Legendre and Chebyshev dual-Petrov–Galerkin methods for Hyperbolic equations

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.