In this paper, we apply discontinuous Galerkin (DG) methods to solve two model equations: Krause's consensus models and pressureless Euler equations. These two models are used to describe the collisions of particles, and the distributions can be identified as density functions. If the particles are placed at a single point, then the density function turns out to be a $\delta$-function and is difficult to be well approximated numerically. In this paper, we use DG method to approximate such a singularity and demonstrate the good performance of the scheme. Since the density functions are always positive, we apply a positivity-preserving limiter to them. Moreover, for pressureless Euler equations, the velocity satisfies the maximum principle. We also construct special limiters to fulfill this requirement.

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.

In this paper we analyze the explicit Runge-Kutta discontinuous Galerkin (RKDG) methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta (TVDRK3) time discretization and upwinding numerical fluxes. By using the energy method, under a standard CFL condition, we obtain $L^2$ stability for general solutions and a priori error estimates when the solutions are smooth enough. The theoretical results are proved for piecewise polynomials with any degree $k \geq 1$. Finally, since the solutions to this system are non-negative, we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy. Numerical results are provided to demonstrate these RKDG methods.

We use a suite of cosmological hydrodynamic simulations, run by two fixed grid codes, in the context of LCDM model to investigate the properties of solenoidal and dilatational, i.e, curl and compressive, motions of the intergalactic medium (IGM), and the impact of numerical viscosity on turbulence. The codes differ only in the spatial difference discretization, one is a second-order TVD scheme and the other is a fifth-order positivity-preserving WENO scheme. We find that (1) The vortical motion grows rapidly since $z=2$, and reaches $\sim 10 km/s -75 km/s$, depending on the gas density, at low redshifts. The small-scale compressive ratio $r_{CS}$ drops from $0.8$ to $0.5$ during the same period. (2) Power spectra of the solenoidal velocity possess two regimes, $\propto k^{-0.89}$ and $\propto k^{-2.02}$, while the total velocity and dilatational velocity follow the scaling $k^{-1.88}$ and $k^{-2.20}$ respectively all through the turbulent scale $\sim 2-0.2 h^{-1}Mpc$. The IGM turbulence may contain two different phases, the supersonic phase and the post-supersonic phase, simultaneously along the hierarchical structure formation history. (3) The non-thermal pressure support, measured by the vortical kinetic energy, is comparable with the thermal pressure for $\rho_b \sim 10$, or $T \sim 10^5 K$ at $z=0.25$. The deviation of the baryon fraction from the cosmic mean shows a positive correlation with the turbulence pressure support. (4) A relatively higher numerical viscosity would artificially dissipate both the compressive and vortical motions in the IGM more seriously, resulting in less developed density fluctuation and vorticity, and leading to remarkably shortened turbulence scale. Shocks in regions out of the clusters are outstandingly suppressed by the numerical viscosity since $z=2$, which could directly cause the different levels of turbulence in the IGM between the codes. We also discuss the impact of ingredients not included in our simulation.

Discontinuous Galerkin (DG) methods combine features in finite element methods (weak formulation, finite dimensional solution and test function spaces) and in finite volume methods (numerical fluxes, nonlinear limiters) and are particularly suitable for simulating convection dominated problems, such as linear and nonlinear waves including shock waves. In this article we will give a brief survey of DG methods, emphasizing their applications in computational fluid dynamics (CFD). We will discuss essential ingredients and properties of DG methods, and will also give a few examples of recent developments of DG methods which have facilitated their applications in CFD.