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The weighted essentially nonoscillatory (WENO) schemes are popular high-order numerical methods for hyperbolic conservation laws. When dealing with hyperbolic systems, WENO schemes are usually used in cooperation with the local characteristic decomposition, as the componentwise WENO reconstruction/interpolation procedure often produces oscillatory approximations near shocks. In this paper, we investigate local characteristic decomposition-free WENO schemes for a special class of hyperbolic systems endowed with a coordinate system of Riemann invariants. We apply the WENO procedure to the coordinate system of Riemann invariants instead of the local characteristic fields to save the expensive computational cost on local characteristic decomposition but meanwhile maintain the essentially nonoscillatory performance. Due to the nonlinear algebraic relation between the Riemann invariants and conserved variables, it is difficult to obtain the cell averages of Riemann invariants directly from those of conserved variables and vice versa; thus, we do not use the finite volume WENO schemes in this work. The same difficulty is also faced in the traditional Shu-Osher lemma [C.-W. Shu and S. Osher, J. Comput. Phys., 83 (1989), pp. 32-78]-based finite difference schemes, as the computation of fluxes is based on reconstruction as well. Therefore, we adopt the alternative formulation of the finite difference WENO scheme [Y. Jiang, C.-W. Shu, and M. Zhang, SIAM J. Sci. Comput., 35 (2013), pp. A1137-A1160, C.-W. Shu and S. Osher, J. Comput. Phys., 77 (1988), pp. 439-471] in this paper, which is based on interpolation for nodal values. The efficiency and good performance of our method are demonstrated by extensive numerical tests which indicate that the coordinate system of Riemann invariants is a good alternative of local characteristic fields for the WENO procedure. [ABSTRACT FROM AUTHOR]

Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one from among the infinite many weak solutions. Recently, several high order discontinuous Galerkin (DG) methods satisfying entropy inequalities were proposed; see [T. Chen and C.-W. Shu, J. Comput. Phys., 345 (2017), pp. 427-461; J. Chan, J. Comput. Phys., 362 (2018), pp. 346-374; T. Chen and C.-W. Shu, CSIAM Trans. Appl. Math., 1 (2020), pp. 1-52] and the references therein. However, high order numerical methods typically generate spurious oscillations in the presence of shock discontinuities. In this paper, we construct a high order entropy stable essentially oscillation-free DG (OFDG) method for hyperbolic conservation laws. With some suitable modification on the high order damping term introduced in [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299-1324; Y. Liu, J. Lu, and C.-W. Shu, SIAM J. Sci. Comput., 44 (2022), pp. A230-A259], we are able to construct an OFDG scheme with dissipative entropy. It is challenging to make the damping term compatible with the current entropy stable DG framework, that is, the damping term should be dissipative for any given entropy function without compromising high order accuracy. The key ingredient is to utilize the convexity of the entropy function and the orthogonality of the projection. Then the proposed method maintains the same properties of conservation, error estimates, and entropy dissipation as the original entropy stable DG method. Extensive numerical experiments are presented to validate the theoretical findings and the capability of controlling spurious oscillations of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and has been actively studied in recent years. This is, however, still a challenging task for many systems, especially those involving nonlinear constraints. Based on some key insights from geometry, we systematically propose an innovative and general framework, referred to as geometric quasilinearization (GQL), which paves a new effective way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transform all nonlinear constraints to linear ones, by properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, using diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches. [ABSTRACT FROM AUTHOR]

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124-2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov's modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta. [ABSTRACT FROM AUTHOR]

