In this paper, we analyze the Lax-Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al., where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax-Wendroff time discretization. We show that, under the standard CFL condition $\tau\leq \lambda h$ (where $\tau$ and $h$ are the time step and the maximum element length respectively and $\lambda>0$ is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the $L^2$ norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution $u$ and its first order time derivative $u_t$ in one dimension, and numerical examples are given to validate our analysis.

This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for 2-D linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the ($2k+1$)-th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree $k$ are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of ($k+1$)-th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of ($k+2$)-th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.

The numerical solution of the one-dimensional nonlinear Klein-Gordon equation on an unbounded domain is studied in this paper. Split local absorbing boundary (SLAB) conditions are obtained by the operator splitting method, then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to show the advantages and effectiveness of the given method, and some interesting collision behaviors are also observed.

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