce Galerkin criterion locally, within each element.

Two flavors of DG. Polynomial structure within each element is eitherModal: Legendre polynomials

Nodal: Lagrange polynomials interpolating theGauss-Legendre-Lobatto (GLL) points


Hybrid Methods

Discontinuous Galerkin Method

Enforce Galerkin criterion locally, within each element.

Two flavors of DG. Polynomial structure within each element is eitherModal: Legendre polynomialsNodal: Lagrange polynomials interpolating theGauss-Legendre-Lobatto (GLL) points


Hybrid Methods

Modal DG

First 5 Legendre Polynomials

(a)-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-0.5

Truly orthogonal


Hybrid Methods

Nodal DG

Lagrange Polynomials Interpolating 5 GLL Nodes

(b)-1 -0.5 0 0.5 1

-0.5

0

0.5

1

Discrete orthogonality using quadrature at the nodes


Hybrid Methods

DG Considerations

Advantages:Suitable for massively parallel computing

Communicates via fluxes with nearest neighbors onlyFor high order-polynomials, solution relatively insensitive to fluxformulationLots of work to do within each element

Rapid convergence for smooth solutions


Hybrid Methods

DG Considerations


Communicates via fluxes with nearest neighbors only

For high order-polynomials, solution relatively insensitive to fluxformulationLots of work to do within each element



Hybrid Methods

DG Considerations


Communicates via fluxes with nearest neighbors onlyFor high order-polynomials, solution relatively insensitive to fluxformulation

Lots of work to do within each element



Hybrid Methods

DG Considerations


Communicates via fluxes with nearest neighbors onlyFor high order-polynomials, solution relatively insensitive to fluxformulationLots of work to do within each element



Hybrid Methods

DG Considerations II

Disadvantages:Requires very short time step

Grid spacing reduced toward element boundaries

DiscontinuitiesCan be accommodated across element boundaries by limiting thefluxes

Cannot naturally be accommodated within each element


Hybrid Methods



Grid spacing reduced toward element boundariesDiscontinuities

Can be accommodated across element boundaries by limiting thefluxes

Cannot naturally be accommodated within each element


Hybrid Methods



Grid spacing reduced toward element boundariesDiscontinuities

Can be accommodated across element boundaries by limiting thefluxesCannot naturally be accommodated within each element


Coda

Closing Opinions

The practical distinction between finite-difference andfinite-volume methods is most consequential in (particularly whensplitting along each spatial dimension)

In the simulation of shocks!In the time differencing.

“Spectral convergence" is hard to beat if the solution is smooth.When simulating low-viscosity flow, finite elements are tooinefficient except for irregular geometriesh-p methods hold great promise, but rely on massively parallelcomputers to be competitive.


Coda

Closing Opinions



“Spectral convergence" is hard to beat if the solution is smooth.

When simulating low-viscosity flow, finite elements are tooinefficient except for irregular geometriesh-p methods hold great promise, but rely on massively parallelcomputers to be competitive.


Coda

Closing Opinions



“Spectral convergence" is hard to beat if the solution is smooth.When simulating low-viscosity flow, finite elements are tooinefficient except for irregular geometries

h-p methods hold great promise, but rely on massively parallelcomputers to be competitive.


Coda

Closing Opinions



“Spectral convergence" is hard to beat if the solution is smooth.When simulating low-viscosity flow, finite elements are tooinefficient except for irregular geometriesh-p methods hold great promise, but rely on massively parallelcomputers to be competitive.


Coda

Reference

Durran, D.R., 2010: Numerical Methods for Fluid Dynamics: WithApplications to Geophysics. 2nd Ed. Springer.


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