CS520: Stabiliy of constant coefficient problems (Ch. 4)

CS520: Stabiliy of constant coefficientproblems (Ch. 4)

Uri Ascher

Department of Computer ScienceUniversity of British Columbia

[email protected]

people.cs.ubc.ca/∼ascher/520.html

Uri Ascher (UBC) CPSC 520: Difference methods (Ch. 4) Fall 2012 1 / 28

Constant coefficient PDE

Outline

General stability

Fourier stability analysis

Explicit, implicit and semi-Lagrangian methods

Constant coefficient PDE Recall

Full discretization

Cauchy problem: pure IVP (for simplicity)

ut = Lu, u(0, x) = u0(x).

Explicit one-step scheme

i=−l

βivnj+i , concisely vn+1 = Qvn.

Implicit one-step scheme

i=−l

γivn+1

j+i =r

i=−l

βivnj+i , concisely Q1v

n+1 = Q0vn.

Stability and convergence

Method is

stable if there are constants K and α such that

‖v(t)‖ ≤ K eαt‖v(0)‖.

convergent if

u(t, x)− v(t, x) → 0, k , h → 0.

Lax Equivalence Theorem:If the linear evolutionary PDE is well-posed and the difference method isconsistent then

convergence ⇐⇒ stability.

In fact, if the method is stable then the solution error inherits the order ofaccuracy.

Method is

‖v(t)‖ ≤ K eαt‖v(0)‖.

convergent if

u(t, x)− v(t, x) → 0, k , h → 0.

Method is

‖v(t)‖ ≤ K eαt‖v(0)‖.

convergent if

u(t, x)− v(t, x) → 0, k , h → 0.

Constant coefficient PDE General stability

General linear stability

Generally, with Q = Q−1

1Q0, can write

vn+1 = Q(nk , h)vn = · · · = Πnl=0Q(lk , h)v0 = Sk,h(tn+1, 0)v

The stability condition is

‖Πn−1

l=0Q(lk , h)‖ ≤ K eαnk ∀n, k , nk ≤ tf .

If Q does not depend on t

‖Q(h)n‖ ≤ K eαnk .

This is satisfied if for all k and h small enough,

‖Q‖ ≤ eαk = 1 + O(k).

Simplify to get verifiable criteria

In some cases can ignore low order terms in PDE

Special cases for simple methods

Special case for simple scheme

An explicit method is monotone if

βi ≥ 0, i = −l ,−l + 1, . . . , r .

For a monotone method

j | ≤ max{|vnj−l |, . . . , |vnj+r |}(note consistency implies

∑ri=−l βi = 1) hence

‖vn+1‖∞ ≤ ‖vn‖∞.

– A very strong stability property indeed (no offshoots in solutioneither).For instance, the Lax-Friedrichs scheme for advection equationut + aux = 0 is

vnj+1 + vnj−1

vnj+1 − vnj−1

Special case for simple scheme

An explicit method is monotone if

βi ≥ 0, i = −l ,−l + 1, . . . , r .

For a monotone method

j | ≤ max{|vnj−l |, . . . , |vnj+r |}(note consistency implies

∑ri=−l βi = 1) hence

‖vn+1‖∞ ≤ ‖vn‖∞.

– A very strong stability property indeed (no offshoots in solutioneither).For instance, the Lax-Friedrichs scheme for advection equationut + aux = 0 is

vnj+1 + vnj−1

vnj+1 − vnj−1

Lax-Friedrichs scheme

The Lax-Friedrichs scheme for the advection equation ut + aux = 0 is

vnj+1 + vnj−1

vnj+1 − vnj−1

It has accuracy order (1, 1), provided µ = k/h is fixed.

Assuming that the CFL condition holds, µ|a| ≤ 1, we have

β0 = 0, β±1 =1

1± µa)

≥ 0.

This is indeed a very stable scheme for hyperbolic PDEs

But it’s not very accurate. In fact, a monotone scheme is at mostfirst order accurate. Need a general-purpose recipe.

Lax-Friedrichs scheme

The Lax-Friedrichs scheme for the advection equation ut + aux = 0 is

vnj+1 + vnj−1

vnj+1 − vnj−1

It has accuracy order (1, 1), provided µ = k/h is fixed.

Assuming that the CFL condition holds, µ|a| ≤ 1, we have

β0 = 0, β±1 =1

1± µa)

≥ 0.

This is indeed a very stable scheme for hyperbolic PDEs

But it’s not very accurate. In fact, a monotone scheme is at mostfirst order accurate. Need a general-purpose recipe.

