Rational Approximation Method for Stiff Initial Value Problems

Artur Karimov,Valery Andreev,Erivelton G Nepomuceno,Denis Butusov

doi:10.3390/math9243185

Abstract

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.

Highlights

The basic principle in designing numerical methods for solving the initial value problem ẏ = f (t, y), y(t0 ) = y0 (1)is that the numerical method must fit the Taylor series expansion of the solution in a given point with the desired accuracy [1]
The main drawback of the explicit Taylor series method is that it can fail on stiff ordinary differential equations (ODE), i.e., lose stability when the step size is not appropriate or may select small step sizes in adaptive time step implementation, being guided more by stability requirements rather than accuracy limits, which results in performance loss
Its stability region grows with its order [9], which may increase the performance of high-order Taylor series methods on moderately stiff problems

Summary

Introduction

Its high computational efficiency on conventional data types can be achieved by using automatic differentiation for computing the derivatives in the Taylor expansion [3,6]. The main drawback of the explicit Taylor series method is that it can fail on stiff ODEs, i.e., lose stability when the step size is not appropriate or may select small step sizes in adaptive time step implementation, being guided more by stability requirements rather than accuracy limits, which results in performance loss. A modification of the Taylor series method for semi-linear problems called the exponential Taylor series method was proposed [10]. This idea can be used to derive Taylor-like methods, which are A-stable and L-stable [11]

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Results

Conclusion