An upwind differencing scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Journal of Computational Physics
Piecewise solenoidal vector fields and the Stokes problem
SIAM Journal on Numerical Analysis
The origin of spurious solutions in computational electromagnetics
Journal of Computational Physics
Developing numerical fluxes with new sonic fix for MHD equations
Journal of Computational Physics
A Nonconforming Finite Element Method for the Stationary Navier--Stokes Equations
SIAM Journal on Numerical Analysis
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
A simple finite difference scheme for multidimensional magnetohydrodynamical equations
Journal of Computational Physics
Journal of Computational Physics
A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
Divergence correction techniques for Maxwell solvers based on a hyperbolic model
Journal of Computational Physics
The &Dgr; • = 0 constraint in shock-capturing magnetohydrodynamics codes
Journal of Computational Physics
Divergence-free adaptive mesh refinement for Magnetohydrodynamics
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Hyperbolic divergence cleaning for the MHD equations
Journal of Computational Physics
AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension)
AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension)
Journal of Computational Physics
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
Runge--Kutta Discontinuous Galerkin Method Using WENO Limiters
SIAM Journal on Scientific Computing
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In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588-610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.