A novel difference equation approach for the stability and robustness of compact schemes for variable coefficient PDEs

Abstract

Fourth-order accurate compact schemes for variable coefficient convection diffusion equations are considered in this paper. Despite superior efficiency due to the compact stencils, the scheme’s stability analysis is much harder for the cumbersome expression of amplification matrix. We present a theoretical investigation of spectral radius using matrix method, as the popular von Neumann stability analysis is not applicable to the schemes for variable coefficient PDEs. Thereby a sufficient condition for the stability of the compact scheme is derived using a difference equation based approach. Subsequently, the constant coefficient PDEs are considered as a special case, and the unconditional stability of the schemes for such case is proved theoretically. An estimate of condition number of the amplification matrix is derived to study the robustness of the scheme. As an application, the Black–Scholes PDE for option pricing is numerically solved in both variable and constant coefficient frameworks. The numerical illustrations evidently support the theoretical findings.