Exact Difference Schemes for Time-dependent Problems

Abstract

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.

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

“…However, as has been shown in [12], see also [11], there is no such simple and practically useful relation for the inhomogeneous equation (1).…”

“…Based on [11], in Section 2 an elementary derivation for a 1D elliptic problem with variable coefficients is reviewed, and the use of harmonic averages is advocated; we also comment on their use in higher dimensional problems. Then we discuss a time-dependent convection-diffusion-reaction equation of singular perturbation type, where the diffusion coefficient is small relative to the other coefficients.…”

Exact Difference Schemes for Multidimensional Heat Conduction Equations

“…Let us stress here that the iteration method (2.13) converges after two-three iterations performed. This is due to the calculation of the initial approximation 0 y i,j from the exact but unstable explicit scheme (2.14) [9]. Tables 4.1 4.2.…”

“…where f 2 (u) = 0, is exact [9]. In [24], the exact difference scheme is constructed for the equation…”

Well-Posedness and Blow Up for IBVP for Semilinear Parabolic Equations and Numerical Methods

“…Note that the discrete estimate (7.17) of the time of the possible blow up of the solution is consistent with differential (6.32) due to the fact that the grid inequality (7.15) is exactly consistent with differential (6.30). Indeed, in [14] was shown that the difference scheme…”