Runge Kutta Chebyshev methods for parabolic problems

Runge Kutta Chebyshev methods for parabolic problems Xueyu Zhu Division of Appied Mathematics, Brown University December 2, 2009 Xueyu Zhu 1/18

Outline Introdution Consistency condition Stability Properties Integration formula Numerical experiments and Performance issues Xueyu Zhu 2/18

Introduction Target Problem (semi-discrete, multi-space): U(t) = F (t, U(t)), 0 < t T, U(0) given F (t, U(t)) is originated from spatial discretization of parabolic equaiton, such as u t = u u t = ɛ u + f(u, x, t) Stiff system: explicit method : easy but time-step restrictive implicit method : larger time steps but expensive What s remedy? Xueyu Zhu 3/18

RKC formula S-stage RKC scheme: Y 0 = U n, Y 1 = Y 0 + ũ 1 τf 0, Y j = µ j Y j 1 + ν j Y j 2 + (1 µ j ν j )Y 0 + ũ j τf j 1 + γ j τf 0, (2 j s) U n+1 = Y s, n = 0, 1,.. which can be rewritten in the general form of Runge Kutta methods: j 1 Y j = U n + τ a jl F (t + c l τ, Y l ) l=0 where τ = t n t n 1 to specify a method, we need to specify s and c l, a jl Xueyu Zhu 4/18

Consistency condition Order 1: Suppose U(t) is sufficiently smooth, all Y j statisfy the following expansion: Y j = U(t n ) + c j τ U(t n ) + X j τ 2 U (2) (t n )) + O(τ 3 ), consistent of order 1 if c s = 1 based on the Taylor s expansion (Y j approximates U(t n + c j τ)). Subsititute this expression into RKC formula, we get X 0 = X 1 = 0 X j = µ j X j 1 + ν j X j 2 + µ j c j 1 Xueyu Zhu 5/18

Order 2: consistent of order requires X j = 1 2 c2 j, which leads to c 2 2 = 2ũ 2 c 1, c 2 3 = µ 3 c 2 2 + 2ũ 3 c 2, c 2 j = µ j c 2 j 1 + ν j c 2 j 2 + 2ũ j c j 1, 4 j s Xueyu Zhu 6/18

Stabiliy region and Stability function Scalar test problem: U = λu(t) For each stage: Y j = P j (z)y 0, z = λτ where P j is a polynomial of degree j At the final stage, U n+1 = P s (z)u n Put it back into the RKC formula, we get the recursion foruma for P j (z) without any assumption on P j (z): P 0 (z) = 1, P 1 (z) = 1 + µ 1 z P j (z) = (1 µ j ν j ) + γ j z + (µ j + µ j z)p j 1 (z) + ν j P j 2 (z) Xueyu Zhu 7/18

Stability Optimal stability per step: Recalling the consistency condition, each stability function approximates e c jz for z 0 as : P j (z) = 1 + c j z + X j z 2 + O(z 3 ) The stability region requires that S = {z C : P s 1} we want the largest boundary value on negative real axis: β(s) = max{ z : z 0, P s 1} Ease of implementation: need analytical form for integration coefficients Xueyu Zhu 8/18

Shifted Chebyshev polynomial Largest negative real stability boundary β(s) leads to Chebyshev polynomial of the first kind: T s (x) = cos(s arccos x), 1 x 1 for 1st order, the stability function : P s (z) = T s (1 + z ), β(s) z 0 s2 gives β(s) = 2s 2. Recall three-terms recursion formula for Chebyshev polynomial: P 0 (z) = 1, P 1 (z) = 1+ z s 2, P j(z) = 2(1+ z s 2 )P j 1 P j 2 (z), j 2 compare to the recursive formua, it is easy to get the analytical form of µ j, ν j, µ j, γ j : µ 1 = 1 s 2, µ j = 2, µ j = 2 s 2, ν j = 1, γ j = 0 Xueyu Zhu 9/18

