Abstract:
In this paper, we propose a new numerical method namely, the trivariate spectral collo-
cation 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. Nu-
merical results confirm that the proposed solution approach is highly accurate and compu-
tationally 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.
