Abstract
In this paper we systematically investigate explicit strong stability preserving (SSP) multistage integration methods, a subclass of general linear methods (GLMs), of order p and stage order q ≤ p. Characterization of this class of SSP GLMs is given and examples of SSP methods of order p ≤ 4 and stage order q = 1, 2, . . . , p are provided. Numerical tests are reported which confirm that the constructed methods achieve the expected order of accuracy and preserve monotonicity.
Highlights
Many practical problems in sciences and engineering are modeled by systems of ordinary differential equations (ODEs) which arise from semidiscretization of partial differential equations (PDEs) of mathematical physics
In this paper we systematically investigate the strong stability preserving (SSP) property for a subclass of explicit general linear methods (GLMs) of order p and stage order q = 1, 2, . . . , p, characterized by having the number of external stages r and the number of internal stages s, which equal the order of the method p (i.e. p = r = s), and no restriction on the rank of V is forced
In order to further validate the order preservation for high stage order SSP multistage integration methods we report in Figs. 4 and 5 the results of numerical tests that point out that the constructed high order stages SSP multistage integration methods preserve the theoretical order of convergence p, while low stage order SSP Runge–Kutta methods suffer from the well known order reduction phenomenon
Summary
Many practical problems in sciences and engineering are modeled by systems of ordinary differential equations (ODEs) which arise from semidiscretization of partial differential equations (PDEs) of mathematical physics. SSP Runge–Kutta (RK) and linear multistep methods (LMMs) have been first studied, using the terminology total variation diminishing (TVD) time discretizations, by Shu and Osher [50] They were further investigated by Gottlieb et al [22, 23, 24, 25, 27, 43], Spiteri and Ruuth [53], Hundsdorfer and Ruuth [32], Hundsdorfer, Ruuth and Spiteri [33], Higueras [28, 29, 30, 31] and Ferracina and Spijker [18, 19, 20, 21].
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