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

Abstract

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 $uu_{x}$ implicitly and attain a system of nonlinear algebraic equations. The Taylor expansion with integral remainder is used to deal with the fourth-order term $u_{xxxx}$ 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 $\alpha$-robust stability, that is, when $\alpha \rightarrow 1^{-}$, 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.