Partially explicit splitting method for a multi-physics problem

Abstract

Multi-physics are widely studied and used in numerical analysis and simulation. Physics-based splitting methods are introduced, in which different methods are applied to solve each individual aspect of a physical system in multiple steps. The implicit methods are typically used for temporal discretization and are unconditionally stable such that the time step size can be large during the computation. However, the implicit methods are more complicated to solve, especially for nonlinear problems. In contrast, the explicit methods are relatively easier to compute, while they may require a much smaller time step size. In this paper, we propose a partially explicit scheme with physics-based splitting. We take the convection diffusion equation as an example. In this scheme, the convection equation is solved using the exact solution, and then we solve the diffusion equation via finite element methods. Some degrees of freedom in the diffusion equation are treated implicitly while others are treated explicitly. We can change the large-scale implicit system into one smaller-scale implicit system and one smaller-scale explicit system, which provides saving in computation. We analyze the stability and the order of accuracy. We also present some numerical results to show the performance of our proposed scheme.