A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

Juan A Acebrón,Marco A Ribeiro

doi:10.1016/j.jcp.2015.10.027

Abstract

A Monte Carlo algorithm is derived to solve the one-dimensional telegraph equations in a bounded domain subject to resistive and non-resistive boundary conditions. The proposed numerical scheme is more efficient than the classical Kac's theory because it does not require the discretization of time. The algorithm has been validated by comparing the results obtained with theory and the Finite-difference time domain (FDTD) method for a typical two-wire transmission line terminated at both ends with general boundary conditions. We have also tested transmission line heterogeneities to account for wave propagation in multiple media. The algorithm is inherently parallel, since it is based on Monte Carlo simulations, and does not suffer from the numerical dispersion and dissipation issues that arise in finite difference-based numerical schemes on a lossy medium. This allowed us to develop an efficient numerical method, capable of outperforming the classical FDTD method for large scale problems and high frequency signals.

Highlights

Probabilistic methods based on Monte Carlo simulations have been used already to solve problems in Science and Engineering modeled by partial differential equations
In order to asses the validity of our method, a comparison was made solving the problems by the classical Finite-difference time domain (FDTD) method
A new probabilistic method to solve the one-dimensional telegraph equations has been presented. What distinguishes this algorithm is the fact that the classical Kac’s theory, and subsequent works based on this theory, were developed analyzing merely the single second order partial differential equation, while the method presented here is based on the corresponding system of first order differential equations

Summary

Introduction

Probabilistic methods based on Monte Carlo simulations have been used already to solve problems in Science and Engineering modeled by partial differential equations. Unless one is interested to compute the solution at single points, as happens in some specific applications on the analysis of systems and networks, they are typically not competitive enough compared with classical numerical algorithms, when used to compute the solution at every point inside a given computational domain This is basically due to the well-known slow convergence rate of the Monte Carlo method. An alternative consists in combining the Monte Carlo method with other classical techniques, such as the domain decomposition method, computing merely the solution in a few points along some chosen interfaces inside the domain This method is called probabilistic domain decomposition method (PDD), and was successfully used to solve a variety of problems modeled by elliptic, and linear and semilinear parabolic partial differential equations [2,3]

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Conclusion