For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax-Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.

We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure, in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

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.

In this paper, we aim to solve one and two dimensional hyperbolic conservation laws on arbitrarily distributed point clouds. The initial condition is given on such a point cloud, and the algorithm solves for point values of the solution at later time also on this point cloud. By using the Voronoi technique and by introducing a grouping algorithm, we divide the computational domain into non-overlapping cells. Each cell is a polygon and contains a minimum number of the given points to ensure accuracy. We carefully select points in each cell during the grouping procedure, and hence are able to interpolate or fit the discrete initial values with piecewise polynomials. By adapting the traditional discontinuous Galerkin method on the constructed polygonal mesh, we obtain a stable, conservative and high order method. Numerical results for both one and two dimensional scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of our mesh generation algorithm as well as the numerical scheme.

The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes, however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric flexibility, suitability for $h$- and $p$-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. In this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivity-preserving limiter in \cite{ZS2} and a new rotational positivity-preserving limiter. This combined limiter is simple to implement and we prove the properties of positivity-preserving and high order accuracy rigorously. One- and two-dimensional numerical results are provided to verify the good properties of the positivity-preserving DG schemes.

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem (IBVP) are treated by the simplified inverse Lax-Wendroff (SILW) procedure. For the fully discrete case, a third order explicit Runge-Kutta method is used as an example for the analysis. Stability is analyzed by both the GKS (Gustafsson, Kreiss and Sundstr\"om) theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

Based on the double shockwave approximation procedure, a local Riemann solver for strongly nonlinear equations of state (EOS) such as the Jones-Wilkins-Lee (JWL) EOS is presented, which has suppressed successfully numerical oscillation caused by high-density ratio and high-pressure ratio across the interface between explosion products and air. The real ghost fluid method (RGFM) and the level set method have been used for converting multi-medium flows into pure flows and for implicitly tracking the interface, respectively. A fifth order finite difference weighted essentially non-oscillatory (WENO) scheme and a third order TVD Runge-Kutta method are utilized for the spatial discretization and the time advance, respectively. An enclosed-type MPI-based parallel methodology for the RGFM procedure on a uniform structured mesh is presented to realize the parallelization of three-dimensional {\color{red}(3D)} air explosion. The overall process of 3D air explosion of both TNT and aluminized explosives has been successfully simulated.

In this paper, we focus on error estimates to smooth solutions of semi-discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu. We show that, with ${P}^k$ (piecewise polynomials of degree $k$) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree $(2k)$, error estimates of $O(h^{k+1/2})$ are obtained for general monotone fluxes, and optimal estimates of $O(h^{k+1})$ are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree $(2k+1)$, error estimates of $O(h^k)$ are obtained for general monotone fluxes, and $O(h^{k+1/2})$ are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis.

In this paper we generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.

Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient.