All-speed numerical methods for the Euler equations via a sequential explicit time integration

Abstract

This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type ("sequential explicit") time integration, a fully explicit method. This time integration takes inspiration from time-explicit staggered grid numerical methods. In this way, advantages of staggered methods carry over to collocated methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme.

References in corpus (9)

• Mechanism and Improvement of the Harten-Lax-van Leer Scheme for All-Speed Flows
• On the Mechanism of Roe-type Schemes for All-Speed Flows
• New numerical solver for flows at various Mach numbers
• Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system
• High order pressure-based semi-implicit IMEX schemes for the 3D Navier-Stokes equations at all Mach numbers
• Towards a new generation of multi-dimensional stellar evolution models: development of an implicit hydrodynamic code
• Energy-conserving Discontinuous Galerkin Methods for the Vlasov-Maxwell System
• An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity
• Truly multi-dimensional all-speed schemes for the Euler equations on Cartesian grids