Abstract

A family of difference schemes on an explicit five-point stencil for the numerical solution of the linear transfer equation is considered. To construct and study the properties of difference schemes, a generalized approximation condition is used. Difference schemes in the space of undefined coefficients are analyzed. In this case, the problem of constructing the optimal difference scheme is reduced to a linear programming problem. A family of hybrid difference schemes is also considered. For them, the switching parameter will be the locally calculated dimensionless wavenumber. The analysis also shows that when constructing schemes with a higher approximation order, their local properties will be determined by the halved dimensionless wavenumber (in comparison with the scheme of the first approximation order). For the transfer equation with a linear sink, a family of difference schemes is also constructed based on such an analysis. In this case, more solutions to the linear programming problem are possible: schemes of a higher order of approximation on an unexpanded template (compact schemes) are among the optimal ones. The properties of the optimal schemes of a higher order of approximation in the case of an equation with a sink are determined by a dimensionless parameter that depends on both the wavenumber and the sink’s coefficient. Considering that for an equation with a sink, difference schemes have somewhat better computational qualities than for a homogeneous linear equation, when solving systems of the hyperbolic type by the splitting method, it is advisable to select the part with a linear sink and it is for this that a hybrid difference scheme is constructed that has a variable order of approximation on the solution of the differential problem. Numerical examples of the implemented schemes are given for the simplest linear equation.

中文翻译：

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基于未定义系数空间分析的排水传递方程差分格式

摘要

考虑了线性传递方程数值解的显式五点模板上的一系列差分格式。为了构造和研究差分方案的性质，使用了广义逼近条件。分析了未定义系数空间中的差分格式。在这种情况下，构造最优差分方案的问题就简化为线性规划问题。还考虑了一系列混合差分方案。对于它们，开关参数将是本地计算的无量纲波数。分析还表明，在构造具有更高近似阶数的方案时，它们的局部特性将由减半的无量纲波数决定（与一阶近似阶数的方案相比）。对于具有线性汇的传递方程，也基于这样的分析构造了差分格式族。在这种情况下，线性规划问题的更多解决方案是可能的：未扩展模板上更高阶近似的方案（紧凑方案）是最佳方案之一。在具有汇的方程的情况下，更高阶近似的最优方案的性质由取决于波数和汇系数的无量纲参数确定。考虑到对于具有汇的方程，差分格式比齐次线性方程具有更好的计算质量，当通过分裂方法求解双曲型系统时，建议选择具有线性接收器的部分，为此构建了一个混合差分方案，该方案在差分问题的解上具有可变的近似阶数。针对最简单的线性方程给出了所实现方案的数值示例。