The numerical study of a continuous Petrov-Galerkin method for the nonlinear convection-diffusion equation

Abstract

This paper aims to use the continuous Petrov-Galerkin (CPG) method to study the nonlinear convection-diffusion equation. This method discretizes the time and space variables simultaneously with the finite element (FE) method, thus it is convenient to derive high order accuracy in time and space and has better numerical stability. In addition, the Petrov-Galerkin method is employed to approximate the model problem, which can reduce the computational scale in comparison with the usual Galerkin method. We demonstrate the existence and uniqueness of the CPG solution and give the convergence analysis without the constraints of spatial grid parameter. Several numerical tests are performed to access the validity and the numerical stability of the CPG method. Also, numerical tests illustrate that the CPG method is superior to the standard finite element (SFE) method and the continuous Galerkin (CG) method in solving the nonlinear convection-diffusion equation.