Solving the MHD equations by the space-time conservation element and solution element method

We apply the space-time conservation element and solution element (CESE) method to solve the ideal MHD equations with special emphasis on satisfying the divergence free constraint of magnetic field, i.e., @?.B=0. In the setting of the CESE method, four approaches are employed: (i) the original CESE method without any additional treatment, (ii) a simple corrector procedure to update the spatial derivatives of magnetic field B after each time marching step to enforce @?.B=0 at all mesh nodes, (iii) a constraint-transport method by using a special staggered mesh to calculate magnetic field B, and (iv) the projection method by solving a Poisson solver after each time marching step. To demonstrate the capabilities of these methods, two benchmark MHD flows are calculated: (i) a rotated one-dimensional MHD shock tube problem and (ii) a MHD vortex problem. The results show no differences between different approaches and all results compare favorably with previously reported data.

[1]  Sin-Chung Chang.  The Method of Space-Time Conservation Element and Solution Element-A New Approach for Solving the Navier-Stokes and Euler Equations , 1995 .

[2]  Sin-Chung Chang,et al.  Regular Article: The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws , 1999 .

[3]  G. Tóth.  The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes , 2000 .

[4]  Phillip Colella,et al.  A Higher-Order Godunov Method for Multidimensional Ideal Magnetohydrodynamics , 1994, SIAM J. Sci. Comput..

[5]  A Roe Scheme for Ideal MHD Equations on 2D Adaptively Refined Triangular Grids , 1999 .

[6]  Guang-Shan Jiang,et al.  A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics , 1999 .

[7]  D. Balsara,et al.  A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations , 1999 .

[8]  Francesco Miniati,et al.  A Divergence-free Upwind Code for Multidimensional Magnetohydrodynamic Flows , 1998 .

[9]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[10]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[11]  K. Powell.  An Approximate Riemann Solver for Magnetohydrodynamics , 1997 .

[12]  P. Londrillo,et al.  On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method , 2004 .

[13]  Wai Ming To,et al.  A brief description of a new numerical framework for solving conservation laws — The method of space-time conservation element and solution element , 1991 .

[14]  Paul R. Woodward,et al.  A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations , 1998 .

[15]  Sin-Chung Chang,et al.  A space-time conservation element and solution element method for solving the two- and three-dimensional unsteady euler equations using quadrilateral and hexahedral meshes , 2002 .

[16]  Rho-Shin Myong,et al.  On Godunov-Type Schemes for Magnetohydrodynamics , 1998 .

[17]  Kun Xu,et al.  A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics , 2000 .

[18]  J. Brackbill,et al.  The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations☆ , 1980 .

[19]  Sin-Chung Chang,et al.  Application of the Space-Time Conservation Element and Solution Element Method to the Ideal Magnetohydrodynamic Equations , 2002 .

[20]  S. Orszag,et al.  Small-scale structure of two-dimensional magnetohydrodynamic turbulence , 1979, Journal of Fluid Mechanics.