Analytical and numerical methods for the Riemann problem

Abstract

In this survey we consider the classical and new methods for computing the analytical and numerical solutions to the Riemann problem, a class of boundary value problems for analytic functions, in a simply connected region Ω+ with smooth boundary Γ = ∂Ω+ in the complex plane. The classical methods for solving this problem based on reducing the Riemann problem to the Dirichlet problem or to the Hilbert problem where it is required the availability of a suitable conformal mapping from Ω+ onto the unit disk D. Recently, the authors introduce a new method for solving the Riemann problem by transforming its boundary condition to a Fredholm integral equation of the second kind with the generalized Neumann kernel. This method has several advantages in terms of numerical operations as well as ease in programming. This paper sketches these classical and new methods and shows the advantages of our method for solving the Riemann problem using Fredholm integral equations.