Compact finite-difference schemes are well known and provide high accuracy order for differential equation with constant coefficients.
Algorithms for constructing compact schemes of the 4-th order for boundary value problems with variable (smooth or jump) coefficient are developed. For the diffusion equations with a smooth variable coefficient and the Levin – Leontovich equation, compact finite-difference schemes are also constructed and their 4-th order is experimentally confirmed. The method of constructing compact schemes of the 4-th order can be generalized to partial differential equations and systems with weak nonlinearity, for example, for the Fisher – Kolmogorov – Petrovsky – Piskunov equation, for the nonlinear Schrödinger equation or for the Fitzhugh – Nagumo system. For such nonlinear problems, a combination of simple explicit schemes and relaxation is used. Richardson’s extrapolation increases the order of the circuits to the 6-th.
To approximate multidimensional problems with discontinuous coefficients, for example, the two-dimensional stationary diffusion equation in inhomogeneous media, it is necessary to estimate the possible asymptotics of solutions in the vicinity of the boundary line’s breaks. To do this, we use generalized eigen-functions in the angle, which can be used as a set of test functions and build compact difference schemes approximating the problem on triangular grids with high order of accuracy. The asymptotics along the radius of these generalized eigen-functions (in polar coordinates in the vicinity of the vertex of the angle) have irrational indices which can be found from a special dispersion equation and which determine the indices of the corresponding Bessel functions along 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 imitating the Cauchy problem (ICP) on the whole space. These conditions depend not only on the original equation, but also on the type of the difference scheme, and even on the coefficients of the corresponding differential equation. The ICP conditions are determined with accuracy to a gauge. But the choice of this gauge turns out to be essential with numerical implementation. The role of rational approximations of the Pade – Hermite type of the symbol of the corresponding pseudo-differential operator is important. Examples of movie solutions of problems with ICP conditions for various finite-difference schemes approximating the basic mathematical physics equations, see https://cs.hse.ru/mmsg/transbounds.
The study was realized within the framework of the Academic Fund Program at the National Research University – Higher School of Economics (HSE) in 2016–2017 (grant No. 16-05-0069) and by the Russian Academic Excellence Project «5–100».
Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-Order Compact Schemes