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SCHWARZ WAVEFORM RELAXATION ALGORITHM FOR HEAT EQUATIONS WITH DISTRIBUTED DELAY

ABSTRACT
Heat equations with distributed delay are a class of mathematic models that has wide applications in many fields. Numerical computation plays an important role in the investigation of these equations, because the analytic solutions of partial differential equations with time delay are usually unavailable. On the other hand, duo to the delay property, numerical computation of these equations is time-consuming. To reduce the computation time, we analyze in this paper the Schwarz waveform relaxation algorithm with Robin transmission conditions. The Robin transmission conditions contain a free parameter, which has a significant effect on the convergence rate of the Schwarz waveform relaxation algorithm. Determining the Robin parameter is therefore one of the top-priority matters for the study of the Schwarz waveform relaxation algorithm. We provide new formula to fix the Robin parameter and we show numerically that the new Robin parameter is more efficient than the one proposed previously in the literature.
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PAPER SUBMITTED: 2016-02-01
PAPER REVISED: 2016-03-11
PAPER ACCEPTED: 2016-03-22
PUBLISHED ONLINE: 2016-09-24
DOI REFERENCE: https://doi.org/10.2298/TSCI16S3659W
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2016, VOLUME 20, ISSUE Supplement 3, PAGES [S659 - S667]
REFERENCES
  1. Hristov, J., Diffusion Models with Weakly Singular Kernels in the Fading Memories: How the Integral Balance Method Can be Applied? Thermal Science, 19 (2015), 3, pp. 947-957
  2. Metzler, R., Klafter, J., The Random Walk's Guide to Anomalous Diffusion: a Fractional Dynamics Approach, Physics Reports, 339 (2000), 1, pp. 1-77
  3. Bellen, A.,Zennaro, M., Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, U. K., 2003
  4. Liu, C. X., et al., Chaos in Discrete Fractional Cubic Logistic Map and Bifurcation Analysis, Journal of Computational Complexity and Applications, 1 (2015), 2, pp. 105 -111
  5. Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, Mass, USA, 1993
  6. Ruehli, A. E., Johnson, T. A., Circuit Analysis Computing by Waveform Relaxation, Wiley Encyclopedia of Electrical Electronics Engineering, New York, USA, 1999
  7. Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, USA, 1996
  8. Wu, F., Liu, J.-F., Discrete Fractional Creep Model of Salt Rock, Journal of Computational Complexity and Applications, 2 (2016), 1, pp. 1-6
  9. Yang, X. J., Srivastava, N. M., An Asymptotic Perturbation Solution for a Linear Oscillator of Free Damped Vibrations in Fractal Medium Described by Local Fractional Derivatives, Communications in Nonlinear Science and Numerical Simulation, 29 (2015), 1-3, pp.499-504
  10. Yang, X. J., et al., Local Fractional Similarity Solution for the Diffusion Equation Defined on Cantor Sets, Applied Mathematics Letters, 47 (2015), Sept. pp. 54-60
  11. Zubik-Kowal, B., Vandewalle, S., Waveform Relaxation for Functional Differential Equations, SIAM Journal on Scientific Computing, 21 (1999), 1, pp. 207-226
  12. Gander, M. J., Stuart, A. M., Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation, SIAM Journal on Scientific Computing, 19 (1998), 6, pp. 2014-2031
  13. Giladi, E., Keller, N. B., Space-Time Domain Decomposition for Parabolic Problems, Numerische Mathematik, 93 (2002), 2, pp. 279-313
  14. Gander, M. J., Zhao, N., Overlapping Schwarz Waveform Relaxation for the Heat Equation in n-Dimensions, BIT Numerical Mathematics, 42 (2002), 4, pp. 779-795
  15. Gander, M. J., Helpern, L., Optimized Schwarz Wave Form Relaxation (SWR) for the One-Dimensional Heat Equation (in French), Comptes Rendus Mathematique, 336 (2003), 6, pp. 519-524
  16. Bennequin, D., et al., A Homographic Best Approximation Problem with Application to Optimized Schwarz Waveform Relaxation, Mathematics and Computations, 78 (2009), 265, pp. 185-223 Wu, S. L.: Schwarz Waveform Relaxation Algorithm for Heat Equations with … THERMAL SCIENCE, Year 2016, Vol. 20, Suppl. 3, pp. S659-S667 S667
  17. El Bouajaji, M., et al., Optimized Schwarz Methods for the Time-Harmonic Maxwell Equations with Damping, SIAM Journal on Scientific Computing, 34 (2012), 4, pp. A2048-A2071
  18. Dolean, V., et al., Optimized Schwarz Methods for Maxwell's Equations, SIAM Journal on Scientific Computing, 31 (2009), 3, pp. 2193-2213
  19. Gander, M. J., Helpern, L., Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Problems, SIAM Journal on Numerical Analysis, 45 (2007), 2, pp. 666-697
  20. Vandewalle, S., Gander, M. J., Optimized Overlapping Schwarz Methods for Parabolic PDE with Time-Delay, Lecture Notes in Computer Science, 40 (2004), V, pp. 291-298

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