Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids
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Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids Wasilij Barsukow1 Received: 28 December 2019 / Revised: 27 July 2020 / Accepted: 4 September 2020 © The Author(s) 2020
Abstract Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state. Keywords Structure preserving · Stationarity preserving · Active flux Mathematics Subject Classification 35L65 · 35L45 · 65M08
1 Introduction Hyperbolic conservation laws in several space dimensions show a number of phenomena which are absent in one-dimensional situations. E.g., in case of the Euler equations, these additional phenomena include vortices, nontrivial stationary states (themselves governed by PDEs), involutions, and the incompressible (low Mach number) limit. Design principles for numerical methods in one spatial dimension have been subject of successful investigation over several decades (correct approximation of weak solutions, stability, entropy, ⋯ ). Numerical methods able to capture truly multi-dimensional phenomena on coarse grids still lack a comprehensive set of design principles.
The author was supported by the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF) and the European Union (FP7-PEOPLE2013-COFUND—Grant agreement no. 605728). * Wasilij Barsukow [email protected] 1
Institute of Mathematics, University of Zurich, 8057 Zurich, Switzerland
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Communications on Applied Mathematics and Computation
There exist examples of numerical methods that can cope with particular multi-dimensional phenomena without requiring excessive refinement of the computational grid. Numerical methods able to reproduce the low Mach number limit are referred to as low Mach number compliant, numerical methods that keep stationary a discretization of an involution—involution-preserving, numerical methods with stationary states being discretizations of all the stationary states of the PDE—stationarity preserving. A standard approach for constructing numerical methods in multiple spatial dimensions is to use a one-dimensional method in a direction-by-direction fashion, i.e., solving a sequence of one-dimensional problems in different directions. Such metho
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