Asymptotic preserving IMEX Active Flux schemes for a hyperbolic approximation of the Korteweg-de Vries equation

Abstract

We present a theoretical and numerical analysis of novel IMEX Active Flux schemes for a hyperbolic approximation of the Korteweg-de Vries (KdV) equation. For this approximation, it is important that the numerical schemes recover the limit equations (asymptotic preserving) and retain their order of accuracy in the limit (asymptotically accurate). Using formal asymptotic analysis and numerical experiments, we show that an Active Flux spatial discretisation (closely related to the standard semi-discrete Active Flux scheme), combined with an implicit-explicit Runge-Kutta scheme, is asymptotic preserving but not asymptotically accurate. We therefore propose alternative spatial discretisations that differ in the update equations of the point values. First, we derive a scheme by rescaling and eliminating terms in the upwind matrices. Second, we develop a second-order scheme by using central discretisation for the terms treated implicitly in time and present two modifications of it to third-order. Finally, we formulate a scheme with predefined upwind directions that are not chosen based on characteristics. We show that the alternative schemes are not only asymptotic preserving but also asymptotically accurate, and compare their accuracy, stability, and energy conservation numerically.