A numerical method for self-similar solutions of ideal magnetohydrodynamics

Abstract

We present a numerical method to obtain self-similar solutions of the ideal magnetohydrodynamics (MHD) equations. Under a self-similar transformation, the initial value problem (IVP) is converted into a boundary value problem (BVP) by eliminating time and transforming the system to self-similar coordinates (ξ≡x/t,η≡y/t). The ideal MHD system of equations is augmented by a generalized Lagrange multiplier (GLM) to maintain the solenoidal condition on the magnetic field. The self-similar solution to the BVP is solved using an iterative method, and implemented using the p4est adaptive mesh refinement (AMR) framework. Existing Riemann solvers (e.g., Roe, HLLD etc.) can be modified in a relatively straightforward manner and used in the present method. Numerical tests illustrate that the present self-similar solution to the BVP exhibits sharper discontinuities than the corresponding one solved by the IVP. We compare and contrast the IVP and BVP solutions in several one dimensional shock-tube test problems and two dimensional test cases include shock wave refraction at a contact discontinuity, reflection at a solid wall, and shock wave diffraction over a right angle corner.