Abstract. A Legendre and Chebyshev dual-Petrov-Galerkin method for
hyperbolic equations is introduced and analyzed. The
dual-Petrov-Galerkin method is based on a natural variational
formulation for hyperbolic equations. Consequently, it enjoys some
advantages which are not available for methods based on other
formulations. More precisely, it is shown that (i) the
dual-Petrov-Galerkin method is always stable without any restriction
on the coefficients; (ii) it leads to sharper error estimates which
are made possible by using the optimal approximation results developed
here with respect to some generalized Jacobi polynomials; (iii) one
can build an optimal preconditioner for an implicit time
discretization of general hyperbolic equations.