Runge Kutta Chebyshev Method for parabolic PDEs · Introduction PropertiesPros and...

Introduction Properties Pros and Cons Examples References

Runge Kutta Chebyshev Method forparabolic PDEs

Zheng ChenBrown UniversityApril 25th, 2012


Overview

1 Introduction

2 PropertiesConsistency conditionsStability PropertiesIntegration formula

3 Pros and Cons

4 Examples

5 References


Introduction

initial value problem for the ODE systems:

u̇(t) = F(t,u(t)), 0 < t 6 T , u(0) = u0 (1)

which originate from spatial discretization of parabolic PDEs.Restrictions:

The eigenvalues of the Jacobian matrix should lie in anarrow strip along the negative axis of the complex plane

Jacobian matrix should not deviate too much from anormal matrix.

Example:

model heat equation

u̇(t) = ∆u (2)

reaction-diffusion problem

u̇(t) = ε∆u+ f(u, x, t), 0 < ε� 1 (3)


Introduction


u̇(t) = F(t,u(t)), 0 < t 6 T , u(0) = u0 (1)


The eigenvalues of the Jacobian matrix should lie in anarrow strip along the negative axis of the complex planeJacobian matrix should not deviate too much from anormal matrix.

Example:

model heat equation

u̇(t) = ∆u (2)


u̇(t) = ε∆u+ f(u, x, t), 0 < ε� 1 (3)


Introduction


u̇(t) = F(t,u(t)), 0 < t 6 T , u(0) = u0 (1)



Example:model heat equation

u̇(t) = ∆u (2)


u̇(t) = ε∆u+ f(u, x, t), 0 < ε� 1 (3)


Introduction

Stiff problems:standard explicit RK methods:easy application, restrictive time-step for stability

implicit RK methods:expensive to implement, unconditionally stableRKC methods:explicit, considerable time-step restrictionextended real stability interval with a length β ∝ s2


Introduction

Stiff problems:standard explicit RK methods:easy application, restrictive time-step for stabilityimplicit RK methods:expensive to implement, unconditionally stable

RKC methods:explicit, considerable time-step restrictionextended real stability interval with a length β ∝ s2


Introduction

Stiff problems:standard explicit RK methods:easy application, restrictive time-step for stabilityimplicit RK methods:expensive to implement, unconditionally stableRKC methods:explicit, considerable time-step restrictionextended real stability interval with a length β ∝ s2


RKC formula

Y0 = Un, (4)Y1 = Y0 + µ̃1τF0, (5)Yj = µjYj−1 + νjYj−2 + (1 − µj − νj)Y0 (6)

+ µ̃jτFj−1 + γ̃jτF0 (2 6 j 6 s), (7)Un+1 = Ys, n = 0, 1, . . . , (8)

This can be rewritten in the standard RK form:

Yj = Un + τ

j−1∑l=0

ajlF(tn + clτ, Yl), (0 6 j 6 s) (9)

where Fj = F(tn + cjτ, Yj), cj are defined by:

c0 = 0, (10)c1 = µ̃1, (11)cj = µjcj−1 + νjcj−2 + µ̃j + γ̃j, (2 6 j 6 s) (12)


Outline

1 Introduction


3 Pros and Cons

4 Examples

5 References


Consistency conditions

Suppose Un = U(tn), where U(t), t > tn is a sufficientlysmooth solution. All Yj satisfy an expansion

Yj = U(tn) + cjτU̇(tn) + Xjτ2U(2)(tn) +O(τ

3) (13)

Substitute this into the RKC formula, we have

X0 = X1 = 0, (14)Xj = µjXj−1 + νjXj−2 + µ̃jcj−1 (2 6 j 6 s) (15)


Consistency conditions

consistent of order 1: if

cs = 1. (16)

consistent of order 2: if

c22 = 2µ̃2c1, (17)

c23 = µ3c

22 + 2µ̃2c2, (18)

c2j = µjc

2j−1 + νjc

2j−2 + 2µ̃jcj−1 (4 6 j 6 s) (19)


Outline

1 Introduction


3 Pros and Cons

4 Examples

5 References


Stability function

Scalar test equation:U̇(t) = λU(t) (20)

Un+1 = Ps(z)Un, z = τλ (21)

Ps is defined recursively:

P0(z) = 1, (22)P1(z) = 1 + µ̃1z, (23)Pj(z) = (1 − µj − νj) + γ̃jz+ (µj + µ̃jz)Pj−1(z) + νjPj−2(z) (2 6 j 6 s)

(24)

for each stage, we have

Uj = Pj(z)Un, (0 6 j 6 s) (25)


Stability boundary

According to the consistency condition, Pj(z) approximates ecjz

for z→ 0 asPj = 1 + cjz+ Xjz

2 +O(z3). (26)

The choice of the stability function Pj(z) is the cental issue indeveloping the RKC methods.

