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In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes.
相似文献2.
Waixiang Cao Hailiang Liu Zhimin Zhang 《Numerical Methods for Partial Differential Equations》2017,33(1):290-317
This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one‐dimensional linear convection‐diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k‐th and ‐th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ‐th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017 相似文献
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Waixiang Cao;Chi-Wang Shu;Zhimin Zhang 《Mathematical Modelling and Numerical Analysis》2017,51(6):2213-2235
In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of O (h k +2)k O (h k +3/2)
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Waixiang Cao;Dongfang Li;Yang Yang;Zhimin Zhang 《Mathematical Modelling and Numerical Analysis》2017,51(2):467-486
In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2k + 1) th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree k k + 1k + 2
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We analyze finite volume schemes of arbitrary order r for the one-dimensional singu- larly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as （N-11n（N 4- 1））r, where 2N is the number of subinter- vals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order （N-11n（N ＋ 1））2r, while at the Gauss points, the derivative error super-converges with order （N-11n（N ＋ 1））r＋1. All the above conver- gence and superconvergence properties are independent of the perturbation parameter e. Numerical results are presented to support our theoretical findings. 相似文献
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In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions. 相似文献
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In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. Two classes of SV methods are constructed by letting a piecewise $k$-th order ($k ≥ 1$ is an integer) polynomial tosatisfy the conservation law in each control volume, which is obtained by refining spectral volumes (SV) of the underlying mesh with $k$ Gauss-Legendre points (LSV) or Radauspoints (RSV) in each SV. The $L^2$-norm stability and optimal order convergence propertiesfor both methods are rigorously proved for general non-uniform meshes. Surprisingly, wediscover some very interesting superconvergence phenomena: At some special points, theSV flux function approximates the exact flux with $(k+2)$-th order and the SV solution itselfapproximates the exact solution with $(k+3/2)$-th order, some superconvergence behaviorsfor element averages errors have been also discovered. Moreover, these superconvergencephenomena are rigorously proved by using the so-called correction function method. Ourtheoretical findings are verified by several numerical experiments. 相似文献
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