The Legendre–Burgers equation: When artificial dissipation fails

Artificial viscosity is a common device for stabilizing flows with shocks and fronts. The computational diffusion smears the frontal zone over a small distance μ where μ is chosen so that the discretization has a couple of grid points in the front, and thus is able to resolve the shock. Spectral element methods use a Legendre spectral viscosity whose effect is to damp the coefficient of P n ( x ) by some amount that depends only on the degree n of the Legendre polynomial. Legendre viscosity is better than ordinary diffusion because it does not require spurious boundary conditions, does not increase the temporal stiffness of the differential equations, and can be applied locally on an element-by-element basis. Unfortunately, Legendre diffusion is equivalent to a diffusion with a spatially-varying coefficient that goes to zero at the boundaries. Using the simplest example, one in which the second derivative of Burgers equation is replaced by the Legendre operator to give the “Legendre–Burgers” equation, u t + uu x = ν [(1 − x 2 ) u x ] x , we show that the width of the computational front can similarly tend to zero at the endpoints, causing a numerical catastrophe.