Difference Schemes for a Drain Transfer Equation Based on Analysis in the Space of Undefined Coefficients

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.