Comparative Analysis of Parallel Methods for Solving SLAEs in Three-Dimensional Initial-Boundary Value Problems

Abstract

AbstractIterative methods for solving systems of linear algebraic equations with high-order sparse matrices that arise in absolutely stable implicit finite-volume approximations of three-dimensional initial-boundary value problems for the heat and mass transfer equation on unstructured grids in computational domains with a complex configuration of multiply connected piecewise smooth boundary surfaces and contrasting material properties are considered. At each time step, algebraic systems are solved using parallel preconditioned algorithms for conjugate directions in Krylov subspaces. To speed up the iterative processes, variational methods for choosing initial approximations are applied using numerical solutions from previous time steps. It is discussed how the proposed approaches can be more general formulations of problems, as well as how to increase the productivity of computational methods and technologies in the multiple solution of algebraic systems with sequentially determined different right-hand sides and with the scalable parallelization of algorithms based on the additive methods of domain decomposition. The efficiency of the proposed approaches is investigated for the implicit Euler and Crank–Nicholson schemes based on the results of numerical experiments on a representative series of methodological problems.Keywordsinitial-boundary value problemimplicit schemesiterative processesKrylov subspacesleast squares methodnumerical experiments