ABSTRACT

All possible symmetric two-level difference schemes on arbitrary extended stencils are considered for the Schrödinger equation and for the heat conduction equation. The coefficients of the schemes are found from conditions under which the maximum possible order of approximation with respect to the main variable is attained. A class of absolutely stable schemes is considered in a set of maximally exact schemes. To investigate the stability of the schemes, the von Neumann criterion is verified numerically and analytically. It is proved that the schemes are absolutely stable or unstable depending on the order of approximation with respect to the evolution variable. As a result of the classification, absolutely stable schemes up to the tenth order of accuracy with respect to the main variable have been constructed.

中文翻译：

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Schrödinger方程和热传导方程在扩展对称模具上最大可能精度的差分方案的分类

摘要

对于Schrödinger方程和热传导方程，考虑了在任意扩展模具上的所有可能的对称两级差分方案。该方案的系数是从这样的条件下得出的，在该条件下，相对于主变量获得了最大可能的近似值。在一组最大精确方案中考虑了一类绝对稳定的方案。为了研究方案的稳定性，对冯·诺伊曼准则进行了数值和分析验证。证明了这些方案是绝对稳定或不稳定的，这取决于相对于演化变量的近似顺序。作为分类的结果，已经构造了相对于主变量的精度达到十分精度的绝对稳定的方案。