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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 in [38] 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 [11] 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.

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do not have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.

In this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [29]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. The modification in this paper improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter.

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is (k+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is (k + 2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for P k polynomials with arbitrary k � 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.

In this paper, we extend the high order finite-difference method with sub- cell resolution (SR) in (34) for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinu- ity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propaga- tion speed of discontinuities in very coarse meshes. AMS subject classifications: 65M06, 76V05, 80A32

In this paper, we survey our recent work on designing high order positivity- preserving well-balanced finite difference and finite volume WENO (weighted essen- tially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condi- tion, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.

Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.

In, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire in to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

In this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.

A conservative modification to the ghost fluid method (GFM) is developed for compressible multiphase flows. The motivation is to eliminate or reduce the conservation error of the GFM without affecting its performance. We track the conservative variables near the material interface and use this information to modify the numerical solution for an interfacing cell when the interface has passed the cell. The modification procedure can be used on the GFM with any base schemes. In this paper we use the fifth order finite difference WENO scheme for the spatial discretization and the third order TVD Runge-Kutta method for the time discretization. The level set method is used to capture the interface. Numerical experiments show that the method is at least mass and momentum conservative and is in general comparable in numerical resolution with the original GFM.

In this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions. The proof of the total variation stability of the schemes for the piecewise constant P0 case is also given. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the LDG method.

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.