A High Order Cartesian Grid, Finite Volume Method for Elliptic Interface Problems
Abstract
We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.
Cite
@article{arxiv.2302.09161,
title = {A High Order Cartesian Grid, Finite Volume Method for Elliptic Interface Problems},
author = {Will Thacher and Hans Johansen and Daniel Martin},
journal= {arXiv preprint arXiv:2302.09161},
year = {2023}
}