Abstract
This paper describes progress on the TLM modelling of the Laplace equation, in particular, how the rate of convergence is influenced by the choice of scattering parameter for a particular discretisation. The hypothesis that optimum convergence is achieved when the real and imaginary parts for the lowest harmonic in a Fourier solution cancel appears to be upheld. The Fourier solution for the problem has been advanced by a better understanding of the nature of the initial excitation. The relationship between the form of the initial condition used in this and many other numerical solutions of the Laplace equation and oscillatory behavior in the results is given a firmer theoretical basis. A correlation between TLM numerical results and those obtained from matrix spectral radius calculations has confirmed much previous work.
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References
Southwell, R.V.: Relaxation Methods in Engineering Science. Oxford University Press, Oxford, UK (1940)
Dusinberre, G.M.: Numerical Analysis of Heat Flow, pp. 121–125. McGraw-Hill, New York (1949)
Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic, New York (1977)
Morton, K.W., Mayers, D.F.: Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge, UK (2005)
Christopoulos, C.: The Transmission Line Modelling Method. IEEE, Piscataway, NJ (1995)
de Cogan, D.: Transmission Line Matrix (TLM) Techniques for Diffusion Applications. Gordon and Breach, Reading, UK (1998)
Smith, A.: Transmission line matrix modelling, optimisation and application to adsorption phenomena. B.Eng. thesis, Nottingham University (1988)
Al-Zeben, M.Y., Saleh, A.H.M., Al-Omar, M.A.: TLM modelling of diffusion, drift and recombination of charge carriers in semiconductors. Int. J. Numer. Model. 5, 219–225 (1992)
de Cogan, D.: The relationship between parabolic and hyperbolic transmission line matrix models for heat-flow. Microelectron. J. 30, 1093–1097 (1999)
Johns, P.B.: A simple, explicit and unconditionally stable routine for the solution of the diffusion equation. Int. J. Numer. Methods Eng. 11, 1307–1328 (1977)
de Cogan, D.: Propagation analysis for thermal modelling. IEEE Trans. Compon. Packaging Manuf. Technol. Part A 21, 418–423 (1998)
de Cogan, D., Chakrabarti, A., Harvey, R.W.: TLM algorithms for Laplace and Poisson fields in semiconductor transport. SPIE Proc. 2373, 198–206 (1995)
de Cogan, D., O’Connor, W.J., Gui, X.: Accelerated convergence in TLM algorithms for the Laplace equation. Int. J. Numer. Methods Eng. 63, 122–138 (2005)
Frankel, S.: Convergence rates of iterative treatments of partial differential equations. Math. Tables Other Aids Comput. 4, 65–75 (1950)
de Cogan, D., O’Connor, W.J., Pulko, S.H.: Transmission Line Matrix (TLM) in Computational Mechanics (See Chapter 2 and 3). CRC, Boca Raton, FL (2006)
Gui, X., de Cogan, D.: Boundary conditions in TLM diffusion modeling. Int. J. Numer. Model. 19, 69–82 (2006)
Flaherty, J.E.: Rensselaer polytechnic institute units CSCI-6840/MATH-6840 “partial differential equations” lecture 9. http://www.cs.rpi.edu/~flaherje/
Flaherty, J.E.: Rensselaer polytechnic institute units CSCI-6840/MATH-6840 “partial differential equations” lecture 4. http://www.cs.rpi.edu/~flaherje/
Chardaire, P., de Cogan, D.: Distribution in TLM models for diffusion (part I: one-dimensional treatment). Int. J. Numer. Model. 15, 317–327 (2002)
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de Cogan, D., Gui, X. & Rak, M. Some Observations on the TLM Numerical Solution of the Laplace Equation. J Math Model Algor 8, 363–385 (2009). https://doi.org/10.1007/s10852-009-9112-6
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DOI: https://doi.org/10.1007/s10852-009-9112-6
Keywords
- Transmission line matrix (TLM) method
- Differential equations
- Laplace equation
- Spectral radius
- Fourier solutions