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Time evolution of discrete fourth‐order elliptic operators
Author(s) -
BenArtzi Matania,
Croisille JeanPierre,
Fishelov Dalia
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22358
Subject(s) - biharmonic equation , mathematics , nonlinear system , convergence (economics) , operator (biology) , constant (computer programming) , mathematical analysis , elliptic operator , parabolic partial differential equation , partial differential equation , computer science , biochemistry , chemistry , physics , repressor , quantum mechanics , transcription factor , programming language , economics , gene , boundary value problem , economic growth
The evolution equation∂ ∂ t u = −∂ ∂ x4 u + A x∂ ∂ x2 u + A ′ x∂ ∂ xu − B x u + f , x ∈ Ω = 0 , 1 , t ≥ 0 , is considered. A discrete parabolic methodology is developed, based on a discrete elliptic (fourth‐order) calculus. The main ingredient of this calculus is a discrete biharmonic operator (DBO). In the general case, it is shown that the approximate solutions converge to the continuous one. An “almost optimal” convergence result ( O ( h 4 − ϵ )) is established in the case of constant coefficients, in particular in the pure biharmonic case. Several numerical test cases are presented that not only corroborate the theoretical accuracy result, but also demonstrate high‐order accuracy of the method in nonlinear cases. The nonlinear equations include the well‐studied Kuramoto–Sivashinsky equation. Numerical solutions for this equation are shown to approximate remarkably well the exact solutions. The numerical examples demonstrate the great improvement achieved by using the DBO instead of the standard (five‐point) discrete bilaplacian.