Hyperviscous shock layers and diffusion zones: Monotonicity, spectral viscosity, and pseudospectral methods for very high order differential equations

Abstract

We solve two problems of x ∈[−∞, ∞] for arbitrary order j . The first is to compute shock-like solutions to the hyperdiffusion equation, u 1=(−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.