Stability Region S = {z ∈ C : |Ps| 6 1}

Stability Boundary β(s) = max{−z : z 6 0, |Ps| 6 1}

Design rules:

β(s) is as large as possibleall coefficients must be known in analytic form


Stability boundary



2 +O(z3). (26)


Stability Region S = {z ∈ C : |Ps| 6 1}Stability Boundary β(s) = max{−z : z 6 0, |Ps| 6 1}

Design rules:

β(s) is as large as possibleall coefficients must be known in analytic form


Stability boundary



2 +O(z3). (26)



Design rules:β(s) is as large as possible

all coefficients must be known in analytic form


Stability boundary



2 +O(z3). (26)



Design rules:β(s) is as large as possibleall coefficients must be known in analytic form


Outline

1 Introduction


3 Pros and Cons

4 Examples

5 References


shifted Chebyshev polynomials

Chebyshev polynomial of the first kind

Ts(x) = cos(sarccosx),−1 6 x 6 1 (27)

Eg: for the 1st order consistent polys, the shifted Chebyshevpoly

Ps(z) = Ts(1 +z

s2 ),−β(s) 6 z 6 0 (28)

yields the largest value: β(s) = 2s2. From the three-termsrecursion formula for Chebyshev polynomials, we get:

P0(z) = 1,P1(z) = 1+z

s2 ,Pj(z) = 2(1+z

s2 )Pj−1(z)−Pj−2(z), j > 2,(29)

which gives the analytical form of the integration coeffs

µ̃1 = 1/s2,µj = 2, µ̃j = 2/s2,νj = −1, γ̃j = 0, 0 6 j 6 s (30)


1st order case: RKC1

For 1st and 2nd order RKC, we have this general form

Pj(z) = aj + bjTj(w0 +w1z), 0 6 j 6 s (31)

RKC1:

aj = 0,bj = T−1j (w0),w0 = 1 +

ε

s2 ,w1 =Ts(w0)

T ′s(w0)

, (0 6 j 6 s)

(32)Therefore,

β(s) ' (w0 + 1)T ′s(w0)

Ts(w0)' (2 −

4ε3)s2, ε→ 0 (33)

choose ε = 0.05, then β(s) = 1.90s2.


1st order case: RKC1

Then compare these with the recursive definition of Pj, we get,

µ̃1 =w1

w0, (34)

µj = 2w0bj

bj−1, νj = −

bj

bj−2, (35)

µ̃j = 2w1bj

bj−1, γ̃j = 0, (2 6 j 6 s) (36)

cj =Ts(w0)T

′j (w0)

T ′s(w0)Tj(w0)

' j2/s2 (37)


2nd order case: RKC2

aj = 1 − bjTj(w0), bj =T ′′j (w0)

(T ′j (w0))2 , (2 6 j 6 s) (38)

w0 = 1 +ε

s2 , w1 =T ′s(w0)

T ′′s (w0)

, (39)

a0 = 1 − b0, a1 = 1 − b1w0, b0 = b1 = b2 (40)

β(s) ' (w0 + 1)T ′′s (w0)

T ′s(w0)

' 23(s2 − 1)(1 −

215ε), ε→ 0 (41)

From the pictures,we can tell ε = 213 is a suitable choice.


2nd order case: RKC2

Then compare these with the recursive definition of Pj, we get,

µ̃1 = b1w1, (42)

µj = 2w0bj

bj−1, νj = −

bj

bj−2, (43)

µ̃j = 2w1bj

bj−1, γ̃j = −(1 − bj−1Tj−1(w0))µ̃j, (2 6 j 6 s)

(44)

c1 =c2

T ′2(w0)

' c2

4, cj =

T ′s(w0)T

′′j (w0)

T ′′s (w0)T

′j (w0)

' j2 − 1s2 − 1

(2 6 j 6 s)

(45)


Pros and Cons

Pros:Explicit and designed for modestly stiff problems

Quadratic increase of β(s) with the number of stagesRequires at most seven vectors of storageno particular difficulties for vectorization and/orparallelization

Cons: Restrictions on the problem



Pros and Cons

Pros:Explicit and designed for modestly stiff problemsQuadratic increase of β(s) with the number of stages

Requires at most seven vectors of storageno particular difficulties for vectorization and/orparallelization




Pros and Cons

Pros:Explicit and designed for modestly stiff problemsQuadratic increase of β(s) with the number of stagesRequires at most seven vectors of storage

no particular difficulties for vectorization and/orparallelization




Pros and Cons

Pros:Explicit and designed for modestly stiff problemsQuadratic increase of β(s) with the number of stagesRequires at most seven vectors of storageno particular difficulties for vectorization and/orparallelization




Pros and Cons


Cons: Restrictions on the problemThe eigenvalues of the Jacobian matrix should lie in anarrow strip along the negative axis of the complex plane

Jacobian matrix should not deviate too much from anormal matrix.


