2025 , Volume 30, № 4, p.108-118

The study of the main terms of the truncation error for difference schemes plays rather important role. Firstly, for a stable difference scheme, the value of the main term of the truncation error determines the order of convergence of its grid solution. Secondly, when coupled with the original equation, they form the so-called first differential approximation [1, 2], which at the differential level methodically suggests the properties of the difference scheme. In order to illustrate the use of the first differential approximation, we present an abstract of one of the works by Yu.I. Shokin [1]. “The issues of stability and approximation viscosity of difference schemes for hyperbolic systems of equations are considered. It is shown that the stability and approximation viscosity of schemes are determined by their first differential approximation. For a number of schemes (simple, majorant, splitting), the sufficiency (and the necessity in some cases) of incomplete parabolicity of first differential approximations for the stability of the schemes is proved. In addition, necessary and sufficient conditions are given for the schemes to have an approximation viscosity that does not act on one of the invariants of the system (property)”. Thus, the main error terms, missing in the explicit form of the difference scheme, in combination with the original equation, nevertheless, affect fundamental properties such as stability, convergence, and the fulfillment of conservation laws in a number of cases. Note that attempts to add grid approximations for some main error terms to the difference scheme usually lead to instability of the resulting difference schemes. In our approach, to improve the accuracy of a numerical solution, we do not change the difference scheme operator, thus leaving the first differential approximation unchanged. We improve the accuracy of the approximation by correcting the right-hand side to compensate the contribution of one of the main error terms. This correction is determined using the already calculated grid solution of the lower order difference scheme. Despite the fact that the lower order difference scheme has only the second order convergence in time, the scheme with the corrected right-hand side already is of the fourth order convergence in time. This property is theoretically proved and is confirmed by a computational experiment. For ordinary differential equations, the application of this method is described in the collective monograph edited by K. Bohmer and H.J. Stetter [3], and was also used in the work by V.V. Shaidurov and A.E. Novikov [4] for systems of such equations. For partial differential equations, the application was restricted by difference schemes for the Poisson equation [5, 6]. In our previous work [7], this method was applied to a parabolic equation, where the correction of the right-hand side of a difference scheme of the first order convergence in time results in the second order convergence on the same difference grid in both time and space.