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Legendre and Chebyshev dual-Petrov-Galerkin methods for hyperbolic equations. (English) Zbl 1173.65342
Summary: 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.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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