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A best finite-difference scheme for the Fisher equation. (English) Zbl 0810.65131

“Nonstandard” difference schemes are constructed for the Fisher partial differential equation \(u_ t = u_{xx} + \lambda u (1 - u)\), \(\lambda \geq 0\), by combining (in the author’s terminology) “exact” or “best” finite difference schemes for the simpler equations \(u_ t = \lambda u(1 - u)\), \(u_ t = u_{xx}\), \(u_{xx} + \lambda u(1 - u) = 0\). By “exact” the author means that the scheme is able to reproduce the exact values of the solution of the differential equation in the grid points, by “best” he means that certain important properties (e.g. conservation laws) of the solution are discretely imitated. He claims that his schemes permit elimination of chaos and of elementary instabilities, caused by “incorrect” discrete modelling.

MSC:

65Z05 Applications to the sciences
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q80 Applications of PDE in areas other than physics (MSC2000)
92D25 Population dynamics (general)
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