Application of analytical methods to computing numerical flux Jacobians

Overview This paper compares the accuracy, convergence, and performance of two different methods for determining numerical flux jacobians for the ideal MHD (magnet ohydrodynamic) equations. This paper's algorithm is a Roe approximate Riemann solver. The numerical flux jacobian is the derivative of the flux with respect to the conserved variables. Implicit solvers, which use a form of Newton iteration, use numerical flux jacobians to find the correct solution. This paper uses two methods to formulate the numerical flux jacobian. The limit formulation perturbs each conserved variable and measures the change in the flux. The analytic formulation evaluates the analytic derivative for the numerical flux. A similar comparison for the Navier-Stokes equations has been published by Orkwis and Venden, that paper suggests that the analytic formulation may be futile effort. This paper reassesses this judgement. The accuracy and convergence of the two methods are identical. The analytical formulation requires less computational time than the limit formulation. This paper describes the algorithm and theory involved. It presents the results from a solver that implements both methods.