Constant coefficient PDE

Outline

General stability

Constant coefficient PDE Fourier stability analysis

Assume:

Constant coefficients

Infinite (or periodic BC), uniform mesh

Then can consider v in Fourier space, satisfying

v(t, x) =1√2π

∫ ∞

−∞

eıξx v(t, ξ)dξ.

Obtain (as in Chapter 1)

v(t + k , ζ) = g(ζ)v(t, ζ), ζ = ξh,

g(ζ) =r

i=−l

βie−ıiζ .

Stability follows if‖g(ζ)‖ ≤ eαk ∀|ζ| ≤ π.

Assume:

Constant coefficients

Infinite (or periodic BC), uniform mesh

Then can consider v in Fourier space, satisfying

v(t, x) =1√2π

∫ ∞

−∞

eıξx v(t, ξ)dξ.

Obtain (as in Chapter 1)

v(t + k , ζ) = g(ζ)v(t, ζ), ζ = ξh,

g(ζ) =r

i=−l

βie−ıiζ .

Stability follows if‖g(ζ)‖ ≤ eαk ∀|ζ| ≤ π.

One PDE - amplification factor

Stability if

|g(ζ)| ≤ eαk = 1 +O(k) ∀|ζ| ≤ π,

where g is the amplification factor.

Explicit scheme:

g(ζ) =r

i=−l

βieıiζ .

Implicit scheme:

g(ζ) =(

i=−l

γieıiζ)−1

i=−l

βieıiζ .

Example: forward Euler for heat equation

For the simple heat equation

ut = uxx ,

apply centered 2nd order discretization in space:

h2D+D−vj .

Next discretize in time using forward Euler:

j = vnj + µD+D−vnj ,

where µ = k/h2.

For the amplification factor, note

D+D−eıξx = (eıζ − 2 + e−ıζ)eıξx = 2(cos ζ − 1)eıξx

= −4 sin2(ζ/2)eıξx .

For the simple heat equation

ut = uxx ,

apply centered 2nd order discretization in space:

h2D+D−vj .

Next discretize in time using forward Euler:

j = vnj + µD+D−vnj ,

where µ = k/h2.

For the amplification factor, note

= −4 sin2(ζ/2)eıξx .

Calculate amplification factor:

= −4 sin2(ζ/2)eıξx ,

sog(ζ) = 1− 4µ sin2(ζ/2).

To ensure stability require |g(ζ)| ≤ 1, so we must require

µ ≤ 1/2 i.e., k ≤ h2/2.

Note it is the high wave numbers ζ near ±π that cause problems firstif µ > 1/2. Also, although method order (1,2) is conducive to takingk ∼ h2, restricting µ ≤ .5 is for stability, not accuracy reasons.

Example: trapezoidal for heat equation

Using explicit RK2 in time gives order (2,2), but harsh stabilitylimitation remains.

Crank-Nicolson method: use implicit trapezoidal in time.

(1− µ

2D+D−)v

j = (1 +µ

2D+D−)v

Obtain method of order (2,2) with amplification factor

g(ζ) =1− 2µ sin2(ζ/2)

1 + 2µ sin2(ζ/2).

Here |g(ζ)| ≤ 1 for all µ > 0, so the scheme is unconditionally stable.

Must solve a tridiagonal system of equations at each time step. Still,this requires O(J) flops whereas forward Euler requires O(J2) flops.

(1− µ

2D+D−)v

j = (1 +µ

2D+D−)v

g(ζ) =1− 2µ sin2(ζ/2)

1 + 2µ sin2(ζ/2).

(1− µ

2D+D−)v

j = (1 +µ

2D+D−)v

g(ζ) =1− 2µ sin2(ζ/2)

1 + 2µ sin2(ζ/2).

Difference operator amplification

Fourier analysis of usual difference operators is standard (need not startfrom scratch each time):

Difference operator Contribution to g(ζ)

D+ eıζ − 1D− 1− e−ıζ

D0 2ı sin ζδ 2ı sin(ζ/2)

D+D− −4 sin2(ζ/2)

Several PDEs - amplification matrix

Amplification factor becomes an s × s amplification matrix G (ζ)defined in the same way as g(ζ).

But checking ‖G (ζ)‖ for infintiely many ζ is still not very simple.

von Neumann condition: check the spectral radius

ρ(G (ζ)) ≤ eαk ∀|ζ| ≤ π.

This is necessary and alomst sufficient for stability.

Example: leap frog for advection

Consider hyperbolic system

ut + Aux = 0,

with A constant and diagonalizable with all real eigenvalues.Leap-frog scheme with µ = k/h

j = vn−1

j − µA(vnj+1 − vnj−1).

