On the convergence of a second-order nonlinear SSP Runge–Kutta method for convex scalar conservation laws

A class of strong stability-preserving (SSP) high-order time discretization methods, first developed by Shu [18] and by Shu and Osher [19], has been demonstrated to be very effective in solving time-dependent partial differential equations (PDEs), especially hyperbolic conservation laws. In this paper, we consider an optimal second order SSP Runge–Kutta method, of which the spatial discretization is based on Sweby’s flux limiter construction [21] with minmod flux limiter and the $E$-scheme as the building block. For one-dimensional scalar convex conservation laws, we make minor modification to one of Yang’s convergence criteria [24] and then use it to show the entropy convergence of this SSP Runge–Kutta method.

Keywords

conservation law, SSP Runge–Kutta method, entropy convergence

2010 Mathematics Subject Classification

Primary 65M12. Secondary 35L60.

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Received 24 May 2021

Accepted 7 February 2022

Published 1 March 2023