Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

• Published in: Journal of computational physics Vol. 228; no. 15; pp. 5574 - 5591
• Main Authors: Acebrón, Juan A., Rodríguez-Rozas, Ángel, Spigler, Renato
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.08.2009; Elsevier
• Subjects: ALGORITHMS; APPROXIMATIONS; Computational techniques; Domain decomposition; Exact sciences and technology; Fault tolerant algorithms; FUNCTIONALS; INTERPOLATION; MATHEMATICAL METHODS AND COMPUTING; Mathematical methods in physics; MONTE CARLO METHOD; Monte Carlo methods; Nonlinear parabolic problems; NONLINEAR PROBLEMS; Parallel computing; PARTIAL DIFFERENTIAL EQUATIONS; Physics; PROBABILISTIC ESTIMATION; Random trees; RANDOMNESS; SPACE-TIME; STOCHASTIC PROCESSES; TWO-DIMENSIONAL CALCULATIONS; Monte Carlo methods; Parallel computing; 65C30; Fault tolerant algorithms; Nonlinear parabolic problems; 65C05; 65N55; Domain decomposition; Random trees; Partial differential equations; Stochastic processes; Decomposition method; Nonlinear problems; Calculation methods; Space-time; Interpolation; Parabolic equations; Algorithms; Functionals; Numerical solution; Calculation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.04.034

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.