Pros and Cons


Cons: Restrictions on the problemThe eigenvalues of the Jacobian matrix should lie in anarrow strip along the negative axis of the complex planeJacobian matrix should not deviate too much from anormal matrix.


Eg 1: linear heat conduction problem

ut = ∆u+ f(x,y, z, t), 0 < x,y, z < 1, 0 6 t 6 0.7 (46)u(x,y, z, t) = tanh(5(x+ 2y+ 1.5z− 0.5 − t)) (47)

Take uniform grid with h = 0.025, then 393 = 59319 equations:

Table: Results for RKC and BDF

Tol maxError #steps #F-evals CPU(s)RKC BDF RKC BDF RKC BDF RKC BDF

1.E-1 8.9E-3 9.9E-1 6 7 402 46 186 351.E-2 1.7E-3 8.3E-2 15 16 729 160 338 1221.E-3 3.7E-4 1.0E-2 27 34 786 237 366 1851.E-4 3.9E-5 1.2E-3 57 70 1087 474 507 3711.E-5 4.3E-6 1.3E-5 129 112 1682 984 787 7701.E-6 6.5E-7 1.9E-5 262 168 2445 1151 1149 913


Eg 2: combustion problem

ct = ∆c−Dce−δ/T ,LTt = ∆T+αDce−δ/T , 0 < x,y, z < 1, 0 6 t 6 0.3

(48)L = 0.9,α = 1, δ = 20,D = Reδ/αδ, with R = 5The grid spacing is h = 1/(N+ 0.5), with N = 40,2× 403 = 128000 equations

Tol maxError #steps #F-evals CPU(s)RKC BDF RKC BDF RKC BDF RKC BDF

1.E-4 5.4E-1 8.7E-1 51 33 525 285 420 4121.E-5 1.8E-1 7.6E-1 124 91 781 659 630 9571.E-6 3.9E-2 1.2E-1 270 201 1270 1141 1030 17021.E-7 8.7E-3 1.2E-3 581 286 2147 1548 1758 2376

The low accuracy is expected from the local instability of theproblem.


References

J.G. Verwer, W.H. Hundsdorfer and B.P. Sommeijer,Convergence properties of the Runge-Kutta-Chebyshevmethod, Numer. Math. 57, 157-178, 1990.

B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: Anexplicit solver for parabolic PDEs, J. Comp. Appl. Math.88, 315-326, 1997.J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKCtime-stepping for Advection-Diffusion-Reaction Problems,J. Comput. Phys. 201, 61-79, 2004.J.G. Verwer and B.P. Sommeijer, An Implicit-ExplicitRunge-Kutta-Chebyshev Scheme for Diffusion-ReactionEquations, SIAM J. Scientific Computing 25, 1824-1835,2004.


References

J.G. Verwer, W.H. Hundsdorfer and B.P. Sommeijer,Convergence properties of the Runge-Kutta-Chebyshevmethod, Numer. Math. 57, 157-178, 1990.B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: Anexplicit solver for parabolic PDEs, J. Comp. Appl. Math.88, 315-326, 1997.

J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKCtime-stepping for Advection-Diffusion-Reaction Problems,J. Comput. Phys. 201, 61-79, 2004.J.G. Verwer and B.P. Sommeijer, An Implicit-ExplicitRunge-Kutta-Chebyshev Scheme for Diffusion-ReactionEquations, SIAM J. Scientific Computing 25, 1824-1835,2004.


References

J.G. Verwer, W.H. Hundsdorfer and B.P. Sommeijer,Convergence properties of the Runge-Kutta-Chebyshevmethod, Numer. Math. 57, 157-178, 1990.B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: Anexplicit solver for parabolic PDEs, J. Comp. Appl. Math.88, 315-326, 1997.J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKCtime-stepping for Advection-Diffusion-Reaction Problems,J. Comput. Phys. 201, 61-79, 2004.

J.G. Verwer and B.P. Sommeijer, An Implicit-ExplicitRunge-Kutta-Chebyshev Scheme for Diffusion-ReactionEquations, SIAM J. Scientific Computing 25, 1824-1835,2004.


References

J.G. Verwer, W.H. Hundsdorfer and B.P. Sommeijer,Convergence properties of the Runge-Kutta-Chebyshevmethod, Numer. Math. 57, 157-178, 1990.B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: Anexplicit solver for parabolic PDEs, J. Comp. Appl. Math.88, 315-326, 1997.J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKCtime-stepping for Advection-Diffusion-Reaction Problems,J. Comput. Phys. 201, 61-79, 2004.J.G. Verwer and B.P. Sommeijer, An Implicit-ExplicitRunge-Kutta-Chebyshev Scheme for Diffusion-ReactionEquations, SIAM J. Scientific Computing 25, 1824-1835,2004.


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