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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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