On a Class of Uniformly Accurate IMEX Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation

Abstract

In this paper we consider hyperbolic systems with relaxation in which the relaxation time $\varepsilon$ may vary from values of order one to very small values. When $\varepsilon$ is very small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. In such cases it is important to have schemes that work uniformly with respect to $\varepsilon$. IMplicit-EXplicit (IMEX) Runge–Kutta (R-K) schemes have been widely used for the time evolution of hyperbolic partial differential equations but the schemes existing in literature do not exhibit uniform accuracy with respect to the relaxation time. We develop new IMEX R-K schemes for hyperbolic systems with relaxation that present better uniform accuracy than the ones existing in the literature and in particular produce good behavior with high order accuracy in the asymptotic limit, i.e., when $\varepsilon$ is very small. These schemes are obtained by imposing new additional order conditions to guarantee better accuracy over a wide range of the relaxation time. We propose the construction of new third-order IMEX R-K schemes of type CK [S. Boscarino, SIAM J. Numer. Anal., 45 (2008), pp. 1600–1621]. In several test problems, these schemes, with a fixed spatial discretization, exhibit for all range of the relaxation time an almost uniform third-order accuracy.