Orthogonal cubic spline collocation method for the extended Fisher–Kolmogorov equation. (English) Zbl 1067.65107

Danumjaya, P.; Pani, Amiya K.

J. Comput. Appl. Math. 174, No. 1, 101-117 (2005).

The extended Fisher-Kolmogorov equation, first transformed into a system of second order equations, is discretized in space using a collocation method with orthogonal cubic splines. The resulting semidiscrete system leads to nonlinear differential algebraic equations of index one. A priori bounds are derived which guarantee the global existence of a unique solution. Finally, numerical experiments are presented which confirm the theoretical 4th-order convergence of the scheme. Time integration is performed with a three stage implicit Runge-Kutta scheme of 5th-order.

Reviewer: Kai Schneider (Marseille)

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
62M20	Inference from stochastic processes and prediction
35Q35	PDEs in connection with fluid mechanics
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L80	Numerical methods for differential-algebraic equations
34A09	Implicit ordinary differential equations, differential-algebraic equations

Keywords:

extended Fisher-Kolmogorov equation; Second-order splitting; Orthogonal cubic spline collocation method; Lyapunov functional; convergence; monomial basis functions; Gaussian quadrature rule; error bounds; semidiscretization; nonlinear differential algebraic equations; numerical experiments; implicit Runge-Kutta scheme

Software:

RODAS; ABDPACK

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