Two-Dimensional Mesh Embedding for B-spline Methods

Abstract

Many factors motivate consideration of B-splines as basis functions for solving partial differential equations. These are arbitrary orders of accuracy and high resolving powers similar to those of compact schemes. Furthermore, if one uses a Galerkin scheme one gets, in addition to conservation of the discretized quantities, conservation of quadratic invariants such as energy. This work develops another property, namely, the ability to treat semi-structured embedded or zonal meshes for two-dimensional geometries. This can drastically reduce the number of grid points in many applications. An algorithm is presented for constructing a global spline basis that automatically hasd-1 continuous derivatives at mesh-block boundaries as everywhere else (heredis the polynomial degree). The basis functions are simply suitable products of one-dimensional B-splines. Both integer and noninteger refinement ratios are allowed across mesh blocks. Finally, test cases for linear scalar equations such as the Poisson and advection equation are presented.