A compact scheme for solving a superdiffusion equation\\ with several variable delays

V. G. Pimenov ; A. V. Lekomtsev

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 148, pp. 440-454

Résumé

A superdiffusion equation with Riesz fractional derivatives with respect to space with several delay variables is considered. The problem is discretized. For this purpose, an analog of the Crank–Nicolson difference method with piecewise linear interpolation to account for the effect of variable delay and with extrapolation by continuation is constructed in time so that the implicitness of the method becomes finite-dimensional. An analog of a compact scheme with a special replacement of Riesz fractional derivatives by fractional central differences is constructed in space. As a result, the method is reduced to solving a system of linear algebraic equations with symmetric and positive-definite main matrix at each time step. The order of smallness with respect to the discretization time-steps $\Delta$ and space-steps $h$ of the residual of the method without interpolation and with interpolation is studied; it is equal to $O(\Delta^2+h^4)$. The main result consists in proving that the method converges with the order $O(\Delta^2+h^4)$ in the energy and compact norm of the layered error vector. The results of test examples for superdiffusion equations with constant and variable delay are presented. The computable orders of convergence for each discretization step in the examples turned out to be close to the theoretically obtained orders of convergence for the corresponding discretization steps.

Détail

Mots-clés : superdiffusion equation
Keywords: several variable delays, compact scheme, piecewise linear interpolation