Spatially partitioned embedded Runge-Kutta methods

TL;DR

This paper introduces spatially partitioned embedded Runge-Kutta schemes for PDEs, addressing efficiency and conservation issues by proposing flux-based partitioning and demonstrating their effectiveness through numerical experiments.

Contribution

It presents novel SPERK schemes that improve efficiency and conservation in PDE time integration, including flux partitioning techniques and high-order embedded pairs.

Findings

Partitioned schemes can cause non-physical effects without conservation.

Flux-based partitioning restores conservation and physical accuracy.

Numerical experiments confirm the effectiveness of proposed methods.

Abstract

We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory.