The mass-preserving solution-flux scheme for multi-layer interface parabolic equations

Abstract

Interface partial differential equations (PDEs) are very important in science and engineering. A new mass preserving solution-flux scheme is proposed in this paper for solving parabolic multi-layer interface problems. In the scheme, the domain is divided into staggered meshes for layers. At grid points in each subdomain, the solution-flux scheme is proposed to approximate the equation. However, due to the interface jump conditions, it is difficult to define the approximate fluxes at the irregular points next to interfaces for satisfying mass conservation for the scheme across the interfaces. The important feature of our work is that at the irregular grid points, the novel corrected approximate fluxes from two sides of the interface are proposed by combining with the interface jump conditions at interfaces, which ensure the developed solution-flux scheme mass conservative while keeping the same accuracy. We prove theoretically that our scheme satisfies mass conservation in the discrete form over the whole domain for the parabolic interface equations with multi-layers. Numerical experiments show mass conservation and convergence orders of our scheme and numerical results in multi-layer media show its excellent performance.