A Robust Second Order Invariant-Domain Preserving Approximation of the Compressible Euler Equations with an Arbitrary Equation of State

Abstract

For many physical problems, robust numerical methods for solving the compressible Euler equations are essential. For the Euler equations to accurately describe the fluid behavior a suitable equation of state (EOS), which describes the relationship between the thermodynamic variables, must be chosen. However, a robust numerical method which can handle an arbitrary equation of state has been unavailable. In this thesis, we present a second order invariant-domain preserving method for the compressible Euler equations with an arbitrary equation of state. The description of the second order method first requires the development of a first order method that preserves certain thermodynamic properties of the fluid. A method which preserves this physical aspect is referred to as an invariant-domain preserving method. The fundamental methodology of the first order method relies on estimating the maximum wave speed of local Riemann problems. For an arbitrary equation of state this estimation can be impossible. We circumvent the issue by extending the system with an interpolatory EOS, and rigorously justify that the use of the max wave speed of this extended problem implies the invariant-domain preserving properties of the method. Using a higher order graph viscosity cannot guarantee the invariant-domain preserving property. We resolve this issue through the use of quasiconcave limiting on the density and a surrogate entropy. For an arbitrary equation of state, access to the entropy may not be possible, therefore, this surrogate entropy is used to guarantee that a local approximation of the entropy will increase across shock waves. Furthermore, limiting on the surrogate entropy guarantees that the specific internal energy satisfies the invariant domain constraint.