| An analysis of dispersion and dissipation properties of Hermite methods applied to linear hyperbolic equations in one and two dimensional space is discussed. We employed two approaches for this analysis. First, we derive a modified equation to approximate the leading order dissipation errors in the Hermite scheme. Second, we numerically compute the dispersion relation of discrete Fourier modes of the scheme by a modal analysis. Numerical results are presented to illustrate the resolution requirements of Hermite methods. We also discuss an implementation of Hermite methods on overlapping grids. Here we investigate the accuracy of the solution to linear acoustics in a two-dimensional disk. Numerical results show that Hermite methods accurately propagate waves without a need of artificial dissipation on overlapping regions. In support of large-scale computations of compressible flows, we study the stability of Hermite methods for hyperbolic-parabolic equations and also experiment with various methods for treating product of nonlinearities. |
