Legendre and Chebyshev dual-PetrovGalerkin methods for Hyperbolic equations
A Legendre and Chebyshev dual-PetrovGalerkin method for hyperbolic equations is introduced and analyzed. The dual-PetrovGalerkin 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-PetrovGalerkin 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.
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