Publication: Simultaneous space-time adaptive wavelet solution of nonlinear partial differential equations

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Title	Simultaneous space-time adaptive wavelet solution of nonlinear partial differential equations
Authors/Editors*	J. Alam, N. Kevlahan, O. Vasilyev
Where published*	J. Comput. Phys.
How published*	Journal
Year*	2006
Volume	214
Number	-1
Pages	829-857
Publisher	xxxxx
Keywords
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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.