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Preconditioned implicit solution of linear hyperbolic equations with adaptivity

Lötstedt, Per and Ramage, Allison and von Sydow, Lina and Söderberg, Stefan (2004) Preconditioned implicit solution of linear hyperbolic equations with adaptivity. Journal of Computational and Applied Mathematics, 194 (2). pp. 269-289. ISSN 0377-0427

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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.