Construction of Boundary Conditions for Hyperbolic Relaxation Approximations II: Jin-Xin Relaxation Model

TL;DR

This paper develops boundary conditions for the Jin-Xin relaxation model, ensuring accurate approximation of hyperbolic conservation laws with non-characteristic boundaries, and analyzes their effectiveness and compatibility.

Contribution

It constructs and analyzes boundary conditions for the Jin-Xin relaxation model that approximate given conservation laws, addressing non-uniqueness and effectiveness.

Findings

Boundary conditions satisfy the generalized Kreiss condition.

Constructed boundary conditions are compatible with initial data.

Asymptotic expansion proves the effectiveness of the approximations.

Abstract

This is our second work in the series about constructing boundary conditions for hyperbolic relaxation approximations. The present work is concerned with the one-dimensional linearized Jin-Xin relaxation model, a convenient approximation of hyperbolic conservation laws, with non-characteristic boundaries. Assume that proper boundary conditions are given for the conservation laws. We construct boundary conditions for the relaxation model with the expectation that the resultant initial-boundary-value problems are approximations to the given conservation laws with the boundary conditions. The constructed boundary conditions are highly non-unique. Their satisfaction of the generalized Kreiss condition is analyzed. The compatibility with initial data is studied. Furthermore, by resorting to a formal asymptotic expansion, we prove the effectiveness of the approximations.