A robust semi-explicit difference scheme for the Kuramoto–Tsuzuki equation

In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in L ∞ -norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in L 2 -norm of the difference scheme by an induction argument, then obtain the estimate in L ∞ -norm of the numerical solutions. Furthermore, based on the estimate in L ∞ -norm, we prove that the scheme is also convergent with second order in L ∞ -norm. Numerical examples verify the correction of the theoretical analysis.