When discretizing a PDE, one often wishes to discretize continuum objects in such a way that important structures, for example symmetries, are preserved. The construction of hamiltonian or energy preserving pseudospectral discretizations of hamiltonian PDEs has been impeded by the fact that the Chebyshev differentiation matrices are not antisymmetric, unlike the Fourier spectral ones. This adversely impacts on the linear and nonlinear stability of the usual methods. We first address the issue of when it is reasonable to expect antisymmetric matrices. Then, we show that Chebyshev differentiation matrices are, in fact, skew-adjoint with respect to an appropriate inner product, and how they can therefore provide conservative discretizations of some wave equations, with corresponding nonlinear stability and Hamiltonian structure. For more complicated (e.g. Euler) equations, normal pseudospectral methods are not conservative, but with proper tweaks---discretizing the conserved energy and the equation compatibly, and using antialiasing and fast transforms---they can be made to be.

MINISIMPOSIUM SESSION: 10. Symplectic methods