The numerical solution of the non-uniquely solvable Riemann problem using a Fredholm integral equation

Abstract

This paper is devoted to the computational aspect of the Riemann problem with non-unique solution in a simply connected region with smooth boundary. The boundary condition of the Riemann problem is transformed to a Fredholm integral equation of the second kind with continuous kernel. By imposing some side conditions, the integral equation can be solved using Nystrom method. The equivalence of the integral equation and the Riemann problem will be established for any smmoth Jordan curve. Typical examples illustrate that numerical results of high accuracy can be obtained provided that the boundaries are sufficiently smooth.