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Journal of Scientific Computing
Published in: Journal of Scientific Computing 3/2019

02-02-2019

New Third Order Low-Storage SSP Explicit Runge–Kutta Methods

Authors: I. Higueras, T. Roldán

Published in: Journal of Scientific Computing | Issue 3/2019

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Abstract

When a high dimension system of ordinary differential equations is solved numerically, the computer memory capacity may be exhausted. Thus, for such systems, it is important to incorporate low memory usage to some other properties of the scheme. In the context of strong stability preserving (SSP) schemes, some low-storage methods have been considered in the literature. In this paper we study 5-stage third order \(2N^*\) low-storage SSP explicit Runge–Kutta schemes. These are SSP schemes that can be implemented with 2N memory registers, where N is the dimension of the problem, and retain the previous time step approximation. This last property is crucial for a variable step size implementation of the scheme. In this paper, first we show that the optimal SSP methods cannot be implemented with \(2N^*\) memory registers. Next, two non-optimal SSP \(2N^*\) low-storage methods are constructed; although their SSP coefficients are not optimal, they achieve some other interesting properties. Finally, we show some numerical experiments.
previous article Generalized Uniformly Optimal Methods for Nonlinear Programming
next article Optimal Resolution Methods for the Klein–Gordon–Dirac System in the Nonrelativistic Limit Regime

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Appendix
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Literature
1.
Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: On some new low storage implementations of time advancing Runge–Kutta methods. J. Comput. Appl. Math. 236(15), 3665–3675 (2012)MathSciNetCrossRefMATH
2.
Calvo, M., Franco, J.M., Rández, L.: Minimum storage Runge–Kutta schemes for computational acoustics. Comput. Math. Appl. 45(1), 535–545 (2003)MathSciNetCrossRefMATH
3.
Cavaglieri, D., Bewley, T.: Low-storage implicit/explicit Runge–Kutta schemes for the simulation of stiff high-dimensional ode systems. J. Comput. Phys 286, 172–193 (2015)MathSciNetCrossRefMATH
4.
Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge–Kutta methods. SIAM J. Numer. Anal. 42(3), 1073–1093 (2004)MathSciNetCrossRefMATH
5.
Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58(11), 1675–1686 (2008)MathSciNetCrossRefMATH
6.
Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)CrossRefMATH
7.
Gottlieb, S., Ketcheson, D.I., Shu, C.W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)MathSciNetCrossRefMATH
8.
9.
Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)MathSciNetCrossRefMATH
10.
Happenhofer, N., Koch, O., Kupka, F.: IMEX methods for the ANTARES code. ASC report, 27 (2011)
11.
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, Nonstiff Problems, 2 revised edn. Springer, Berlin (1993)MATH
12.
Higueras, I.: Representations of Runge–Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43(3), 924–948 (2005)MathSciNetCrossRefMATH
13.
14.
Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Math. 35(3), 177–219 (2000)MathSciNetCrossRefMATH
15.
Ketcheson, D.I.: Highly efficient strong stability preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30(4), 2113–2136 (2008)MathSciNetCrossRefMATH
16.
17.
Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge–Kutta methods. Appl. Numer. Math. 59(2), 373–392 (2009)MathSciNetCrossRefMATH
18.
Ketcheson, D.I., Parsani, M., Ahmadia, A.J.: Rk-opt: software for the design of Runge–Kutta methods, version 0.2 (2013)
19.
Kraaijevanger, J.F.B.M.: Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems. Numer. Math. 48(3), 303–322 (1986)MathSciNetCrossRefMATH
20.
21.
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefMATH
22.
Ruuth, S.J.: Global optimization of explicit strong-stability-preserving Runge–Kutta methods. Math. Comput. 75(253), 183–208 (2006)MathSciNetCrossRefMATH
23.
24.
Shu, C.W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)MathSciNetCrossRefMATH
25.
26.
Spijker, M.N.: Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems. Math. Comput. 45(172), 377–392 (1985)MathSciNetCrossRefMATH
27.
Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45(3), 1226–1245 (2008)MathSciNetCrossRefMATH
28.
Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong stability preserving time discretization methods. SIAM J. Numer. Anal. 40(2), 469–491 (2002)MathSciNetCrossRefMATH
29.
Spiteri, R.J., Ruuth, S.J.: Non-linear evolution using optimal fourth-order strong-stability-preserving Runge–Kutta methods. Math. Comput. Simulat. 62(1–2), 125–135 (2003)MathSciNetCrossRefMATH
30.
Van Der Houwen, P.J.: Construction of integration formulas for initial value problems. North Holland (1977)
31.
Metadata
Title
New Third Order Low-Storage SSP Explicit Runge–Kutta Methods
Authors
I. Higueras
T. Roldán
Publication date
02-02-2019
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 3/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-019-00916-3

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