The higher-Order CESE Method for two-dimensional Shallow Water Magnetohydrodynamics Equations

The numerical solution of one and two dimensional shallow water magnetohydrodynamics model is obtained using the 4th-order conservation element solution element method (CESE). The method is based on unified treatment of spatial and temporal dimensions contrary to the finite difference and finite volume methods. The higher-order CESE scheme is constructed using same definitions of conservation and solution elements that are used for 2nd-order CESE scheme formulation. Hence it is more convenient to increase accuracy of CESE methods as compared to the finite difference and finite volume methods. Moreover the scheme is developed using the conservative formulation and does not require change in the source term for treating the degenerate hyperbolic nature of shallow water magnetohydrodynamics system due to divergence constraint. The spatial and temporal derivatives have been obtained by incorporating 3rd-order Taylor expansion and the projection method is used to handle the divergence constraint. The accuracy and robustness of the extended method is tested by performing benchmark numerical tests taken from the literature. Numerical experiments revealed the accuracy and computational efficiency of the scheme.