Semi-decoupling hybrid asymptotic and augmented finite volume method for nonlinear singular interface problems

Abstract

An accurate semi-decoupling numerical method has been proposed for nonlinear singular differential equation with interface, which combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to decouple the singular interface problem and get high order accurate numerical solution. Key to our proposed new method is introducing three augmented variables involving the interface and the singularity, and reconstructing the representative of the solution as the Puiseux series expansions on the interface to decouple original problem as two standard nonlinear singular problems with the second jump condition. In this way, the augmented variables related with semi-analytic solutions near the interface and numerical solutions can be simultaneously solved in the remaining interval on both sides of the interface. It demonstrates that our method does not take more heavier works for handling jump conditions like other methods, and is independent of the interface and jump ratio. A rigorous error estimate for the solution of nonlinear singular differential equation with interface and augmented variables is obtained. Numerical experiments for those singular differential equations with interface confirm the theoretical analysis and accuracy of the new approach. In particular, an interesting example with blow-up coefficient at singular point shows that our approach can be extended to numerically solve strongly singular interface problem.