arXiv subjects

Simon Krotsch

Publications and source records attributed to Simon Krotsch.

3 recordsLinked to original sources

arXiv2026

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

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.

math.NA↗

arXiv2026

On Enhancing the Dissipative Behavior of Active Flux Advection Schemes

In this work, the traditional third-order Active Flux advection scheme is modified by reformulating the method and introducing four additional parameters R, S, T, and U that allow the numerical dissipation and dispersion of the scheme to be adjusted. The influence of these parameters is studied using Fourier analysis and parameter choices that lead to schemes with improved dissipative properties are identified. These improvements are validated by numerical experiments.

math.NA↗

arXiv2026

On the equivalence of semi-discrete Active Flux and Discontinuous Galerkin methods and a comparison of their performance

The Active Flux (AF) method employs a globally continuous approximation, like continuous Finite Element methods. This is achieved through the placement of point values at cell interfaces which are shared between adjacent cells. With, on average, K+1 degrees of freedom per cell, Active Flux achieves a polynomial approximation of degree K+1, while the Discontinuous Galerkin (DG) method uses only polynomials of degree K, i.e. one degree less with the same number of degrees of freedom. Despite all the differences, in this paper we show, however, that for linear problems in one and several dimensions as well as -- in some sense -- for nonlinear ones, semi-discrete AF and DG are the same method. We identify a mapping between their respective degrees of freedom, upon which the updates of these degrees of freedom turn out to agree. On the one hand, AF therefore seems more economical then DG for a given value of the error, and we confirm this in numerical experiments. On the other hand, this is a way to understand superconvergence of DG in a natural way, and we show how Radau polynomials and their zeros appear in the mapping between DG and AF: In the Radau points, AF "shines through" as the background high-order scheme behind DG.

math.NA↗