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.
翻译:我们在2D Cartesian 网格内嵌入的界面上,提出了解决椭圆形PDE的更高层次有限数量的方法。第二、第四和第六级准确性在各种测试中都得到了证明,包括高相位和空间差异系数的问题、源术语中的大不一致性和复杂的界面地理特征。我们包括基于以细胞为中心的泰勒系列扩展的普遍脱轨错误分析,然后根据本地离散溶解数据和几何信息来定义线性差值。在这个过程中,我们根据Green的理论,从嵌入界面的隐含功能定义中直接计算精确的几何时点。这个方法可以产生简单的双线代表的电线性电线性电流,可以很容易地包括空间变化系数和跳跃条件,并且可以执行有限的量保护。