A validated parallel across time and space solution of the heat transfer equation

• Published in: Applied numerical mathematics Vol. 31; no. 1; pp. 65 - 79
• Main Author: JEZEQUEL, F
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.1999; Elsevier
• Subjects: CADNA library; CESTAC method; Collocation methods; Computer Science; Exact sciences and technology; Finite difference methods; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical validation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Collocation methods; CESTAC method; Partial differential equations; Numerical validation; CADNA library; Finite difference methods; Validation; Algorithm; Partial differential equation; Equation system; Heat transfer equation; Finite element method; Linear system; Parallel computation; Software; Tridiagonal matrix; Collocation method; Performance; Block matrix; Comparative study
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/S0168-9274(98)00122-6

In this paper, we present a parallel across time and space algorithm, based on an implicit collocation method, for the heat transfer equation. The solution is approximated by polynomials, the coefficients of which are computed from a block-tridiagonal linear system. The optimal degree of the polynomials is the highest degree for which all the coefficients are significant. It is obtained with the use of the CADNA library. If the time step and the space step increase, this optimal degree may increase and the time interval where the solution can be computed in parallel becomes larger. Once the polynomials have been determined for a given time interval, the solution can be accurately computed in parallel at any point of this interval. If the number of points where the solution is computed in the considered time interval is sufficient, it appears that the run-time performances of the implicit collocation method on a parallel machine are better than those of an explicit finite difference method. Numerical experiments on an MIMD architecture are presented.