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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.

A b stra ct-There are considered elliptic and parabolic equations of arbitrary dimension with alternating coefficients at mixed derivatives. For such equations, monotone difference schemes of the second order of local approximation are constructed. Schemes suggested satisfy the principle of maximum. A priori estimates of stability in the norm С without limitation on the grid steps r and ha , a = 1,2,... ,p are obtained (unconditional stability).

-The subject of this paper is the maximum principle and its application for investigating the stability and convergence of finite difference schemes. To some extent, this is a survey of the works on constructing and investigating certain new classes of monotone difference schemes. In this connection the maximum principle for the derivatives discussed in this paper is of fundamental importance. It is used as a basis for proving the coefficient stability of difference schemes in Banach spaces and the monotonicity of economical schemes of full approximation. New results on unconditional stability of monotone difference schemes with weights, conservative explicit-implicit schemes (staggered schemes), monotone schemes of second-order approximation in arbitrary domains, and monotone difference schemes for multidimensional elliptic equations with mixed derivatives are given.2000 Mathematics Subject Classification: 65M06; 65M12; 65M50.

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