Abstract A new method is proposed for numerically solving the Poisson equation for non-continuous scalar fields on a uniform Cartesian grid. The sharp discontinuity in both the magnitude and the gradient of the scalar field normal to the interface is represented by the numerical solution with second order accuracy at the interface. This is achieved by setting up a composite solution, which is a weighted average of two fictitious scalar fields that together produce the required discontinuity within each interfacial grid cell. A smooth treatment of the Poisson coefficient in a narrow band around the interface allows sharp interfacial jumps to be expressed with second order accuracy on regular grid points around the interface using a standard signed distance function. Moreover, the jump in the gradient tangent to the interface is not needed to enforce the jump in the gradient normal to the interface. The resulting linear system is symmetric and leads to second order accurate solutions on grid points adjacent to the interface. The accuracy of the new framework is compared with other methods.

中文翻译：

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均匀笛卡尔网格上泊松方程不连续解的局部二阶对称方法

摘要 提出了一种数值求解均匀笛卡尔网格上非连续标量场泊松方程的新方法。垂直于界面的标量场的幅度和梯度的急剧不连续性由界面处具有二阶精度的数值解表示。这是通过设置复合解决方案来实现的，该解决方案是两个虚构标量场的加权平均值，它们共同在每个界面网格单元内产生所需的不连续性。对界面周围窄带中泊松系数的平滑处理允许使用标准有符号距离函数在界面周围的规则网格点上以二阶精度表示急剧的界面跳跃。而且，与界面相切的梯度跳跃不需要强制垂直于界面的梯度跳跃。由此产生的线性系统是对称的，并在与界面相邻的网格点上产生二阶精确解。新框架的准确性与其他方法进行了比较。