Symbolic-numerical computation of the stability regions for Jameson's schemes

We describe a symbolic-numerical method for the Fourier stability analyses of difference initial-value problems approximating the initial-value problems for hyperbolic or parabolic PDEs. The Fourier method is reduced to the algebra of the resultants. We further use the REDUCE computer algebra system for the symbolic computation of the resultant and for the generation of a FORTRAN function to compute the value of the resultant. Basing on this FORTRAN function we further construct a binary function to characterize the stability and instability points. Using this function we generate a bilevel digital picture, and the stability region boundaries are then detected in this picture with the aid of the efficient algorithm proposed previously by Pavlidis (1982). The above symbolic-numerical method has enabled us to obtain for the first time the stability regions of the considered Jameson's schemes as applied to the two-dimensional advection-diffusion equation. Analytical formulas are proposed for the approximation of the boundaries of the obtained stability regions. It is shown that these formulas underestimate insignificantly the actual sizes of the stability regions. Therefore, these formulas can efficiently be used in practical computations by Jameson's schemes.