Strong Stability-Preserving (SSP) High-Order Time Discretization Methods Xinghui Zhong 12/09/ 2009
Outline 1 Introduction Why SSP methods Idea History/main reference 2 Explicit SSP Runge-Kutta Methods SSP Theory Optimal SSPRK Methods for Nonlinear Problems Optimal SSPRK methods for Linear Operator Low Storage Methods 3 Explicit SSP Multi Step Methods SSP Theory Order Barriers 4 Implicit SSP Methods Diagonally Implicit Runge-Kutta methods Implicit SSP Multi Step Methods 5 Summary
Why SSP methods Time-dependent PDE = ODE Lax equivalence theorem: A linear method consistent with a linear problem stability convergence. Strang s theorem For nonlinear problems with sufficiently smooth solution, if an approximation is consistent and its linearized version is L 2 stable, = convergence problems with discontinuous solutions??? high order spatial discretization + forward Euler time stepping method
Why SSP methods hyperbolic conservation law u t = f (u) x. ODE u t = L(u) L(u)+ forward Euler: stability properties L(u)+ high order time discretization??
Why SSP methods Burger s equation Initial condition ( ) u 2 u t + 2 x = 0. u(x, 0) = { 1, if x 0, 0.5, if x > 0, Spatial discretization 2nd order minmod based Monotone Upstream-centered Schemes for Conservation Law (MUSCL)
Why SSP methods Time discretization SSP 2nd order RK non-ssp method u (1) = u n + tl(u n ) u n+1 = 1 2 un + 1 2 u(1) + 1 2 tl(u(1) ). u (1) = u n 20 tl(u n ) u n+1 = u n + 41 40 u(1) 1 40 tl(u(1) ).
Idea Idea Assume first order forward Euler time discretization of the method of lines ODE is strongly stable under, when t t FE, and then try to find a higher order time discretization that maintains the strong stability for the same norm, under t t. strong stability A sequence u n is said to be strongly stable in a given norm, provided u n+1 u n total variation diminishing (TVD) property TV (u n+1 ) TV (u n ) where TV (u) = j u j+1 u j.
History/main reference 1988 C.-W. Shu and S. Osher, Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes, II 1988 C.-W. Shu, Total-Variation-Diminishing Time Discretizations 1998 S. Gottlieb and C.-W. Shu Total-Variation-Diminishing Runge-Kutta Schemes 2001 S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong Stability Preserving High-Order Time Discretization Methods 2002 S. J. Ruuth and R. J. Spiteri. Two Barriers on Strong-Stability-Preserving Time Discretization Methods 2005 S. Gottlieb. On High Order Strong Stability Preserving Runge-Kutta and Multi Step Time Discretizations. J.S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-Dependent Problems
SSP Theory A general m-stage Runge-Kutta method u (0) = u n, i 1 ( ) u (i) = α i,k u (k) + tβ i,k L(u (k) ), α i,k 0, i = 1,, m, k=0 u n+1 = u (m). Consistency = i 1 k=0 α i,k = 1 β i,k 0, t β i,k α i,k t β i,k < 0, L is replace by L t β i,k α i,k t L approximates the same spatial derivative as L Strong stability property where u n+1 u n u n+1 = u n t L(u n ).
SSP Theory Theory (C.-W. Shu and S. Osher) If under CFL restriction and if u n + t L(u n ) u n t t FE, (1) u n t L(u n ) u n under the CFL restriction (1), Then the RK method is SSP u n+1 u n, under the CFL restriction, t c t FE, c = min i,k α i,k β i,k Provided L is replaced by L whenever β i,k is negative.
Optimal SSPRK Methods for Nonlinear Problems Optimal c: as large as possible L and L: avoid negative β i,k whenever possible definition effective CFL c eff = c l, where c: CFL coefficient l: the number of computations of L and L required per time step.
Optimal SSPRK Methods for Nonlinear Problems SSPRK (2,2): If we require β i,k 0, then u (1) = u n + t L(u n ) u n+1 = 1 2 un + 1 2 u(1) + 1 2 t L(u(1) ). with c = 1 and c eff = 1/2. SSPRK (3,3): If we require β i,k 0, then u (1) = u n + tl(u n ) u (2) = 3 4 un + 1 4 u(1) + 1 4 t L(u(1) ) u n+1 = 1 3 un + 2 3 u(2) + 2 3 t L(u(2) ) with c = 1 and c eff = 1/3. Shu-Osher method
Optimal SSPRK Methods for Nonlinear Problems Proposition (S.Gottlieb and C.-W. Shu) The four-stage, fourth-order SSP Runge-Kutta scheme with a nonzero CFL coefficient c must have at least one negative β i,k. Spiteri and Ruuth proved that any SSPRK with nonzero CFL of order p > 4 will have negative β i,k. Spiteri and Ruuth developed fourth order methods with m = 5, 6, 7 and 8 stages.
Optimal SSPRK Methods for Nonlinear Problems SSPRK(5,4) u (1) =u n + 0.391752226571890 t L(u n ), u (2) =0.444370493651235 u n + 0.555629506348765 u (1) +0.368410593050371 tl(u (1) ), u (3) =0.620101851488403 u n + 0.379898148511597 u (2) +0.251891774271694 t L(u (2) ), u (4) =0.178079954393132 u n + 0.821920045606868 u (3) +0.544974750228521 t L(u (3) ), u n+1 =0.517231671970585 u (2) +0.096059710526147 u (3) + 0.063692468666290 t L(u (3) +0.386708617503269 u (4) + 0.226007483236906 t L(u (4) ) with c = 1.508 and c eff = 0.377.
