Runge Kutta Methods Optimized For Advection Problems

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1 Runge Kutta Methods Optimized For Advection Problems Brown University December 9, 2009

2 Introduction Classical Runge Kutta Methods Overview Standard Runge Kutta Schemes for Advection Equation Low Dissipation and Low Dispersion Runge Kutta() Schemes

3 Introduction Classical Runge Kutta Methods Introduction Computational acoustics is a recently emerging tool for acoustic problems Minimal dispersion and dissipation errors are desired and so high order spatial discretization schemes are used Need for accurate and efficient time advancing schemes

4 Introduction Classical Runge Kutta Methods 4th Order Runge Kutta Method In 4th order Runge Kutta method, we have for stability k t 2.83 Consider the Wave Equation u t + u x = 0 For k t = 1.6, method is stable but exhibit serious dispersion and dissipation errors dt=1.6; dx=

5 Introduction Classical Runge Kutta Methods Amplification Ũ n+1 k = Ũ n k (1 + p j=1 c j(ick t) j ) where c j are related to w i, β ij in U n+1 = u n + p i w i K i K i = tf (u n + i 1 j=1 β ijk j )i = 1, 2...p Amplification Factor r = Ũk n+1 Ũ k n = 1 + j=p j=1 c j( iσ) j where σ = ck t The exact amplification factor is r e = e ick t = e iσ

6 Introduction Alternating Schemes Low Storage Methods Low Dissipation and Dispersion Runge Kutta Methods Traditional choice of c j maximizes the possible order of accuracy Optimization is carried out instead on Γ p j=1 c j( iσ) j e iσ 2 dσ = MIN where Γ specifies the range of ck t in the optimization

7 Introduction Alternating Schemes Low Storage Methods Two Step Alternating Schemes Dissipation and dispersion errors can be further reduced and higher order of acccuracy can be obtained using alternate stepping schmes Amplification factors of the first and second steps are given by r 1 = 1 + p 1 j=1 a j( iσ) j r 2 = 1 + p 2 j=1 b j( iσ) j p 1, p 2 are the number of stages of the two steps. Combined Amplification factor = r 1 r 2 Optimization criteria Γ 0 (1 + p 1 j=1 a j( iσ) j )(1 + p 2 j=1 b j( iσ) j ) e 2iσ 2 dσ = MIN

8 Introduction Alternating Schemes Low Storage Methods Low Storage Methods 3N storage required for classical RK4 and covered so far. Consider du dt = F (t, U(t)); U(t 0 ) = U 0 For low storage schemes, Runge Kutta algorithm becomes w i = α i w i 1 + hf (t i 1, U i 1 ) U i = U i 1 + β i w i α 1 = 0; u 0 = u n 1 ; u n = u s ; t i = t n 1 + hc i

9 Example, Dissipation and Dispersion can be estimated by the amplification factor G(ω t) = ũn+1 (ω t) ũ n (ω t) = 1 + s j=1 γ j(iω t) j Dissipation = 1 G(ω t) Dispersion = ω t ω t /π

10 Example, Dissipation and Dispersion Limits Scheme Dissipation(10 4 ) Dispersion(10 4 ) RK RK46(S) RK26(B) RK46-NL

11 Numerical Example Example Linear Convective Equation with the methods that have been optimized is considered u t + u x2 x = 0 u(x, 0) = K exp( ) c 2 Analytic Solution u(x, t) = K exp( (x t)2 ) K = 0.5; c = 3; c 2 9 Point central difference of 8th order is used for spatial discretization u x (x j, t) 1 t [ 4 5 (u j 1 u j+1 ) (u j 2 u j+2 ) (u j 3 u j+3 ) (u j 4 u j+4 )] Domain x=-50 to x = 450 and dx = 1 Time interval = [0,400] and dt = 1.26

12 Absolute Errors Example x t LDD46(C) LDD25 RK4 1/ x x x / x x x / x x x / x x x 10 3

13 Example 4th Order Runge Kutta Method Classical RK4 scheme stable but not time accurate Time steps much smaller than stability limit required for long time integrations dt=1.26; dx=

14 5 Stage - Stanescu Example i α β c

15 LDD56-Stanescu Example Two step alternating scheme with five/six stages in the first/second step The scheme has fourth order accuracy in both steps

16 Further Research Advantages And Disadvantages Conclusion References Further Research Few more methods have been proposed to optimize Runge Kutta schemes for wave propogation phenomena Calvo et al - First maximize the stability range of the algorithm and then improve the range of well resolved Courant numbers Bogey and Bailly: Minimize the sum of the norms of dissipation and dispersion errors in a given range of frequencies Ramboer et al: Minimize the total dissipation and dispersion errors deriving from coupling with time integration

17 Further Research Advantages And Disadvantages Conclusion References Advantages/Disadvantages Advantages Minimize dispersion and dissipation Low storage requirements(only two memory slots per variable) Larger time steps can be used Disadvantages More computations/function evaluations as compared to Runge Kutta Fourth order accuracy ensured only for linear operators with large stability regions Weak stablility properties for non linear operators for schemes of order four. Does not strictly apply to problems where grid spacing is dictated by physical constraints.

18 Further Research Advantages And Disadvantages Conclusion References Analysis of the dissipation and dispersion properties of Runge Kutta methods. schemes are proposed for convection problems Efficient implementation of large scale wave propogation problems using 2N storage formulation.

19 Further Research Advantages And Disadvantages Conclusion References References Low Dissipation and Low Dispersion RUnge Kutta Schemes, Hu et al 2N Storage Low Dissipation and Dispersion Runge Kutta Schemes, Stanescu and Habashi A General Strategy for the Optimization of RUnge Kutta Schemes for Wave Propogation Phenomena, Bernardini et al Low Dissipation and Low Dispersion Fourth Order Runge Kutta Algorithm, Berland et al A New Minimum Storage RungeKutta Scheme For Computational Acoustics, Calvo et al

CS205b/CME306. Lecture 9

CS205b/CME306. Lecture 9 CS205b/CME306 Lecture 9 1 Convection Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5; Leveque, Sections 6.7, 8.3, 10.2, 10.4. For a reference on Newton polynomial interpolation via divided