On numerical methods for one problem of mixed type

Abstract

This article is devoted to further investigation of numerical methods for one differential problem of mixed type. We consider a two‐dimensional first‐order differential equation with one complex‐valued and one real constant coefficients. So, we have an elliptic problem with respect to the first argument and a hyperbolic problem with respect to the second one. The equations of such type are generalized transfer equations. Firstly, the correctness of the problem stated is discussed. Secondly, possible difference scheme of the multicomponent modification of the alternating direction method is proposed. Its stability and convergence is investigated. Results of numerical experiments on modelling of nonlinear regime of surface volume free electron laser are analyzed.

Straipsnyje tiriami mišraus tipo uždaviniu sprendimo algoritmai. Nagrinejama dvimate pirmosios eiles diferencialine lygtis, kurios vienas koeflciantas yra kompleksine funkcija, o kitas ‐ realaus tipo skaičius. Taigi uždavinys yra elipsinio tipo vieno kintamojo atžvilgiu ir hiperbolinio tipo kito kintamojo atžvilgiu. Ištirtas baigtiniu skirtumu schemu stabilumas ir konvergavimas. Schemos realizavimui naudojamas dagiakomponentinis iteracinis metodas, ištirtas jo stabilumas. Pateikti matematinio modeliavimo rezultatai.