A Singular Parameterized Finite Volume Method for the Advection-Diffusion Equation in Irregular Geometries

Abstract

Solving the advection-diffusion equation in irregular geometries is of great importance for realistic simulations. To this end, we adopt multi-patch parameterizations to describe irregular geometries. Different from the classical multi-patch parameterization method, $C^1$-continuity is introduced in order to avoid designing interface conditions between adjacent patches. However, singularities of parameterizations can't always be avoided. Thus, in this paper, a finite volume method is proposed based on smooth multi-patch singular parameterizations. It is called a singular parameterized finite volume method. Firstly, we present a numerical scheme for pure advection equation and pure diffusion equation respectively. Secondly, numerical stability results in $L^2$ norm show that the numerical method is not suffered from the singularities. Thirdly, the numerical method has second order accurate in $L^2$ norm. Finally, three numerical tests in different irregular geometries are presented to show efficiency of this numerical method.