Regular and non-regular solutions of the Riemann problem in ideal magnetohydrodynamics

TL;DR

This paper develops a numerical code to solve the Riemann problem in ideal magnetohydrodynamics, revealing the existence and non-uniqueness of non-regular solutions, challenging previous assumptions about the problem's solution space.

Contribution

The authors created a comprehensive code capable of handling both regular and non-regular solutions, demonstrating the existence and multiplicity of non-regular solutions in ideal MHD Riemann problems.

Findings

Existence of non-regular solutions where regular solutions are absent.

Non-regular solutions are not unique, with uncountably many solutions for certain initial conditions.

Regular solutions are insufficient to fully describe the solution space of the ideal MHD Riemann problem.

Abstract

We have built a code to numerically solve the Riemann problem in ideal magnetohydrodynamics (MHD) for an arbitrary initial condition to investigate a variety of solutions more thoroughly. The code can handle not only regular solutions, in which no intermediate shocks are involved, but also all types of non-regular solutions if any. As a first application, we explored the neighborhood of the initial condition that was first picked up by Brio & Wu (1988) and has been frequently employed in the literature as a standard problem to validate numerical codes. Contrary to the conventional wisdom that there will always be a regular solution, we found an initial condition, for which there is no regular solution but a non-regular one. The latter solution has only regular solutions in its neighborhood and actually sits on the boundary of regular solutions. This implies that the regular solutions are not sufficient to solve the ideal MHD Riemann problem and suggests that at least some types of non-regular solutions are physical. We also demonstrate that the non-regular solutions are not unique. In fact, we found for the Brio & Wu initial condition that there are uncountably many non-regular solutions. This poses an intriguing question: why a particular non-regular solution is always obtained in numerical simulations? This has important ramifications to the discussion of which intermediate shocks are really admissible.