Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor...

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Bibliographic Details
Published in:Mathematics (Basel) 2021-12, Vol.9 (24), p.3185
Main Authors: Karimov, Artur, Butusov, Denis, Andreev , Valery, Nepomuceno, Erivelton G.
Format: Article
Language:English
Subjects:
Accuracy
Adaptive control
Algorithms
Approximation
Boundary value problems
Control stability
Decomposition
Differential equations
Mathematical analysis
Mathematics
Methods
Numerical analysis
numerical integration
Numerical methods
ODE
Polynomials
rational approximation
Runge-Kutta method
stiff problem
stiff systems
Taylor series
variable step size
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Summary:While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9243185