The Crank–Nicolson/Adams–Bashforth scheme for the Burgers equation with H2 and H1 initial data

Abstract

In this paper, we consider the stability and convergence results of the Crank–Nicolson/Adams–Bashforth scheme for the Burgers equation with smooth and nonsmooth initial data. The spatial approximation is based on the standard conforming finite element space. The temporal treatment of the spatial discrete Burgers equation is based on the implicit Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Firstly, we prove that the Crank–Nicolson/Adams–Bashforth scheme is almost unconditionally stable with initial data u0∈Hα (α=1,2). Secondly, the optimal error estimates of the numerical solution in L2-norm are derived with initial data u0∈H2, and the error estimates of approximate solution in L2 norm obtained with initial data u0∈H1 is reduced by 12. Finally, some numerical examples are provided to verify the established stability theory and convergence results with H2 and H1 initial data.