We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$-th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.

In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new {\em a posteriori} error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.

In this article, we give a brief overview on high order accurate shock capturing schemes with the aim of applications in compressible turbulence simulations. The emphasis is on the basic methodology and recent algorithm developments for two classes of high order methods: the weighted essentially non-oscillatory (WENO) and discontinuous Galerkin (DG) methods.

In this paper, we investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving conservation laws, with the goal of obtaining a robust and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The idea of this limiter is to reconstruct the entire polynomial, instead of reconstructing point values or moments in the classical WENO reconstructions. That is, the reconstruction polynomial on the target cell is a convex combination of polynomials on this cell and its neighboring cells and the nonlinear weights of the convex combination follow the classical WENO procedure. The main advantage of this limiter is its simplicity in implementation, especially for multi-dimensional meshes. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure.

The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions. The main purpose of the well-balanced schemes designed in this paper is to well resolve small perturbations of the hydrostatic balance state on coarse meshes. The more difficult issue of convergence towards such hydrostatic balance state from an arbitrary initial condition is not addressed in this paper.

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22-44 and Zhang S., Jiang S and Shu C.-W. (2008), J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273-305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216-238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws, we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection-diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in earlier work, we extend the schemes to two-dimensional convection-diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier-Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

In this paper, we develop a three-dimensional parallel solver using the fifth order high-resolution weighted essentially non-oscillatory (WENO) finite difference scheme to perform extensive simulation for three-dimensional gaseousdetonations. A careful study is conducted for the propagation modes of three-dimensional gaseous detonation wave-front structures in a long square duct. The numerical results indicate that, the instability of detonation, overdrive factor, transverse dimension and initial perturbation are the main factors that influence the appearance of spinning detonation.

In this paper, we provide a priori $L^2$ error estimates for the semi-discrete discontinuous Galerkin method and the local discontinuous Galerkin method for one- and two-dimensional nonlinear Hamilton-Jacobi equations with smooth solutions. With a special Gauss-Radau projection, the optimal error estimates on rectangular meshes are obtained.

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $\delta$-singularities. Negative-order norm error estimates for the accuracy of DG approximations to $\delta$-singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $k$-th degree polynomials, at time $t$, the error in the $H^{-(k+2)}$ norm over the whole domain is $(k+1/2)$-th order, and the error in the $H^{-(k+1)}(\mathbb{R}\backslash\mathcal{R}_t)$ norm is $(2k+1)$-th order, where $\mathcal{R}_t$ is the pollution region due to the initial singularity with the width of order $\mathcal{O}(h^{1/2} \log (1/h))$ and $h$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $L^2$ error estimate of $(2k+1)$-th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel's principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $\delta$-singularities.

We develop an alternative formulation of conservative finite difference weighted essentially non-oscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax-Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar and system conservation laws demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

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In this work a simple method to enforce the positivity-preserving property for general high-order conservative schemes is proposed. The method detects critical numerical fluxes which may lead to negative density and pressure, and then imposes a simple flux limiter combining the high-order numerical flux with the first-order Lax-Friedrichs flux to satisfy a sufficient condition for preserving positivity. Though an extra time-step size condition is required to maintain the formal order of accuracy, it is less restrictive than those in previous works. A number of numerical examples suggest that this method, when applied on an essentially non-oscillatory scheme, can be used to prevent positivity failure when the flow involves vacuum or near vacuum and very strong discontinuities.

Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.

In this paper, we design a new class of central compact schemes based on the cell-centered compact schemes of Lele {[}S.K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992) 16-42{]}. These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the function on both the grid points (cell boundaries) and the cell-centers. In our approach, instead of using a compact interpolation to compute the values on cell-centers, the physical values on these half grid points are stored as independent variables and updated using the same scheme as the physical values on the grid points. This approach increases the memory requirement but not the computational costs. Through systematic Fourier analysis and numerical tests, we observe that the schemes have excellent properties of high order, high resolution and low dissipation. It is an ideal class of schemes for the simulation of multi-scale problems such as aeroacoustics and turbulence.

In this paper we generalize a new type of limiters based on the weighted essentially non-oscillatory (WENO) finite volume methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the entire polynomials of the DG solutions from the troubled cell and its immediate neighboring cells, and then apply the classical WENO procedure to form a convex combination of these polynomials based on smoothness indicators and nonlinear weights, with suitable adjustments to guarantee conservation. The main advantage of this new limiter is its simplicity in implementation, especially for the unstructured meshes considered in this paper, as only information from immediate neighbors is needed and the usage of complicated geometric information of the meshes is largely avoided. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good performance of this procedure.

Two-dimensional interaction between two Taylor vortices is simulated systematically through solving the two-dimensional, unsteady compressible Navier-Stokes equations using a fifth order weighted essentially nonoscillatory finite difference scheme. The main purpose of this study is to reveal the mechanism of sound generation in two-dimensional interaction of two Taylor vortices. Based on an extensive parameter study on the evolution of the vorticity field, we classify the interaction of two Taylor vortices into four types. The first type is the interaction of two counter-rotating vortices with similar strengths. The second type is the interaction of two co-rotating vortices without merging. The third type is the merging of two co-rotating vortices. The fourth type is the interaction of two vortices with a large difference in their strengths or scales. The mechanism of sound generation is analyzed.

Homotopy continuation is an efficient tool for solving polynomial systems. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this article, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. A third-order accurate finite difference weighted essentially non-oscillatory (WENO) scheme with Lax-Friedrichs flux splitting is utilized to derive the difference equation. This new approach is free of the CFL condition constraint. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency and robustness of the new method.

We investigate the angular distribution of Ly$\alpha$ photons transferring in or emergent from an optically thick medium. Since the evolutions of specific intensity $I$ in the frequency space and the angular space are coupled with each other, we first develop the WENO numerical solver in order to find the time-dependent solutions of the integro-differential equation of $I$ in frequency and angular space simultaneously. We first show that the solutions with the Eddington approximation, which assume $I$ to be linearly dependent on the angular variable $\mu$, yield similar frequency profiles of the photon flux as that without the Eddington approximation. However, the solutions of the $\mu$ distribution evolution are significantly different from that given by Eddington approximation. First, the angular distribution of $I$ are found to be substantially dependent on the frequency of photons. For photons with the resonant frequency $\nu_0$, $I$ contains only a linear term of $\mu$. For photons with frequency at the double peaks of the flux, the $\mu$-distribution is highly anisotropic, in which most photons are in the direction of radial forward. Moreover, either at $\nu_0$ or at the double peaks, the $\mu$ distributions actually are independent of the initial $\mu$ distribution of photons of the source. This is because the photons with frequency either of $\nu_0$ or of the double peaks undergo the process of forgetting their initial conditions due to the resonant scattering. We also show that the optically thick medium is a collimator of photons at the double peaks. Photons from the double peaks form a forward beam with very small spread angle.

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.