A qualitative numerical solution of the equations of mathematical physics is intimately connected with ensuring a high accuracy of approximation of all differential operators included in these equations. The solution of this problem for the first and second derivatives in the equations of mathematical physics, which are used to describe a wide range of scientific and technical problems has been described in numerous literary publications. At the same time, mixed derivatives are not so often present in the equations of mathematical physics and, therefore, issues related to the quality of finite-difference approximation of these derivatives are not given enough attention in literary publications. One of the main reasons for the appearance of mixed derivatives in the equations of mathematical physics is the use of an affine transformation of the coordinate system, which provides the transition to domain of a substantially simpler form. The solution of this problem is the subject of the present paper. The problem is solved by the example of approximation of mixed derivatives on rectangular domain of definition of the required function with constant steps in each direction. A detailed derivation of the finite-difference relations used for the finite-difference approximation of mixed derivatives in all typical nodes of the function domain is given, which makes it possible to develop the proposed technique on domains of different types.
Горский В.В., Реш В.Г. Конечно-разностная аппроксимация смешанных производных в уравнениях математической физики. Математическое моделирование и численные методы, 2021, № 4, с. 58–79