Malachowski, Michal J. Transition probability coefficients and the stability of finite difference schemes for the diffusion and telegraphers’ equations. (English) Zbl 0996.65097 Int. J. Numer. Model. 15, No. 3, 243-249 (2002). Summary: A probabilistic approach has been used to analyze the stability of the various finite difference formulations for propagation of signals on a lossy transmission line. If the sign of certain transition probabilities is negative, then the algorithm is found to be unstable. We extend the concept to consider the effects of space and time discretizations on the signs of the coefficients in a probabilistic finite difference implementation of the telegraphers’ equation and draw parallels with the transmission line matrix technique. Cited in 1 Document MSC: 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 35L05 Wave equation Keywords:diffusion equation; stability; finite difference; algorithm; telegraphers’ equation; transmission line matrix PDFBibTeX XMLCite \textit{M. J. Malachowski}, Int. J. Numer. Model. 15, No. 3, 243--249 (2002; Zbl 0996.65097) Full Text: DOI References: [1] Simons, American Journal of Physics 54 pp 1048– (1986) [2] Kaplan, American Journal of Physics 52 pp 267– (1984) [3] Transmission Line Matrix (TLM) Techniques for Diffusion Applications. Gordon and Breach: London, 1998. · Zbl 0929.76003 [4] Taitel, International Journal of Heat and Mass Transfer 15 pp 1369– (1972) [5] Goldstein, Quarterly Journal of Mechanics and Applied Mathematics 1(IV) pp 129– (1951) · Zbl 0045.08102 [6] Liebmann, Journal of Applied Physics 6 pp 129– (1995) [7] Enders, International Journal of Numerical Modelling 6 pp 109– (1993) [8] Johns, International Journal of Numerical Methods in Engineering 11 pp 137– (1977) · Zbl 0364.65103 [9] Kaniadakis, Mathematical Computer Modelling 17 pp 31– (1993) · Zbl 0781.60067 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.