In this paper, we investigate the superconvergence property of the local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time dependent fourth order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, $\bar{e}_u$, achieves $(k+\frac32)$-th order superconvergence when polynomials of degree $k$ ($k\ge 1$) are used. Numerical experiments of $P^k$ polynomials, with $1\le k \le3$, are displayed to demonstrate the theoretical results, which show that the error $\bar{e}_u$ actually achieves $(k+2)$-th order superconvergence, indicating that the error bound for $\bar{e}_u$ obtained in this paper is sub-optimal. Initial-boundary value problems, nonlinear equations and solutions having singularities are numerically investigated to verify that the conclusions hold true for very general cases.

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We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier-Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced earlier in the literature, and the natural energy estimate, which is rarely used in the numerical study of this system before.

Jiang et al. (Transportation Research Part B, 2011, 45(2), 343-363) proposed a predictive continuum dynamic user-optimal (PDUO-C) model to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers. The modeled region is a dense urban city, which is arbitrary in shape and has a single central business district (CBD). However, we argue that the model is not well posed due to an inconsistency in the route-choice strategy. To overcome this inconsistency, we revisit the PDUO-C problem, and construct an improved path-choice strategy. The improved model consists of a conservation law to govern the density, in which the flow direction is determined by the improved path-choice strategy, and a Hamilton-Jacobi equation to compute the total travel cost. The simultaneous satisfaction of both equations can be treated as a fixed-point problem. A self-adaptive method of successive averages (MSA) is proposed to solve this fixed-point problem. This method can automatically determine the optimal MSA step size using the least squares approach. Numerical examples are used to demonstrate the effectiveness of the model and the solution algorithm.

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak monotonicity property, which implies that a simple limiter can enforce the bound- preserving property without losing conservation and high order accuracy. Higher order accurate compact finite difference schemes satisfying the weak monotonicity will also be discussed.