Introducing wnj = vn−1

j , this can be written as

− µ

A 00 0

Obtain 2s × 2s matrices βi , i = −1, 0, 1.For scalar advection, amplification matrix reads

G (ζ) =

−2ıaµ sin ζ 11 0

ut + Aux = 0,

j = vn−1

− µ

A 00 0

G (ζ) =

ut + Aux = 0,

j = vn−1

− µ

A 00 0

G (ζ) =

Example: leap frog for scalar advection

G (ζ) =

The eigenvalues are

λ = −ıaµ sin ζ ±√

1− a2µ2 sin2 ζ.

So, |λ|2 = a2µ2 sin2 ζ + (1− a2µ2 sin2 ζ) = 1 provided square rootexpression stays real: the von Neumann condition holds if CFL holds,

µ|a| ≤ 1.

Is this sufficient for stability?If µ|a| < 1 then λ1 6= λ2. Hence G is diagonalizable and the schemeis stable.

Example: leap frog for scalar advection

G (ζ) =

The eigenvalues are

λ = −ıaµ sin ζ ±√

1− a2µ2 sin2 ζ.

So, |λ|2 = a2µ2 sin2 ζ + (1− a2µ2 sin2 ζ) = 1 provided square rootexpression stays real: the von Neumann condition holds if CFL holds,

µ|a| ≤ 1.

Is this sufficient for stability?If µ|a| < 1 then λ1 6= λ2. Hence G is diagonalizable and the schemeis stable.

But if µ|a| = 1 then λ1 = λ2 and G is not diagonalizable.Indeed, there is a mild instability in this case, ‖Gn(π/2)‖ growslinearly in n.So require

For a PDE system with eigenvalues (wave speeds) ai require forstability

max1≤l≤s |al |.

For a problem with variable coefficients, the analysis does not hold,but it’s common practice to use

µn <1

maxx , 1≤l≤s |al(tn, x)|and cross our fingers.Uri Ascher (UBC) CPSC 520: Difference methods (Ch. 4) Fall 2012 19 / 28

max1≤l≤s |al |.

µn <1

max1≤l≤s |al |.

µn <1

Upwind scheme

For advection equation ut + aux = 0 recall one-sided scheme

j = vnj − µa(

vnj+1 − vnj)

Fourier analysis gives amplification factor g(ζ) = 1− µa(

eıζ − 1)

,hence for stability, must require 0 ≤ µ(−a) ≤ 1.

What if a > 0? The forward difference in space gives anunconditionally unstable method, but backward difference in spacegives the scheme

j = vnj − µa(

vnj − vnj−1

which for a > 0 is the mirror image of the forward scheme for a < 0.

So, more generally, use the upwind scheme

j = vnj − µa

vnj+1− vnj

, a < 0(

vnj − vnj−1

, a > 0.

Upwind scheme

j = vnj − µa(

vnj+1 − vnj)

eıζ − 1)

j = vnj − µa(

vnj − vnj−1

j = vnj − µa

vnj+1− vnj

, a < 0(

vnj − vnj−1

, a > 0.

Upwind scheme

j = vnj − µa(

vnj+1 − vnj)

eıζ − 1)

j = vnj − µa(

vnj − vnj−1

j = vnj − µa

vnj+1− vnj

, a < 0(

vnj − vnj−1

, a > 0.

Upwind scheme cont.

j = vnj − µa

vnj+1− vnj

, a < 0(

vnj − vnj−1

, a > 0.

For stability, must require µ|a| ≤ 1, which coincides with the CFLcondition.

For variable coefficients, a = a(t, x), can still do upwinding usinga = a(tn, xj), although stability theory is yet to be developed.

For a hyperbolic PDE system

ut + Aux = 0,

if T−1AT = Λ = diag (a1, a2, . . . , as), let w = T−1u. Obtain

(wi )t + ai (wi )x = 0, i = 1, 2, . . . , s.

Apply upwinding to each of these, then back to v = Tw.

Upwind scheme cont.

j = vnj − µa

vnj+1− vnj

, a < 0(

vnj − vnj−1

, a > 0.

For stability, must require µ|a| ≤ 1, which coincides with the CFLcondition.

For variable coefficients, a = a(t, x), can still do upwinding usinga = a(tn, xj), although stability theory is yet to be developed.

For a hyperbolic PDE system

ut + Aux = 0,

if T−1AT = Λ = diag (a1, a2, . . . , as), let w = T−1u. Obtain

(wi )t + ai (wi )x = 0, i = 1, 2, . . . , s.

Apply upwinding to each of these, then back to v = Tw.

Outline

General stability

Constant coefficient PDE Semi-Lagrangian

Explicit vs implicit methods

The explicit methods we saw, both for ODEs and for PDEs, all havestability restrictions, limiting the time step k for a given spatial step h.

