A formulation of the Taylor expansion with symmetric polynomial algebra allows to compute the coefficients of compact finite difference schemes, which solve the Poisson equation at an arbitrary order of accuracy on a uniform Cartesian grid in arbitrary dimensions. This construction produces original high order schemes which respect the Discrete Maximum Principle : a tenth order scheme in dimension three and several sixth order schemes in arbitrary dimension. Numerical experiments validate the accuracy of these schemes.

中文翻译：

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任意维泊松方程的任意阶紧致有限差分格式

具有对称多项式代数的泰勒展开式的公式允许计算紧凑有限差分格式的系数，该格式在任意维度的均匀笛卡尔网格上以任意精度顺序求解泊松方程。这种构造产生了尊重离散最大原则的原始高阶方案：第三维的十阶方案和任意维度的几个六阶方案。数值实验验证了这些方案的准确性。