IIM-based ADI finite difference scheme for nonlinear convection-diffusion equations with interfaces

Based on Li's immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection-diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams-Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams-Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.