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ESERK5: : A fifth-order extrapolated stabilized explicit Runge–Kutta method

Published: 15 August 2019 Publication History

Abstract

A new algorithm is developed and analyzed for multi-dimensional non-linear parabolic partial differential equations (PDEs) which are semi-discretized in the spatial variables leading to a system of ordinary differential equations (ODEs). It is based on fifth-order extrapolated stabilized explicit Runge–Kutta schemes (ESERK). They are explicit methods, and therefore it is not necessary to employ complicated software for linear or non-linear system of equations. Additionally, they have extended stability regions along the negative real semi-axis, hence they can be considered to solve stiff problems coming from very common diffusion or reaction–diffusion problems.
Previously, only lower-order codes (up to fourth-order) have been constructed and made available in the scientific literature. However, at the same time, higher-order codes were demonstrated to be very efficient to solve equations where it is necessary to have a high precision or they have transient zones that are very severe, and where functions change very fast. The new schemes allow changing the step length very easily and with a very small computational cost. Thus, a variable step length, with variable number of stages algorithm is constructed and compared with good numerical results in relation to other well-known ODE solvers.

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Published In

cover image Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics  Volume 356, Issue C
Aug 2019
407 pages

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 15 August 2019

Author Tags

  1. Higher-order schemes
  2. Multi-dimensional partial differential equations
  3. Stabilized explicit Runge–Kutta methods
  4. Variable-step length ODE solvers

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