Splitting based finite volume schemes for ideal MHD equations

2009 ◽  
Vol 228 (3) ◽  
pp. 641-660 ◽  
Author(s):  
F.G. Fuchs ◽  
S. Mishra ◽  
N.H. Risebro
Keyword(s):  
Finite Volume ◽  
Mhd Equations ◽  
Finite Volume Schemes ◽  
Ideal Mhd

scholarly journals Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

2011 ◽  
Vol 9 (2) ◽  
pp. 324-362 ◽  
Author(s):  
Franz Georg Fuchs ◽  
Andrew D. McMurry ◽  
Siddhartha Mishra ◽  
Nils Henrik Risebro ◽  
Knut Waagan
Keyword(s):  
Finite Volume ◽  
Source Term ◽  
Second Order ◽  
High Order ◽  
Mhd Equations ◽  
Positive Pressure ◽  
Finite Volume Schemes ◽  
Riemann Solvers ◽  
Ideal Mhd ◽  
Approximate Riemann Solvers

AbstractWe design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

Partially Invariant Solutions for Two-dimensional Ideal MHD Equations

1990 ◽  
Vol 45 (11-12) ◽  
pp. 1219-1229 ◽  
Author(s):  
D.-A. Becker ◽  
E. W. Richter
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Mhd Equations ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
First Order ◽  
Partially Invariant Solutions ◽  
Quasilinear Systems ◽  
Ideal Mhd ◽  
Non Linear Equations

AbstractA generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary ideal MHD with a magnetic field perpendicular to the plane of motion.

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