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Компактные разностные схемы для слабо нелинейных задач и граничные условия, имитирующие задачу Коши
Compact difference schemes are well known and demonstrate a high order of accuracy for differential equations with constant coefficients.
Algorithms for constructing compact schemes of the 4-th order for boundary value problems with variable (smooth and with a jump) coefficient have been developed. For the diffusion equations with a smooth variable coefficient and the Levin - Leontovich equations, difference schemes are also constructed and their 4-th order is experimentally confirmed. The method of constructing compact schemes of the 4th order can be generalized to partial differential equations and systems with weak nonlinearity, for example, the Fisher – Kolmogorov – Petrovsky – Piskunov equation, the nonlinear Schrodinger equation or the Fitzhugh - Nagumo system. For nonlinear problems, a combination of simple explicit schemes and relaxation is used. Richardson's extrapolation makes it possible to increase the order of the circuits to the 6-th.
To approximate multidimensional problems with discontinuous coefficients, for example, a two-dimensional stationary diffusion equation in heterogeneous media, it is necessary to estimate the possible asymptotics of solutions in the vicinity of boundary line breaks. To do this, generalized eigen-functions in the angle are used, which can be used as a set of test functions and build compact difference schemes approximating the problem on triangular grids with a high order of accuracy. The asymptotics with respect to the radius of generalized eigenfunctions (in polar coordinates in a vicinity of the corner vertex) have irrational powers, which can be found from a special dispersion equation and which determine the indices of the corresponding Bessel functions with respect to the radius.
For a number of difference schemes approximating the most important evolutionary equations of mathematical physics, it is possible to construct special boundary conditions simulating the Cauchy problem on the entire space. These conditions depend significantly not only on the original equation, but also on the type of difference scheme and even on the coefficients of the corresponding differential equation. The conditions of the ICP are determined up to the gauge. But with numerical implementation, the choice of this gauge turns out to be significant. The role of rational approximations of the Pade - Hermite type of the symbol of the corresponding pseudodifferential operator is important. Examples of solutions' movie to problems with the conditions of ICP for difference schemes approximating the basic equations of mat. physics, see https://cs.hse.ru/mmsg/transbounds.