Hermite methods are arbitrary-order polynomial-based general-purpose methods for solving time dependent PDEs. Noteworthy properties of Hermite methods are high order of accuracy in both space and time combined with the ability to march in time with c∆t≤h, for any order of accuracy. The essential description of Hermite methods is as follows:
(i) The degrees of freedom are tensor-product Taylor polynomials at the cell vertices
(ii) The cell polynomial is the Hermite interpolant of the vertex polynomials. This yields a tensor-product polynomial of degree 2m + 1 in each coordinate.
(iii) P is evolved locally at the cell center to produce the required data on the staggered grid.
This talk will present the basic elements of Hemite methods and their application to hyperbolic systems in general and to compressible flows in particular.