Preconditioned Implicit Solution of Linear Hyperbolic Equations with Adaptivity

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.