ABSTRACT
This chapter presents the method for the stability analysis of difference schemes. This method was rigorously justified only for very limited classes of linear difference schemes for hyperbolic partial differential equations (PDEs). The finite difference methods enable one to obtain only an approximate solution of the original initial-value problem for the PDE. The accuracy of the numerical solution can be checked with the aid of the comparison with some exact analytic solution. Many problems of the mathematical physics involve a domain of spatial variables. The chapter discusses the notion of the convergence of a difference scheme to a PDE. The formulation of the convergence criteria for difference initial-value problems and difference initial- and boundary-value problems is one of the important parts of the theory of difference schemes for PDEs. The chapter utilizes the Fourier method for the stability analysis of difference schemes approximating the scalar PDEs.
