Preconditioned implicit solution of linear hyperbolic equations with adaptivity

• Published in: Journal of computational and applied mathematics 2004-09, Vol.170 (2), p.269-289
• Main Authors: Lötstedt, Per, Ramage, Alison, von Sydow, Lina, Söderberg, Stefan
• Format: Article
• Language: English
• Subjects: Adaptivity; Exact sciences and technology; Finite volume method; GMRES; Linear multistep method; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Parallel computation; 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; Semi-Toeplitz preconditioning
• ISSN: 0377-0427; 1879-1778; 1879-1778
• DOI: 10.1016/j.cam.2004.01.041

This paper describes a method for solving hyperbolic partial differential equations using an adaptive grid: the spatial derivatives are discretised with a finite volume method on a grid which is structured and partitioned into blocks which may be refined and derefined as the solution evolves. The solution is advanced in time via a backward differentiation formula. The discretisation used is second-order accurate and stable on Cartesian grids. The resulting system of linear equations is solved by GMRES at every time-step with the convergence of the iteration being accelerated by a semi-Toeplitz preconditioner. The efficiency of this preconditioning technique is analysed and numerical experiments are presented which illustrate the behaviour of the method on a parallel computer.