Numerical analysis of coupled hydromagnetic wave equations with a finite difference scheme

A finite difference scheme offering second-order accuracy is introduced to solve numerically a system of two mixed-type coupled partial differential equations with variable coefficients. The stability conditions of the scheme have been examined by both the Fourier method and the matrix method. The Fourier method via the local transform is first used to investigate parametrically the stability conditions of the proposed scheme. The stability conditions are checked point by point for the entire domain of interest without involving the convolution of the Fourier transform. These conditions are further verified by the matrix method. Since two different methods are employed, one can ensure that the stability conditions are achieved consistently. Moreover, the optimum parameters increasing the accuracy of the numerical solutions can be determined during the stability analysis. The proposed numerical algorithm has been demonstrated by a boundary value problem which considers the coupling and propagation of hydromagnetic waves in the magnetosphere.