18.336 spring 2009 lecture 16 02/19/08

Modified Equation

Idea: Given FD approximation to PDE

Find another PDE which is approximated better by FD scheme. Learn from new PDE about FD scheme.

Ex.: ut = cux Un+1

j − Ujn Uj

n +1 − Un Uj

n +1 − 2Uj

n + Un

Lax-Friedrichs: Δt

− c 2Δx

j−1 − 2Δt

j−1 = 0

1 1 1 1 Taylor: ut+

2 uttΔt−cux −

6 cuxxxΔx 2 −

2ΔtuxxΔx 2 −

24ΔtuxxxxΔx 4+. . . � 2 �1 Δx

= (ut − cux) + utt Δt − uxx + . . . 2 ���� Δt

2=c uxx� 2 �1 Δx= (ut − cux) + c 2Δt − uxx

2 ΔtModified equation:

Δx2 cΔt ut − cux = (1 − r 2)uxx r =

2Δt ΔxAdvection-diffusion equation with diffusion constant

Δx22D = ( 1 r )���� ����2Δt

− added diffusion antidiffusion by central differencing

Ex.: Upwind: ut − cux =

21 cΔx(1 − r)uxx (exercise)

Compare:

For c = 1, r = 21 DLF =

43 Δx , DUW =

41 Δx−→

Upwind less diffusive than LF.

Ex.: Lax-Wendroff ut − cux = 1 cΔx2(r2 − 1)uxxx (uxx cancels by construction)

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Advection-dispersion equation with dissipation constant

µ = −61 cΔx2(1 − r2)

Disturbances behave like Airy’s equation

Message:

First order methods behave diffusive. Second order methods behave dispersive.

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� �� �

More on Advection Equation

ut + cux = 0

So far:

1. Upwind: ⎧⎪⎪⎪⎨

⎫⎪⎪⎪⎬Un

j − Ujn −1 −c

Δx c > 0

Ujn+1 − Uj

n

e = O(Δt) + O(Δx)= →⎪⎪⎪⎩⎪⎪⎪⎭

Δt Un

−c j+1

Δ

−

x

Ujn

c < 0

2. Lax-Friedrichs/Lax-Wendroff:

Ujn+1 − Uj

n

= Uj

n +1 − Uj

n −1

+ θUj

n +1 − 2Uj

n + Ujn −1

Δt 2Δx (Δx)2

LF: θ = (Δx)2

e = O(Δt) + O(Δx 2)2Δt

→

LW: θ =Δt

c 2 e = O(Δt2) + O(Δx 2)2

→

Semidiscretization:

Central: ux = Uj+1 − Uj−1

+ O(Δx 2)2Δx ⎤⎡⎤⎡ ⎤⎡0 1 −1

(ux)1 u1⎢⎢⎢⎣

⎥⎥⎥⎦

. .1 . .−1⎢⎣

⎥⎦⎢⎣

⎥⎦. ....

...Matrix = ·2Δx . .. . . . 1(ux)k uk

1 −1 0

=A

AT = −A ⇒ eigenvalues purely imaginary

Need time discretization that is stable for u̇ = λu with λ = iµ, µ ∈ R

Linear Stability for ODE:

Region of absolute stability = {λ ∈ C : method stable for u̇ = λu}

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Ex.:

Forward Euler Backward Euler Trapezoidal un+1 = un + λΔtun

n+1 1 n n+1 1 + 21 λΔt

u = u u = = (1 + λΔt)un 1 − λΔt 1 −

21 λΔt

Stable if |1 + λΔt| ≤ 1

RK4 RK2 Adams-Bashforth 3

Can also use higher order discretization of ux

(up to spectral). If central need ODE solver for timestep ⇒that is stable for u̇ = iµu.

Spurious Oscillations

Stable does not imply “no oscillations.”

Ex.: Lax-Wendroff

Overshoots remain bounded stable.⇒Still bad (e.g. density can become negative)

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� � Total Variation:

TV (u) = |uj+1 − uj | ≈ |ux(x)|dx “total up and down” j

Method total variation diminishing (TVD), ifTV (un+1) ≤ TV (un).

Bad News: Any linear method for advection that is TVD,is at most first order accurate.

[i.e.: high order spurious oscillations]→

Remedy: Nonlinear Methods:

1. Flux-/Slope- Limiters

� conservation laws; limit flux TVD→

2. ENO/WENO

(weighted) essentially non-oscillatory(essentially TVD; no noticeable spurious oscillations)

ENO/WENO

Approximate ux by interpolation.

ENO: At each point consider multiple interpolating polynomials (through various choices of neighbors). Select the most “stable” one to define ux.

WENO: Define ux as weighted average of multiple interpolants. Higher order when u smooth, no overshoots when u non-smooth.

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� �

Ex.: Fifth order WENO

s1 = 13 (v1 − 2v2 + v3)2 + 1 (v1 − 4v2 + 3v3)

2 12 4

s2 = 13 (v2 − 2v3 + v4)2 + 1 (v2 − v4)

2 12 4

= 13 + 1 (3v3 − 4v4 + v5)2 =

Uj+1 − Ujs3 12 (v3 − 2v4 + v5)

24 vj

Δx a1 = 1 /(� + s1)

2 10 6 2)a2 = 10 /(� + s2)

2 � = 10−6 max(vj· j

3 a3 = 10 /(� + s3)

2

sa = a1 + a2 + a3 a1w1 = sa

a2w2 = sa

a3w3 = sa

w = 1 (2v1 − 7v2 + 11v3) + w2 · (−v2 + 5v3 + 2v4) + w3 · (2v3 + 5v4 − v5))6 (w1 ·

Left sided approximation to ux at x4

Right sided approximation to ux at x3

ut + cux = 0

Upwind WENO with FE:

Ujn+1 − Uj

n � −c WENOleft Uj

n > 0 �

= ·

Δt −c WENOright Ujn ≤ 0·

TVD time stepping

Consider method that is TVD with FE. Is it also TVD with high order time stepping? In general: “no.” But for special class of ODE schemes: “yes.” Strong Stability Preserving (SSP) methods

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Ex.: FE (un) = un + Δtf(un)

RK3-TVD 1 2 3 1n+1 un + un + FE(FE(un)))FE (u =3 3 4 4

Convex combination of FE steps ⇒ Preserves TVD property

Compare: Classical RK4 cannot by written this way. It is not SSP.

Popular approach for linear advection:ut + cux = 0

RK3-TVD in time, upwinded WENO5 in space.

2D/3D: Tensor product in space.

WENO5-stencil

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18.336 Numerical Methods for Partial Differential Equations Spring 2009

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