Abstract
The present paper deals with the implementation in a variable-step algorithm of general linear methods in Nordsieck form with inherent quadratic stability and large stability regions constructed recently by Braś and Cardone. Various implementation issues such as rescale strategy, local error estimation, step-changing strategy and starting procedure are discussed. Some numerical experiments are reported, which show the performances of the methods and make comparisons with other existing methods.
Highlights
We consider initial value problem for the system of ordinary differential equations (ODEs) in the autonomous form:y = f (y), t ∈ [t0, T ], y(t0) = y0, (1.1)M
This section is devoted to the description of the issues we have considered for the implementation of our methods in a variable stepsize environment
By employing the PI stepsize control, we observe a lower number of rejected steps which provides a lower computational cost: this advantage is commonly acknowledged in the literature
Summary
We consider initial value problem for the system of ordinary differential equations (ODEs) in the autonomous form:. This property guarantees that the stability polynomial of the GLM takes the form p(ω, z) = ωr−2 ω2 − p1(z)ω + p0(z) , its stability properties depend on the quadratic polynomial p(ω, z) = ω2 − p1(z)ω + p0(z) Such a property is useful for the practical derivation of highly stable methods (e.g. A- and L-stable) in the implicit case (compare [3, 11, 19]), and methods with large stability regions in the explicit one case (see [4, 9, 10]).
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