Abstract
We solve two problems ofx∈[−∞, ∞] for arbitrary orderj. The first is to compute shock-like solutions to the hyperdiffusion equation,u1=(−1)j+1 u 2j,x. The second is to compute similar solutions to the stationary form of the hyper-Burgers equation, (−1)j u 2j.x+uu x=0; these tanh-like solutions are asymptotic approximations to the shocks of the corresponding time dependent equation. We solve the hyperdiffusion equation with a Fourier integral and the method of steepest descents. The hyper Burgers equation is solved by a Fourier pseudospectral method with a polynomial subtraction.
Except for the special case of ordinary diffusion (j=1), the jump across the shock zone is described bynonmonotonic, oscillatory functions. By smearing the front over the width of a grid spacing, it is possible to numerically resolve the shock with a weaker and weaker viscosity coefficient asj, the order of the damping, increases. This makes such “hyperviscous” dampings very attractive for coping with fronts since, outside the frontal zone, the impact of the artificial hyperviscosity is much smaller than with ordinary viscosity. Unfortunately, both the intensity of the oscillations and the slowness of their exponential decay from the center of the shock zone decrease asj increases so that the shock zone is muchwider than for ordinary diffusion. We also examined generalizations of Burgers equation with “spectral viscosity”, that is, damping which is tailored to yield exponentially small errors outside the frontal zone when combined with spectral methods. We find behavior similar to high order hyperviscosity.
We conclude that high order damping, as a tool for shock-capturing, offers both advantages and drawbacks. Monotonicity, which has been the holy grail of so much recent algorithm development, is a reasonable goal only for ordinary viscosity. Hyperviscous fronts and shock zones in flows with “spectral viscosity” aresupposed to oscillate.
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Boyd, J.P. Hyperviscous shock layers and diffusion zones: Monotonicity, spectral viscosity, and pseudospectral methods for very high order differential equations. J Sci Comput 9, 81–106 (1994). https://doi.org/10.1007/BF01573179
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DOI: https://doi.org/10.1007/BF01573179