Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition

• Published in: Journal of scientific computing Vol. 43; no. 2; pp. 135 - 157
• Main Authors: Acebrón, Juan A., Rodríguez-Rozas, Ángel, Spigler, Renato
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.05.2010; Springer Nature B.V
• Subjects: Algorithms; Boundary value problems; Computational Mathematics and Numerical Analysis; Domain decomposition methods; Fault tolerance; Interpolation; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Monte Carlo simulation; Nonlinearity; Parabolic differential equations; Partial differential equations; Probabilistic methods; Probability theory; Stochastic processes; Theoretical; Branching stochastic processes; Monte Carlo methods; Parallel computing; Domain decomposition; Nonlinear parabolic problems; Fault-tolerant algorithms
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-010-9349-2

Initial- and initial-boundary value problems for nonlinear one-dimensional parabolic partial differential equations are solved numerically by a probabilistic domain decomposition method. This is based on a probabilistic representation of solutions by means of branching stochastic processes. Only few values of the solution inside the space-time domain are generated by a Monte Carlo method, and an interpolation is then made so to approximate suitable interfacial values of the solution inside the domain. In this way, a fully decoupled set of sub-problems is obtained. This method allows for an efficient massively parallel implementation, is scalable and fault tolerant. Numerical examples, including some for the KPP equation and beyond are given to show the performance of the algorithm.