Optimal Error Estimates of the Legendre--Petrov--Galerkin Method for the Korteweg--de Vries Equation

Abstract

In this paper, the Legendre--Petrov--Galerkin method for the Korteweg--de Vries equation with nonperiodic boundary conditions is analyzed. The nonlinear term is computed with the Legendre spectral method and some pseudospectral methods, respectively. Optimal error estimates in L2 -norm are obtained for both semidiscrete and fully discrete schemes. The method is also applicable to some (2m+1)th-order differential equations.