A new symbolic-numeric approach to stability analysis of difference schemes

Abstract

The numerical stability of most finite difference schemes that are currently in use to attempt to numer-

Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

“…The feature of the symbolic-numeric algorithm proposed by Ganzha et al (1992) is that it makes use of the Newton identities to compute the numerical values of the coefficients c j ( κ, k) of the characteristic equation…”

“…, κ M like the parameters (33). Thus, the symbolic computations were used in Ganzha et al (1992) beginning from the initial stage of deriving the difference scheme in integral steps "n" and "n + 1" and ending with the symbolic computation of the entries g jk , j, k = 1, . .…”

“…This strategy of terminating the symbolic mode of computation at the stage of symbolic computation of the entries of G appeared attractive for us also in the case of difference schemes on curvilinear grids. However, an attempt to use the symbolic stages from Ganzha et al (1992) resulted in the memory exhaustion already at the initial stage of deriving the difference equations in integral steps from the three-stage difference scheme (20) by eliminating the intermediate values u…”

“…The main reason for this was the fact that the symbolic method of Ganzha et al (1992) requires the explicit specification of the entries of the Jacobi matrices A and B (12) already at the early stages of symbolic computations. In the case of curvilinear grid one has to use much more complex matrices A 1 and A 2 (19) in accordance with (18), which immediately leads to memory exhaustion.…”

“…In the method of Ganzha et al (1992) the Newton identities were used to compute numerical values of the coefficients c j ( κ, k) of the characteristic equation (1). We first implemented this approach in a Mathematica program.…”