Theoretical analysis and numerical scheme of local conservative characteristic finite difference for 2-d advection diffusion equations

In this paper, the mass conservative characteristic finite difference scheme for 2-d advection diffusion equations is analyzed. Firstly, along x-direction, we obtain the solutions $\left\{{\tilde{U}}_{i,j}^n\right\}$ by applying the piecewise parabolic method (PPM) on the Lagrangian grid where $\overline{x}$ is solved using the first-order Runge Kutta scheme. Secondly, the mass ${\overline{M}}_{i,j}^n$ over ${\overline{\Omega }}_{i,j}\left(t^n\right)$ are solved by the PPM scheme along y-direction. Finally, the local conservative characteristic finite difference scheme is constructed. By some auxiliary lemmas, we prove our scheme is stable and obtain the optimal error estimate. Our scheme is proved to be of second order convergence in space and of first order in time. Numerical experiments are used to verify the theoretical analysis.