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Explicit Runge–Kutta methods for starting integration of Lane–Emden problem

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  • Tsitouras, Ch.

Abstract

Traditionally, when constructing explicit Runge–Kutta methods we demand the satisfaction of the trivial simplifying assumption. Thus, f1=f(x0,y0) is always used as the first stage of these methods when applied to the Initial Value Problem (IVP): y′(x)=f(x,y),y(x0)=y0. Here we examine the case with f1=f(x0+c1h,y0),(h: the step) and c1 ≠ 0. We derive the order conditions for arbitrary order and construct a 5th order method at the standard cost of six stages per step. This method is found to outperform other classical Runge–Kutta pairs with orders 5(4) when applied to problems with singularity at the beginning (e.g. Lane–Emden problem).

Suggested Citation

  • Tsitouras, Ch., 2019. "Explicit Runge–Kutta methods for starting integration of Lane–Emden problem," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 353-364.
    • Handle: RePEc:eee:apmaco:v:354:y:2019:i:c:p:353-364
    DOI: 10.1016/j.amc.2019.02.047
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    References listed on IDEAS

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    1. D. F. Papadopoulos & T. E. Simos, 2013. "The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, May.
    2. T. E. Simos, 2012. "Optimizing a Hybrid Two-Step Method for the Numerical Solution of the Schrödinger Equation and Related Problems with Respect to Phase-Lag," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-17, April.
    3. T. E. Simos, 2012. "New Stable Closed Newton-Cotes Trigonometrically Fitted Formulae for Long-Time Integration," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-15, May.
    4. Junyan Ma & T. E. Simos, 2016. "Hybrid high algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 27(05), pages 1-20, May.
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    Cited by:

    1. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Tamara V. Karpukhina & Theodore E. Simos & Charalampos Tsitouras, 2022. "Runge–Kutta Embedded Methods of Orders 8(7) for Use in Quadruple Precision Computations," Mathematics, MDPI, vol. 10(18), pages 1-12, September.

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