What is it about?
We propose new techniques to adapt the step-sizes in the step by step integration of systems of differential equations of stiff type, by keeping the local errors under some user-supplied tolerance of error. With this technique, we guarantee some proportionality between the global errors along the integration interval and the user-supplied tolerance. The method used as basic method in the integrations are those ones of the Runge-Kutta family known as Radau IIA methods.
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Why is it important?
The advantage of our approach, regarding other ones classical in the literature is that it is very cheap, that is, it does no increase practically the cost of the underlying method on each integration step. Our technique is based in the last two integration steps accepted for the method to give a guess for the next integration step. Another advantage is that is very easy to program and can be applied on any existing collocation Runge-Kutta method without additional computational cost.
Perspectives
We expect this technique to be incorporated in modern software (Fortran, C++, Matlab) to make more efficient the integrations of stiff problems and Differential Algebraic Equations, when using collocation Runge-Kutta methods with high order and nice stability properties, as underlying formulas.
Severiano Gonzalez Pinto
Universidad de La Laguna
Read the Original
This page is a summary of: Variable Step-Size Control Based on Two-Steps for Radau IIA Methods, ACM Transactions on Mathematical Software, December 2020, ACM (Association for Computing Machinery),
DOI: 10.1145/3408892.
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