For one-dimensional and multidimensional semilinear transport equations of quite a general form with given initial data and boundary conditions the exact difference schemes (EDSs) are constructed. In the case of constant coe±cients, such numerical methods can be created on rectangular grids, while in the case of variable coefficients - on moving grids only. The questions of developing difference schemes of arbitrary order for quasi-linear transport equations with a nonlinear right-hand side are discussed. In this paper, the EDSs are constructed also for certain classes of linear and quasilinear parabolic equations, for convection-diffusion problems with a small parameter, as well as inhomogeneous wave equations with constant coe±cients.

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The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u(x, t) = X(x)T 1 (t) + T 2 (t), ∂u ∂x = 0, it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. The special attention is paid to investigating approximation, stability and convergence of the ADS. The convergence of the iteration method is also considered. The presented numerical examples illustrate theoretical results investigated in the paper. IntroductionIn which cases an EDS or an ADS approximating nonlinear parabolic equation can be constructed? The paper deals with this question. The simple technique is presented and the main features of the constructed scheme are considered. Definition 2. [3]A difference scheme is exact if the truncation error is equal to zero or the exact solution agrees with the numerical solution at the grid nodes.The problem of constructing a difference scheme of high order of accuracy is topical. In papers [1,6] the EDSs and truncated difference schemes of an arbitrary rank were constructed for the nonlinear second order differential equation and for the systems of first-order nonlinear * lapkam@kul.pl Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 16/07/2020 03:37:39 U M C S 94 Difference schemes of arbitrary order of accuracy for semilinear parabolic equations τ m 2M +2 , where m, M are positive, natural constants.