Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity

Abstract

This paper proposes a time two-grid finite difference (TTGFD) technique for computing numerical solution of the one-dimensional (1D) fourth-order Sobolev-type equation with Burgers-type nonlinearity. The proposed strategy mainly contains three computational stages. First, the fully nonlinear problem is approximated over a coarse grid with grid size τC. Second, an approximate solution is obtained on a fine grid with grid size τF based on the solution of the coarse grid using the Lagrangian interpolation formula. Finally, the linear problem is solved on the fine grid. Compared with the standard finite difference (SFD) scheme, an advantage of the TTGFD method is that it can maintain optimal accuracy while reducing the computational cost. Meanwhile, based on the reduced order and the discrete energy method, the conservative invariant and uniqueness of proposed method are demonstrated. In addition, the stability and convergence with the order O(τC2+τF2+h2) are evaluated in the L∞-norm, where h is the spatial step size. Finally, numerical results show the validity of the proposed strategy and support the theoretical results.