There are implicit methods (e.g., Crank-Nicolson) that areunconditionally stable, i.e., k not limited by stability considerations.However, for nonlinear problems in several space dimensions, solvingresulting algebraic problem can become expensive.

Recall the CFL condition: the domain of dependence of the PDEmust be contained in the domain of dependence of the differencescheme. This suggests (for hyperbolic PDEs) that all explicit schemeshave finite stability restrictions, whereas implicit schemes can beunconditionally stable.

Eulerian vs Lagrangian

Eulerian approach: stationary observer on a grid observing fluidflowing by.Pro: easiest to handle book-keeping, develop general methods, etc.Con: stability restriction on time step, occasionally excessive artificialdiffusion/viscosity.

Lagrangian approach: observer sits on a fluid particle flowing along acharacteristic curve.Pro: clean solutions, no stability restrictions, avoid certain boundarydifficulties.Con: may not have good solution everywhere, more difficultbook-keeping.

Semi-Lagrangian: attempt to combine the best of both worlds byfollowing characteristics backwards from a grid at the unknown timelevel.

Example: simplest advection

Recall: for ut + aux = 0, a < 0, have CFL compatibility if µ|a| ≤ 1.

.2 .4 .6 .8 1

Recall one-sided scheme

For advection equation ut + aux = 0 consider one-sided scheme

j = vnj − µa(

vnj+1 − vnj)

eıζ − 1)

For stability, require amplification factor to satisfy

|g(ζ)| ≤ 1, −π ≤ ζ ≤ π

hence must require 0 ≤ µ(−a) ≤ 1, i.e., that CFL hold.

Can interpret this method as semi-Lagrangian: trace characteristic at(tn+1, xj) to its foor at tn, which is at x∗ = (−a)µxj+1 + (1 + aµ)xj .Then interpolate linearly

vn∗ = (−a)µvnj+1 + (1 + aµ)vnj

and set vn+1

j = vn∗ .

j = vnj − µa(

vnj+1 − vnj)

eıζ − 1)

|g(ζ)| ≤ 1, −π ≤ ζ ≤ π

and set vn+1

j = vn∗ .

j = vnj − µa(

vnj+1 − vnj)

eıζ − 1)

|g(ζ)| ≤ 1, −π ≤ ζ ≤ π

and set vn+1

j = vn∗ .

Semi-Lagrangian method

Extend this interpretation of the one-sided scheme by tracing thecharacteristic curve further back!

Imagine the one-sided scheme on a coarse grid with widths k = lk

and h = lh for some l ≥ 1.

Next, instead of interpolating between xj and xj+1 = xj + h, find νsuch that xν ≤ x∗ ≤ xν + h and interpolate v nν and v nν+1 linearly for

v∗. Then set this value to be v n+1

j as before. The new time step istherefore l times larger, with the same spatial mesh as before.

Now, although we may still declare lk(−a) = k(−a) ≤ h = lh, this isno longer binding because can increase l arbitrarily for the (k , h)mesh.

In other words, for a fixed h can take k arbitrarily large withoutstability concerns in this way!

Semi-Lagrangian method

Extend this interpretation of the one-sided scheme by tracing thecharacteristic curve further back!

Imagine the one-sided scheme on a coarse grid with widths k = lk

and h = lh for some l ≥ 1.

Next, instead of interpolating between xj and xj+1 = xj + h, find νsuch that xν ≤ x∗ ≤ xν + h and interpolate v nν and v nν+1 linearly for

v∗. Then set this value to be v n+1

j as before. The new time step istherefore l times larger, with the same spatial mesh as before.

Now, although we may still declare lk(−a) = k(−a) ≤ h = lh, this isno longer binding because can increase l arbitrarily for the (k , h)mesh.

In other words, for a fixed h can take k arbitrarily large withoutstability concerns in this way!

Semi-Lagrangian method cont.

Can obviously extend this method automatically to a of any sign(automatic upwinding).

The described method has similar accuracy characteristics as theupwind method (in particular it is 1st order accurate), but can requiremuch less work.

Technically, need l → ∞ for unlimited stability, so it’s not really anunconditionally stable explicit scheme, mathematically speaking.

The above assessment relies more heavily on the special form of theconstant-coefficient, scalar PDE.

Semi-Lagrangian method cont.

Can obviously extend this method automatically to a of any sign(automatic upwinding).

The described method has similar accuracy characteristics as theupwind method (in particular it is 1st order accurate), but can requiremuch less work.

Technically, need l → ∞ for unlimited stability, so it’s not really anunconditionally stable explicit scheme, mathematically speaking.

The above assessment relies more heavily on the special form of theconstant-coefficient, scalar PDE.

CS520: Stabiliy of constant coefficient problems (Ch. 4)

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