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Theoretical and numerical studies of the \(P_NP_M\) DG schemes in one space dimension. (English) Zbl 1524.65511

The authors prove the existence of a solution of reconstruction operators used in the \(P_NP_M\) DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. They further study the stability of these \(P_NP_M\) DG schemes when applying to the linear advection equation. They computationally explore maximal limits of the Courant numbers for the \(P_NP_M\) DG schemes by applying the von Neumann analysis and using an experimental procedure. A numerical study explains how the stencils used in the reconstruction affect the efficiency of the schemes.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques

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