The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations

In this paper, the conservation characteristic difference method based on the second-order modified upwind scheme is analyzed for solving the two-dimensional convection-diffusion equations by combining the splitting technique. Along each directional, the intermediate numerical solutions are first computed by the piecewise parabolic method (PPM), where the endpoint values over each domain are solved by the second-order modified upwind scheme. Then, the solutions are computed by the splitting implicit characteristic difference method. By some auxiliary lemmas, we can prove that our scheme is stable in \(L^{2}\) $L^2$ -norm. The optimal error estimate is given and the scheme is convergent of second order in space. Numerical experiments are given to verify the theoretical analysis.