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Analysis of a Chebyshev-based backward differentiation formulae and relation with Runge-Kutta collocation methods. (English) Zbl 1229.65115

The paper introduces a method based on the backward differentiation formula, (BDF), in terms of truncated Chebyshev series to solve stiff initial value problems of the form \(y'=f(t,y),\;y(t_0)=y_0\). Instead of approximating \(f\), an interpolating polynomial at Chebyshev-Gauss-Lobatto nodes is taken, while, as in the BDF-technique, imposing the derivative at these nodes to coincide with the values of \(f.\) Truncation error, stability and absolute stability of the method are been studied, provided that the solution \(y\) is sufficiently smooth. The method coincides with a class of Runge-Kutta collocation method. A couple of standard examples are exhibited.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L04 Numerical methods for stiff equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
62L20 Stochastic approximation
34A34 Nonlinear ordinary differential equations and systems

Software:

DIFSUB; VODE; DASSL; RADAU
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Full Text: DOI

References:

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