First order forumla for 1st and 2nd order RKC, the stability functions fit in the general form: 1st order: P j (z) = a j + b j T j (w 0 + w 1 z), 0 j s a j = 0, b j = T 1 (w 0 ), w 0 = 1 + ɛ s 2, w 1 = T s(w 0 ) T s(w 0 ), 0 j s leads to β(s) = (2 4 3 ɛ)s2 as ɛ 0. ɛ is a damping factor choose ɛ = 0.05, then β(s) 1.9s 2. Xueyu Zhu 10/18

with this set of parameters, we can match them to three-terms of recursion formula to get the integration coeffcients: µ 1 = w 1 w 0, µ j = 2w 0 b j b j 1, ν j = b j b j 2 b j µ j = 2w 1, γ j = 0, 2 j s b j 1 c j = T s(w 0 ) T j (w 0) T s(w 0 ) T j (w 0 ) Xueyu Zhu 11/18

Second order formula 2nd order: a j = 1 b j T j (w 0 ), b j = T j (w 0) (T, (2 j s) (w 0 )) 2 w 0 = 1 + ɛ s 2, w 1 = T s(w 0 ) T s (w 0 ) a 0 = 1 b 0, a 1 = 1 b 1 w 0, b 0 = b 1 = b 2 for this set of parameters, we get β(s) (w 0 + 1)T s (w 0 ) 2 T s (w 0 ) 3 (s2 1)(1 2 15 ɛ) as ɛ 0. Xueyu Zhu 12/18

with this set of parameters, we can match them to three-terms of recursion formula to get the integration coeffcients: µ 1 = b 1 w 1, µ j = 2w 0 b j b j 1, ν j = b j b j 2 µ j = 2w 1 b j b j 1, γ j = (1 b j 1 T j 1 (w 0 )) µ j, 2 j s c 1 = c 2 T 2 (w 0), c j = T s(w 0 ) T j (w 0) T s (w 0 ) T j (w 0) Xueyu Zhu 13/18

Checking Consistency Expanding P s (z): P s (z) = 1 + z + z2 2 + 4 + s2 10( 1 + s 2 ) z3... which is indeed 2nd order consistency Xueyu Zhu 14/18

Example 1: linear heat problem u t = u + f(x, y, z, t), 0 < x, y, z < 1, t > 0 take h = 0.025, 39 3 = 59319 equations Table: result of RKC2 and VODPK tol err steps F-evals CPU 1e-01 0.89e-02/0.99e-00 6/7 402/46 186/35 1e-02 0.17e-02/0.83e-01 15/16 729/160 338/122 1e-03 0.37e-03/0.10e-01 27/34 786/237 366/185 1e-04 0.39e-04/0.12e-02 57/70 1087/474 507/371 1e-05 0.43e-05/0.13e-04 129/112 1682/984 787/770 1e-06 0.65e-06/0.19e-04 262/168 2445/1151 1149/913 more accurate; approximate same amout of CPU time with relatively strict tolerence. Xueyu Zhu 15/18

Example 2: Combustion problem c t = c Dce δ T, LTt = T + αdce δ T, 0 < x, y, z < 1, t > 0 take N = 40, 10 6 equations Table: result of RKC2 and VODPK tol err steps F-evals CPU 1e-04 0.54e-00/0.87e-00 51/33 525/285 420/412 1e-05 0.18e-00/0.76e-00 124/91 781/659 630/957 1e-06 0.39e-01/0.12e-00 270/201 1270/1141 1030/1702 1e-07 0.87e-02/0.12e-02 581/286 2147/1548 1758/2376 lower accruacy is expected from the local instablity of the problem. RKC2 is more accurate and faster in general. Xueyu Zhu 16/18

Conclusion Desgined for modestly stiff, semi-discrete problems Construction of the extened stability region by shifted Chebyshev polynomial Qudratic increase of β(s) with number of stages Not good for eigenvalues of Jacobian that is far from negative real axis Xueyu Zhu 17/18

Reference J.G. Verwer, W.H. Hundsdorfer and B.P. Sommeijer, Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math. 57, 157-178, 1990. B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: An explicit solver for parabolic PDEs, J. Comp. Appl. Math. 88, 315-326, 1997. J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKC time-stepping for Advection-Diffusion-Reaction Problems, J. Comput. Phys. 201, 61-79, 2004. J.G. Verwer and B.P. Sommeijer, An Implicit-Explicit Runge-Kutta-Chebyshev Scheme for Diffusion-Reaction Equations, SIAM J. Scientific Computing 25, 1824-1835, 2004. J.G. Verwer, Explicit Runge-Kutta methods for parabolic partial differential equations, Applied Numerical Mathematics 22 (1996) 359-379 Xueyu Zhu 18/18