| This work mainly discusses the dissipative spectral element method (DPSEM) and thediscontinuous dissipative spectral element method (DDSEM) for the first order hyperbolicequations, and the application of the DPSEM to the nonlinear advection-dominant difusionequation.Firstly, the dissipative spectral method (DSM) and the DPSEM for the first order hyper-bolic equations are discussed.We construct Fourier DSM and Legendre DSM for the first order linear variable coefcientshyperbolic equations with the periodic and the Dirichlet boundary conditions respectively, andobtain quasi-optimal error estimates in spacial direction for both schemes. Numerical examplesshow some superiorities of the underlying method over the standard spectral method in dealingwith the singularity nearby the boundaries.Taking the DSM as the foundation, we construct the DPSEM for two-dimensional hyper-bolic equation. Convergencies are established for both the semi-discrete and the Crank-Nicolson(CN) fully discrete schemes. Numerical results verify the spectral accuracy of the scheme ap-plying to smooth solution, the validity in dealing with complex boundary conditions, and somesuperiority over the SEM in dealing with the solutions of limit smoothness.Furthermore, we apply the Legendre DPSEM to the nonlinear periodic hyperbolic equa-tion, treating the coefcients with the Chebyshev-Gauss-Labatto collocation method to reducecomputational capacity. Convergencies are established for both the semi-discrete and the Crank-Nicolson/Leap-Frog (CN/LF) fully discrete schemes, and numerical examples are given. Thesmooth example shows that the DPSEM has spectral accuracy for nonlinear problem as well aslinear equations. An example which has limit smoothness in the interval and weak discontinuityat the boundaries shows that, for such a problem, the DPSEM can obtain better results thanthe SEM when taking long time calculation.Secondly, the DDSEM for the first order hyperbolic equations is discussed.To the linear equation, we apply the DDSEM in space discretization. First, the CNscheme is used in time discretization. The stability and the convergence of the CN-DDSEM areestablished. Furthermore, we devise an algorithm by introducing an additional ‘discontinuous’basis, so that the problem in each element can be solved in parallel. What can be seen fromour numerical results is that, the discontinuous mechanism and the dissipative mechanism can localize and homogenize the errors respectively, and the DDSEM summarizes both the virtues.Next, we use the discontiuous Galerkin (DG) scheme in time discretization to improve theaccuracy in temporal direction. Analysis of the stability and convergence of the DG-DDSEMand numerical results are given. In the examples, the CN-DDSEM and the DG-DDSEM arecompared. After that, we discuss the space-time DDSEM, of which the stability and the quasi-optimal error estimate in both spacial and temporal directions are obtained.To the nonlinear equation, we apply the DDSEM with CN/LF time discretization. Anexample is given to compare the DDSEM with the SEM and the DPSEM discussed before: onone hand, it verifies the validity of the DPSEM for non-periodic problems; on the other hand,the results show that, for the solutions having some singularity, the error in the element withinflow boundary in of the DDSEM are smaller than that of the DPSEM, especially smaller thanthe SEM.At last, the DPSEM is used to solve the nonlinear convection-difusion equation, in whichthe coefcients are treated with the Chebyshev-Gauss-Labatto collocation method. Quasi-optimal error estimate in spacial direction of the fully discrete scheme for the two-dimensionalequation in the case of γ≥γ0>0is obtained. Besides, we give out examples of both one dimen-sion and two dimensions. From the numerical results, we find that, when the coefcient γ of thedifusion term is small enough, the dissipative term, which can reduce the error accumulationto a certain extent, makes the DPSEM perform better than the SEM. |
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