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：我们提出了一种高阶有限体积方法，用于求解嵌入二维笛卡尔网格中具有跳跃条件的椭圆型偏微分方程。通过多种测试案例（包括高对比度与空间变化系数、源项大间断以及复杂界面几何形状的问题），验证了该方法在二阶、四阶和六阶精度上的有效性。我们基于单元中心泰勒级数展开进行了广义截断误差分析，从而根据局部离散解数据与几何信息定义了模板。在此过程中，我们开发了一种基于格林定理的简单方法，直接从嵌入式界面的隐函数定义中计算精确几何矩。该方法生成的模板具有简单的双线性表示形式，能够轻松纳入空间变化系数与跳跃条件，并可强制满足有限体积守恒律。