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Uecker, Hannes

Continuation and bifurcation in nonlinear PDEs - algorithms, applications, and experiments. (English) Zbl 1490.35002

Jahresber. Dtsch. Math.-Ver. 124, No. 1, 43-80 (2022).
The author explains the application of arc-length continuation and the detection of bifurcation points, as well as computing the bifurcating branches for a system of equations with emphasis on partial differential equations, which by discretization lead to a large system of algebraic equations or ordinary differential equations.
As demonstration for the power of these method, he presents the solutions of different demanding problems, that were obtained using the software package pde2path, which was developed by the author. These problems include pattern formation in a 2-component system, pattern formation on a circular disk and pattern formation for a system with “dead core” properties. All these problems display a rich variety of solution patterns and continuation paths.
Reviewer: Alois Steindl (Wien)

Cited in 6 Documents

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B32 Bifurcations in context of PDEs
35B36 Pattern formations in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65-04 Software, source code, etc. for problems pertaining to numerical analysis

Keywords:

numerical continuation and bifurcation; branch switching; interaction of Hopf and Turing modes; patterns on disks; dead core pattern formation

Software:

pde2path; MATCONT; LOCA; psSchur; OOPDE; HomCont; BifurcationKit.jl; COCO; ILUPACK; AUTO; oomph-lib
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Full Text: DOI
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References:

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