Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I: The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics. (English) Zbl 0760.65087
This paper is part I of a series of research papers on a dynamical approach to study numerical methods for nonlinear differential equations and it serves as an introduction to the parts II and III. About 21 pages are devoted to the introduction, motivation and relevance of this approach, and the implications, recommendations and conclusions.
On 30 pages it is demonstrated how differently differential equations and their discretized counterparts can behave. Numerical experiments for a scalar nonlinear ordinary differential equation \(u'=\alpha S(u)\) with \(S(u)=u(1-u)\) (the logistic equation) or \(S(u)=u(1-u)(b-u)\) are pictured in terms of fixpoint diagrams and bifurcation diagrams. Several two level and three level methods are tested. One of the findings is that spurious steady-state solutions can occur below the linearized stability limit. A rough outlook for partial differential equations is given on 9 pages.
I close by citing the authors: “Thus the mission of this paper is not to provide the answer or theory or to illustrate the connection of dynamical behavior of practical partial differential equations to their discretized counterpart, but rather to gain insight into the nonlinear features unconventional to this type of study and concentrate on the fundamentals. In order to bring out the new features, the illustrations concentrate on simple scalar differential equation examples in which the exact solutions of the differential equations are known.”.
On 30 pages it is demonstrated how differently differential equations and their discretized counterparts can behave. Numerical experiments for a scalar nonlinear ordinary differential equation \(u'=\alpha S(u)\) with \(S(u)=u(1-u)\) (the logistic equation) or \(S(u)=u(1-u)(b-u)\) are pictured in terms of fixpoint diagrams and bifurcation diagrams. Several two level and three level methods are tested. One of the findings is that spurious steady-state solutions can occur below the linearized stability limit. A rough outlook for partial differential equations is given on 9 pages.
I close by citing the authors: “Thus the mission of this paper is not to provide the answer or theory or to illustrate the connection of dynamical behavior of practical partial differential equations to their discretized counterpart, but rather to gain insight into the nonlinear features unconventional to this type of study and concentrate on the fundamentals. In order to bring out the new features, the illustrations concentrate on simple scalar differential equation examples in which the exact solutions of the differential equations are known.”.
Reviewer: W.Zulehner (Linz)
MSC:
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
time discretization; algorithm; dynamics of numerics; stability of methods; numerical experiments; nonlinear differential equations; logistic equation; fixpoint diagrams; bifurcation diagrams; spurious steady-state solutionsReferences:
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