Optimal SSPRK methods for Linear Operator Denote If L is a linear constant coefficient operator, then L(u) = L u. Theory (Spiteri and Ruuth) Consider SSPRK (m,p) methods with α i,k, β i,k 0 applied to u t = L u. The CFL restriction then satisfies t (m p + 1) t FE.
Optimal SSPRK methods for Linear Operator Table 1. Optimal CFL coefficients c, and the Corresponding Effective CFL c eff of SSPRK linear (m, p)
Optimal SSPRK methods for Linear Operator SSPRK linear (m,m) u (i) = u (i 1) + t Lu (i 1), i = 1,, m 1, u (m) = m 2 k=0 where α 1,0 = 1 and α m,k u (k) + α m,m 1 (u (m 1) + t L u (m 1)), α m,k = 1 k α m 1,k 1, k = 1,, m 2, α m,m 1 = 1 m!, m 1 α m,0 = 1 α m,k. k=1 with c = 1 and c eff = 1/m.
Optimal SSPRK methods for Linear Operator Table 2. Coefficients α m,j of SSPRK linear (m, m)
Optimal SSPRK methods for Linear Operator SSPRK linear (m, 1) ( u (i) = 1 + t ) m L u (i 1), i = 1,, m. with c = m and c eff = 1. SSPRK linear (m, 2) ( u (i) = 1 + t ) m 1 L u (i 1), i = 1,, m 1, u (m) = 1 m u(0) + m 1 ( 1 + t ) m m 1 L u (m 1), with c = m 1 and c eff = (m 1)/m.
Low Storage Methods The general low-storage RK methods: u (0) = u n, k i = A i k i 1 + t L(u (i 1) ), i = 1,, m u (i) = u (i 1) + B i k i, i = 1,, m, B 1 = c. u n+1 = u (m). M. Carpenter and C. Kennedy Fourth-order 2N-storage Runge-Kutta schemes, all the low-storage RK (3, 3). the best SSP 3rd order with c = 0.92 and c eff = 0.32(S. Gottlieb and C.-W. Shu). less optimal than SSPRK (3,3) useful for large-scale calculations classes of the low-storage RK (5, 4) unable to find SSP methods with β i,k > 0.
The Need for SSP Property in the Intermediate Stages Remark SSPRK methods have also intermediate stage SSP properties. u (i) u (i 1), i = 1, m Consider u t u x = 0, 0 x 1 { 0 if x 1 u(0, x) = 2, 1 if x > 1 2. Spatial discretization u t = L(u) = u(t, x j+1) u(t, x j ) x
Time discretization SSPRK (2, 2) non SSPRK u (1) = u n + tl(u n ) u (1) = u n 20 tl(u n ) u n+1 = 1 2 un + 1 2 u(1) u n+1 = u n + 41 40 u(1) + 1 2 tl(u(1) ). 1 40 tl(u(1) ). Figure: Intermediate stage solution u (1) after 10 time steps.
SSP Theory Explicit SSP multi step methods m ( ) u n+1 = α i u n+1 i + tβ i L(u n+1 i ), α i 0. i=1 Theory (S. Gottlieb, C.-W. Shu and E. Tadmor) If u n + t L(u n ) u n under CFL restriction t t FE, (2) and if u n t L(u n ) u n under the CFL restriction (2), Then the multi step method is SSP u n+1 u n, under the CFL restriction, t c t FE, c = min i,k α i β i Provided L is replaced by L whenever β i is negative.
Order Barriers The explicit SSP multi step methods are p-th order accurate if m α i = 1, i=1 ( m m ) i k α i = k i k 1 β i.k = 1,, p. i=1 i=1 Proposition (S. Gottlieb, C.-W. Shu and E. Tadmor) For m 2, there is no m-step, mth-order SSP method with all nonnegative β i, and there is no m-step SSP method of order (m + 1).
Diagonally Implicit Runge-Kutta methods DIRK method u (0) = u n i 1 u (i) = α i,k u (k) + tβ i L(u (i) ), α i,k 0, i = 1,, m, k=0 u n+1 = u (m). Remark β i < 0, introduce an associated operator L approximate the same spatial derivative(s) as L unconditionally strong stable for first-order implicit Euler, backward in time: u n+1 = u n t L(u n+1 ).
Diagonally Implicit Runge-Kutta methods If u n+1 = u n + t L(u n+1 ) and u n+1 = u n t L(u n+1 ) is unconditionally strong stable, u n+1 u n Then the above DIRK methods are unconditionally strong stable under the same norm. Provided L is replaced by L whenever β i is negative. Proposition (S.Gottlieb, C.-W. Shu and E. Tadmor) If the above DIRK is at least second-order accurate, then α i,k cannot be all nonnegative.
Implicit SSP Multi Step Methods Implicit SSP multi step methods u n+1 = m α i u n+1 i + tβ 0 L(u n+1 ). i=1 If u n+1 = u n + t L(u n+1 ) and u n+1 = u n t L(u n+1 ) is unconditionally strong stable, u n+1 u n. Then this method would be unconditionally strong stable under the same norm Provided L is replaced by L whenever β 0 is negative. Proposition (S.Gottlieb, C.-W. Shu and E. Tadmor) If the above multi-step method is at least second-order accurate, then α i cannot be all nonnegative.
Summary SSP methods preserves the strong stability, in any norm or semi-norm, of the forward Euler (for explicit methods) or the backward Euler (for implicit methods). SSP methods are very useful for method of lines numerical schemes for PDEs, especially in solving hyperbolic problems. The goal to design higher order implicit SSP methods, which share the strong stability properties of implicit Euler, without any restriction on the time step, cannot be realized.
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