We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.

中文翻译：

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粗网格和细网格的精确解-自适应有限差分方案

我们通过最小化最小二乘意义上的截断误差，引入了解相关的有限差分模具，其系数适合当前的数值解。所得的方案具有在欠解析域中保留色散关系的差分模板的分辨能力，以及在分辨率良好的区域中常规中心差分方法的高阶收敛速度。数值实验表明，在从非常粗糙到非常精细的所有网格分辨率上，新的模板都优于传统的模板。