Monotone economical schemes for quasilinear parabolic equations

Abstract

In order to approximate a multidimensional quasilinear parabolic equation with unlimited nonlinearity the economical vector‐additive scheme is constructed. It is shown that its solution satisfies the maximum principle and, hence, the scheme is monotone. The proof is based on the equivalence of the vector‐additive scheme and the scheme of summarized approximation (locally one‐dimensional scheme). The a priori estimates of the difference solution in the uniform norm are obtained.

Šiame straipsnyje pasiūlytas ekonomiškos vektoriškai adityvios schemos, aproksimuojančios daugiamate kvazitiesine paraboline lygti su negriežto tipo netiesiškumu. Irodyta, kad šios lygties sprendinys tenkina maksimumo principa ir todel pasiūlytoji schema yra monotonine. Irodymas yra pagristas vektoriškai‐adityvios schemos ekvivalentiškumu sumines aproksimacijos schemai, kuri yra lokaliai vienmate schema. Gauti skaitinio sprendinio aprioriniai iverčiai maksimumo normoje.

Key words:

• finite‐difference schemes
• nonlinear parabolic problems
• maximum principle