Hyperviscous Shock Layers and Diffusion Zones: Monotonicity, Spectral Viscosity, and Pseudospectral Methods for Very High Order Differential Equations

We solve two problems on x ~ [- or, ~] for arbitrary order j. The first is to compute shock-like solutions to the hyperdiffusion equation, uz = (- 1)J+ ~ u2j, x. The second is to compute similar solutions to the stationary form of the hyper-Burgers equation, ( t) ' u2s, x+ uux = 0: these tanh-like solutions are asymptotic approximations to the shocks of the corresponding time dependent equation. We solve the hyperdifl'usion equation with a Fourier integral and the method of steepest descents. The hyper Burgers equation is solved by a Fourier pseudo-spectral method with a polynomial subtraction. Except for the special case of ordinary diffusion I / = 1), the jump across the shock zone is described by nonmonotonic, 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 as L 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, bot