A Singular Parameterized Finite Volume Method for the Advection-Diffusion Equation in Irregular Geometries (Journal of Computational Mathematics, 2019)
Solving the advection-diffusion equation in irregular geometries is of great importancefor realistic simulations. To this end, we adopt multi-patch parameterizations to describeirregular geometries. Different from the classical multi-patch parameterization method, $C^1$-continuity is introduced in order to avoid designing interface conditions between adjacentpatches. However, singularities of parameterizations can't always be avoided. Thus, inthis paper, a finite volume method is proposed based on smooth multi-patch singularparameterizations. It is called a singular parameterized finite volume method. Firstly,we present a numerical scheme for pure advection equation and pure diffusion equationrespectively. Secondly, numerical stability results in $L^2$ norm show that the numericalmethod is not suffered from the singularities. Thirdly, the numerical method has secondorder accurate in $L^2$ norm. Finally, three numerical tests in different irregular geometriesare presented to show efficiency of this numerical method.