Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation

Mahmoud Saleh; Endre Kovács; Nagaraja Kallur; Mahmoud Saleh; Endre Kovács; Nagaraja Kallur

doi:10.3934/nhm.2023046

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1059-1082. doi: 10.3934/nhm.2023046

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Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation

1.
University of Miskolc, Institute of Physics and Electrical Engineering, Hungary
2.
Department of Mathematics, Amrita School of Engineering, Bengaluru, Amrita Vishwa Vidyapeetham, India

Received: 04 January 2023 Revised: 26 February 2023 Accepted: 02 March 2023 Published: 29 March 2023

We systematically test families of explicit adaptive step size controllers for solving the diffusion or heat equation. After discretizing the space variables as in the conventional method of lines, we are left with a system of ordinary differential equations (ODEs). Different methods for estimating the local error and techniques for changing the step size when solving a system of ODEs were suggested previously by researchers. In this paper, those local error estimators and techniques are used to generate different types of adaptive step size controllers. Those controllers are applied to a system of ODEs resulting from discretizing diffusion equations. The performances of the controllers were compared in the cases of three different experiments. The first and the second system are heat conduction in homogeneous and inhomogeneous media, while the third one contains a moving heat source that can correspond to a welding process.
- diffusion equation,
- explicit adaptive step controller,
- moving heat source,
- Runge-Kutta methods,
- linear-neighbor method
Citation: Mahmoud Saleh, Endre Kovács, Nagaraja Kallur. Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation[J]. Networks and Heterogeneous Media, 2023, 18(3): 1059-1082. doi: 10.3934/nhm.2023046

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Abstract

We systematically test families of explicit adaptive step size controllers for solving the diffusion or heat equation. After discretizing the space variables as in the conventional method of lines, we are left with a system of ordinary differential equations (ODEs). Different methods for estimating the local error and techniques for changing the step size when solving a system of ODEs were suggested previously by researchers. In this paper, those local error estimators and techniques are used to generate different types of adaptive step size controllers. Those controllers are applied to a system of ODEs resulting from discretizing diffusion equations. The performances of the controllers were compared in the cases of three different experiments. The first and the second system are heat conduction in homogeneous and inhomogeneous media, while the third one contains a moving heat source that can correspond to a welding process.

References

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