A highly accurate trivariate spectral collocation method of solution for two-dimensional nonlinear initial-boundary value problems

Abstract

In this paper, we propose a new numerical method namely, the trivariate spectral collocation method for solving two-dimensional nonlinear partial differential equations (PDEs) arising from unsteady processes. The problems considered are nonlinear PDEs defined on regular geometries. In the current solution approach, the quasi-linearization method is used to simplify the nonlinear PDEs. The solutions of the linearized PDEs are assumed to be trivariate Lagrange interpolating polynomials constructed using Chebyshev Gauss-Lobatto (CGL) points. A purely spectral collocation-based discretization is employed on the two space variables and the time variable to yield a system of linear algebraic equations that are solved by iteration. The numerical scheme is tested on four typical examples of nonlinear PDEs reported in the literature as a single equation or system of equations. Numerical results confirm that the proposed solution approach is highly accurate and computationally efficient when applied to solve two-dimensional initial-boundary value problems defined on small time intervals and hence it is a reliable alternative numerical method for solving this class of problems. The new error bound theorems and proofs on trivariate polynomial interpolation that we present support findings from the numerical simulations.