Simultaneous space–time adaptive wavelet solution of nonlinear parabolic differential equations

Abstract

Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space-time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space-time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space-time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D+t)-dimensional and (2D+t)-dimensional test problems. In particular, we found that the space-time method uses roughly 18 times fewer space-time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy.Key words: wavelets; lifting scheme; second generation wavelets; partial differential equations; elliptic problem; adaptive grid; numerical method; multilevel method; multigrid method * Corresponding author Email addresses: alamj@math.mcmaster.ca (Jahrul M. Alam), kevlahan@mcmaster.ca (Nicholas K.-R. Kevlahan), Oleg.Vasilyev@Colorado.EDU (Oleg V. Vasilyev).

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