On Advection by Hermite Methods

Ideally, discretization methods for propagating waves should combine both high resolution for smooth solutions and L1-stability for nonsmooth solutions. Hermite methods are high-order polynomialbased spectral element methods for hyperbolic systems with a number of unique properties. A Hermite method whose degrees-of-freedom are the coefficients of a tensor-product Taylor polynomial of degree m in each coordinate has order 2m + 1 convergence for smooth solutions. Here we examine, both experimentally and analytically, the behavior of Hermite methods applied to the advection of nonsmooth solutions. As m is increased we find that the total variation remains bounded, but does not decay rapidly with time. In addition, the observed convergence rates are consistent with the sublinear order 2m+1 2m+2 as predicted by the analysis of the modified equations. Our conclusion is that Hermite methods are effective for advecting waves with sharp fronts and that increasing the polynomial order leads to markedly improved performance.