Spatially Partitioned Embedded Runge--Kutta 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 nonembedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to nonphysical 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 nonoscillatory spatial discretizations. Numerical experiments are provided to support the theory.

Citation
Spatially Partitioned Embedded Runge--Kutta Methods 2013, 51 (5):2887 SIAM Journal on Numerical Analysis

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/130906258

arXiv
1301.4006

Additional Links
http://epubs.siam.org/doi/abs/10.1137/130906258http://arxiv.org/abs/1301.4006

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