Convergence and stability of a BSLM for advection-diffusion models with Dirichlet boundary conditions

In this paper, we present a concrete analysis of the convergence and stability of a backward semi-Lagrangian method for a non-linear advection-diffusion equation with the Dirichlet boundary conditions. The total time derivative and the diffusion term are discretized by BDF2 and the second-order central finite difference, respectively, together with the local Lagrangian interpolation. The Cauchy problem for characteristic curves is resolved by an error correction method based on a quadratic polygon. Under the mesh restriction ▵t=O(▵x^1/2) between the temporal step size △t and the spatial grid size △x, it turns out that the proposed method has the convergence order O(▵t²+▵x²+▵x^p+1/▵t) in the sense of the discrete H¹-norm, where p is the degree of an interpolation polynomial. Further, the unconditional stability of the method is established. Numerical tests are presented to support the theoretical analyses.