A probabilistic symbolic-numerical method for the stability analyses of difference schemes for PDEs

We present a new symbolic-numerical method for an automatic stability analysis of difference schemes approximating scalar linear or nonlinear partial differential equations (PDEs) of hyperbolic or parabolic type. In this method the grid values of the numerical solution for any fixed moment of time are considered aa random correlated variables obeying the normal distribution law. Therefore, one can apply the notion of the Shannon’s entropy to characterize the stability of a difference scheme. The reduction of this entropy, or uncertainty, is taken as a stability criterion. It is shown at a number of examples that this criterion yields the same stability regions in the cases of linear difference initialvalue problems, as the Fourier method. In the case of two spatial variables the present probabilistic method is computationally by two orders of magnitude faster than the Fourier method.