On numerical methods for hyperbolic PDE with curl involutions

Abstract

In this paper we present three different numerical approaches to account for curl-type involution constraints in hyperbolic partial differential equations for continuum physics. All approaches have a direct analogy to existing and well-known divergence-preserving schemes for the Maxwell and MHD equations. The first method consists in a generalization of the Godunov-Powell terms, which means adding suitable multiples of the involution constraints to the PDE system in order to achieve the symmetric Godunov form. The second method is an extension of the generalized Lagrangian multiplier (GLM) approach of Munz et al., where the numerical errors in the involution constraint are propagated away via an augmented PDE system. The last method is an exactly involution preserving discretization, similar to the exactly divergence-free schemes for the Maxwell and MHD equations, making use of appropriately staggered meshes. We present some numerical results that allow to compare all three approaches with each other.

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Cited by in corpus (5)

• A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics
• High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension
• Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems
• On Hamiltonian continuum mechanics
• Inverse problems for first-order hyperbolic equations with time-dependent coefficients