TY - JOUR

T1 - A numerical method for self-similar solutions of the ideal magnetohydrodynamics

AU - Chen, Fang

AU - Samtaney, Ravi

N1 - KAUST Repository Item: Exported on 2021-09-13
Acknowledged KAUST grant number(s): BAS/1/1349-01-01
Acknowledgements: The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST) under grant no. BAS/1/1349-01-01.

PY - 2021/9/10

Y1 - 2021/9/10

N2 - 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 prob1 lem (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 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 problem 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.

AB - 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 prob1 lem (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 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 problem 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.

UR - http://hdl.handle.net/10754/671151

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021999121005854

U2 - 10.1016/j.jcp.2021.110690

DO - 10.1016/j.jcp.2021.110690

M3 - Article

SP - 110690

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -