ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

An Adaptive Multiresolution Scheme with Second Order Local Time-stepping for Reaction-diffusion Equations

Journal of Applied Nonlinear Dynamics 7(3) (2018) 287--295 | DOI:10.5890/JAND.2018.09.006

Müller Moreira Lopes$^{1}$; Margarete O. Domingues$^{2}$; Odim Mendes$^{3}$; Kai Schneider$^{4}$

$^{1}$ Post-graduation program in Applied Computing, National Institute for Space Research, São José dos Campos, 12227-010, Brazil

$^{2}$ Associated Laboratory for Computing and Applied Mathematics, National Institute for Space Research, São José dos Campos, 12227-010, Brazil

$^{3}$ Space Geophysics Division, National Institute for Space Research, São José dos Campos, 12227-010, Brazil

$^{4}$ Institut de Mathématiques de Marseille (I2M), Aix-Marseille Université, 13453, Marseille, Cedex 13, France

Abstract

For adaptive multiresolution schemes we propose a local timestepping scheme based on natural extensions of Runge–Kutta methods. We consider reaction-diffusion equations in two space dimensions and assess the precision and efficiency of the new method. The obtained results are compared with those using classical finite volume schemes on a uniform grid and multiresolution schemes with global time stepping. It is shown that both CPU time and precision of the adaptive solutions are improved.

Acknowledgments

The authors thank the Brazilian agencies CAPES, CNPq (140626/2014-0, 306038/2015-3, 307083/2017-9), FAPESP (2015/50403-0, 2015/25624-2), FINEP/CT-INFRA (01120527-00) for financial support. The authors are indebted to Prof. C.-D. Munz who motivated the use of NERK method for local time stepping. We thank Dr. Olivier Roussel for developing the original Carmen Code and fruitful scientific discussions. We also thank Eng. V. E. Menconi and Dr. A. K. F. Gomes for their helpful computational assistance. MD thankfully acknowledges financial support from ECM, France. KS acknowledges financial support from the ANR-DFG project AIFIT (Grant 15-CE40-0019), and the Pacific Institute for Mathematical Sciences, Banff, Canada, for hospitality.

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