We generate Gaussian radial function based higher order compact RBF-FD formulas for some differential operators. Analytical expressions for weights associated to first and second derivative formulas (up to order 10) and 2D-Laplacian formulas (up to order 6) are derived. Then these weights are used to obtain analytical expression for local truncation errors. The weights are obtained by symbolic computation of a linear system in Mathematica. Often such linear systems are not directly amenable to symbolic computation. We make use of symmetry of formula stencil along with Taylor series expansions for performing the computation. In the flat limit, the formulas converge to their respective order polynomial based compact FD formulas. We validate the formulas with standard test functions and demonstrate improvement in approximation accuracy with respect to corresponding order multiquadric based compact RBF-FD formulas and compact FD schemes. We also compute optimal value of shape parameter for each formula.

翻译：本文生成了基于高斯径向函数的高阶紧致RBF-FD格式，适用于若干微分算子。推导了一阶与二阶导数格式（最高十阶）及二维拉普拉斯算子格式（最高六阶）对应权系数的解析表达式，并利用这些权系数得到局部截断误差的解析表达式。权系数通过Mathematica对线性方程组进行符号计算获得，此类线性方程组通常难以直接进行符号计算。我们利用格式模板的对称性结合泰勒级数展开完成了计算。在平坦极限下，这些格式收敛至对应阶数的基于多项式的紧致有限差分格式。通过标准测试函数验证了格式的有效性，并证明了其在逼近精度上相对于同阶多重二次径向函数紧致RBF-FD格式及紧致有限差分格式的改进。同时计算了各格式的最优形状参数值。