An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties

In this paper, an implicit robust difference method with graded meshes is constructed for the modified Burgers model with nonlocal dynamic properties. The L1 formula on graded meshes for the fractional derivative in the Caputo sense is employed. The Galerkin method based on piece-wise linear test functions is used to handle the nonlinear convection term uux implicitly and attain a system of nonlinear algebraic equations. The Taylor expansion with integral remainder is used to deal with the fourth-order term uxxxx and the second-order term u(xx). Then the existence and uniqueness of numerical solutions for the proposed implicit difference scheme are proved. Meanwhile, the unconditional stability is also derived. By introducing a new discrete Gronwall inequality, we improve the L-2-stability to the a-robust stability, that is, when a ? 1(-), the bound will not blow up. And we also yield the optimal convergence order in the L-2 energy norm. Finally, we give three numerical experiments to illustrate and compare the efficiency of the proposed robust method on uniform meshes and graded meshes. It shows that the results are consistent with our theoretical analysis.

Automated summary

In this paper, an implicit robust difference method is constructed to handle the modified Burgers model with nonlocal dynamic properties. The L1 formula on graded meshes is employed for the fractional derivative. The existence and uniqueness of numerical solutions are proved, and the unconditional stability is derived. A new discrete Gronwall inequality is introduced to improve the stability, and the optimal convergence order in the L-2 energy norm is also obtained. Numerical experiments demonstrate the efficiency of the proposed robust method on uniform and graded meshes.