In this paper we shall establish an a priori L²-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of onedimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant--Friedrichs--Lewy condition, and the quasi-optimal and/or optimal order of accuracy in space is achieved for many widely used numerical fluxes, no matter whether the solution contains sonic points or not. The main tools used in this paper are the matrix transferring process and the generalized Gauss--Radau projection of the reference functions, depending on the relative upwind effect. Finally some numerical experiments are given to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop an essentially oscillation-free discontinuous Galerkin method for systems of hyperbolic conservation laws. Based on the standard discontinuous Galerkin method, the numerical damping terms are introduced so as to control the spurious oscillations, similar to the scalar case [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299--1324]. We use both the classical Runge--Kutta method and the modified exponential Runge--Kutta method in time discretization. Particularly, the latter one could avoid additional restrictions of time step size due to the numerical damping. Extensive numerical experiments are shown to demonstrate our algorithm is robust and effective. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numerical schemes would generate spurious oscillations when the solution of the hyperbolic conservation laws contains discontinuities. The spurious oscillations may be harmful to the numerical simulation, as it not only generates some artificial structures not belonging to the problems, but also causes many overshoots and undershoots that make the numerical scheme less robust. To overcome this difficulty, in this paper we introduce a numerical damping term to control spurious oscillations based on the classic DG formulation. In comparison to the classic DG method, the proposed DG method still maintains many good properties, such as the extremely local data structure, conservation, L²-boundedness, optimal error estimates, and superconvergence. We also provide some numerical examples to show the good performance of the proposed DG scheme and verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this article a modification of the algorithm for data discretized in the point values introduced in [S. Amat, J. Ruiz, and C.-W. Shu, Appl. Math. Lett., 105 (2020), pp. 106-298] is presented. In the aforementioned work, the authors managed to obtain an algorithm that reaches a progressive and optimal order of accuracy close to discontinuities for WENO-6. For higher orders, i.e., WENO-8, WENO-10, etc., it turns out that the previous algorithm presents some shadows in the detection of discontinuities, meaning that the order of accuracy is better than the one attained by WENO of the same order, but not optimal. In this article a modification of the smoothness indicators used in the original algorithm is presented. It is oriented to solve this problem and to attain a WENO-2r algorithm with progressive order of accuracy close to the discontinuities. We also show proofs for the accuracy and explicit formulas for all the weights used for any order 2r of the algorithm. [ABSTRACT FROM AUTHOR]

The meaning of weak solutions of a nonconservative hyperbolic system with discontinuous coefficients modeling traffic flows involving hysteresis is defined. An upwinding approximation scheme for the model is shown to be total variation diminishing. Vehicles' speeds and drivers' hysteresis states satisfy maximum and minimum principles. The limit of a convergent subsequence generated by the approximation scheme is shown to be a weak solution of the model if it is piecewise C1, establishing the existence of such solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the Runge-Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge- Kutta time-marching is used. By the aid of the equivalent evolution representation with temporal differences of stage solutions, we make a detailed investigation on the matrix transferring process about the energy equations and then present a sufficient condition to ensure the L²-norm stability under the standard Courant-Friedrichs-Lewy condition. If the source term is equal to zero, we achieve the strong (boundedness) stability without the matrix transferring process to multiple-steps time-marching of the RKDG method. By carefully introducing the reference functions and their projections, we obtain the optimal (or suboptimal) error estimate under a mild smoothness assumption on the exact solution, which is independent of the stage number of the RKDG method. Some numerical experiments are also given to verify our conclusions. [ABSTRACT FROM AUTHOR]

In this paper we present a discontinuous Galerkin (DG) method to approximate stochastic conservation laws, which is an efficient high-order scheme. We study the stability for the semidiscrete DG methods for fully nonlinear stochastic equations. Error estimates are obtained for smooth solutions of semilinear stochastic equations with variable coefficients. We also estab- lish a derivative-free second-order time discretization scheme for matrix-valued stochastic ordinary differential equations. Numerical experiments are performed to confirm the analytical results. [ABSTRACT FROM AUTHOR]

Well-balanced and free energy dissipative first- and second-order accurate finitevolume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The variation of the natural Lyapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analogous property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. Performing a careful validation in a battery of relevant test cases, we show that these schemes can accurately analyze the stability properties of stationary states in challenging problems such as phase transitions in collective behavior, generalized Euler--Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories. [ABSTRACT FROM AUTHOR]

In this paper we propose a simple and unified framework to investigate the L2 -norm stability of the explicit Runge--Kutta discontinuous Galerkin (RKDG) methods when solving the linear constant-coefficient hyperbolic equations. Two key ingredients in the energy analysis are the temporal differences of numerical solutions in different Runge--Kutta stages and a matrix transferring process. Many popular schemes, including the fourth order RKDG schemes, are discussed in this paper to show that the presented technique is flexible and useful. Different performances in the L2 - norm stability of different RKDG schemes are carefully investigated. For some lower-degree piecewise polynomials, the monotonicity stability is proved if the stability mechanism can be provided by the upwind-biased numerical fluxes. Some numerical examples are also given. [ABSTRACT FROM AUTHOR]

Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for seminegative autonomous linear systems. The analysis is based on the energy method and can be performed with the aid of a computer. Strong stability of various RK methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under this framework. Based on numerous numerical observations, we further characterize the features of strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of RK methods of odd linear order. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of new nonlinear optimal weights for weighted ENO (WENO) interpolation capable of raising the order of accuracy close to discontinuities. The new nonlinear optimal weights are constructed using a strategy inspired by the original WENO algorithm, and they work very well for corner or jump singularities, leading to optimal theoretical accuracy. This is the first part of a series of two papers. In this first part we analyze the performance of the new algorithms proposed for univariate function approximation in the point values (interpolation problem). In the second part, we will extend the analysis to univariate function approximation in the cell averages (reconstruction problem). Our aim is twofold: to raise the order of accuracy of the WENO type interpolation schemes both near discontinuities and in the interval which contains the singularity. The first problem can be solved using the new nonlinear optimal weights, but the second one requires a new strategy that locates the position of the singularity inside the cell in order to attain adaption. This new strategy is inspired by the ENO-SR schemes proposed by Harten [J. Comput. Phys., 83 (1989), pp. 148-184]. Thus, we will introduce two different algorithms in the point values. The first one can deal with corner singularities and jump discontinuities for intervals not containing the singularity. The second algorithm can also deal with intervals containing corner singularities, as they can be detected from the point values, but jump discontinuities cannot, as the information of their position is lost during the discretization process. As mentioned before, the second part of this work will be devoted to the cell averages and, in this context, it will be possible to work with jump discontinuities as well. [ABSTRACT FROM AUTHOR]

The article focuses on research reproducibility in scientific computing journals, Journal on Scientific Computing (SISC), Journal of Scientific Computing (JSC), and Journal of Computational Physics (JCP), by requiring authors to publicly share their data and credibility in scientific research.

The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desirable but remains a challenge, especially in the multidimensional cases. In this paper, we develop uniformly high-order discontinuous Galerkin (DG) schemes which provably preserve the positivity of density and pressure for multidimensional ideal MHD. The schemes are constructed by using the locally divergence-free DG schemes for the symmetrizable ideal MHD equations as the base schemes, a PP limiter to enforce the positivity of the DG solutions, and the strong stability preserving methods for time discretization. The significant innovation is that we discover and rigorously prove the PP property of the proposed DG schemes by using a novel equivalent form of the admissible state set and very technical estimates. Several two-dimensional numerical examples further confirm the PP property and demonstrate the accuracy, effectiveness, and robustness of the proposed PP methods. [ABSTRACT FROM AUTHOR]

Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been extensively studied in the last several years, but nearly all the developed schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the con- straint for the Courant-Friedrichs-Lewy (CFL) number. This makes explicit methods impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this paper we develop a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. We focus on one spatial dimension. However, the result easily generalizes to multidimensional tensor product meshes and polynomial spaces. This work is based on a generalization of the positivity-preserving limiters in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] and [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 8918-8934] to implicit time discretizations. Both the analysis and numerical experiments indicate that a lower bound for the CFL number is required to obtain the positivity-preserving property. The proposed scheme not only preserves the positivity of the numerical approximation without compromising the designed high-order accuracy, but also helps accelerate the convergence towards the steady-state solution and adds robustness to the nonlinear solver. Numerical experiments are provided to support these conclusions. [ABSTRACT FROM AUTHOR]

We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori L² error estimates of order k + 1 are obtained for the semidiscrete scheme when piecewise polynomials of degree at most k (k ≥ 0) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze the Runge-Kutta discontinuous Galerkin (RKDG) method to solve weakly coupled hyperbolic multidomain problems. Such problems involve transfer type boundary conditions with discontinuous fluxes between different domains, calling for special techniques to prove stability of the RKDG methods. We prove both stability and error estimates for our RKDG methods on simple models, and then apply them to a biological cell proliferation model [N. Echenim, D. Monniaux, M. Sorine, and F. Clément, Math. Biosci., 198 (2005), pp. 57-79]. Numerical results are provided to illustrate the good behavior of our RKDG methods. [ABSTRACT FROM AUTHOR]

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 nonoverlapping 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. [ABSTRACT FROM AUTHOR]

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 exibility, 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 [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 8918-8934] 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. [ABSTRACT FROM AUTHOR]

This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k + 1)th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k + 1)th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k+2)th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge-Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems in the sense that the time-step t is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent of the mesh-size h, even though the advection term is treated explicitly. The last is that under this time step condition, we obtain optimal error estimates in both space and time for the third order IMEX Runge-Kutta time-marching coupled with LDG spatial discretization. Numerical experiments are also given to verify the main results. [ABSTRACT FROM AUTHOR]

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 kth 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 a posteriori error estimate. Our analysis is valid for arbitrary regular meshes and for Pk polynomials with arbitrary k ≥ 1, 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. [ABSTRACT FROM AUTHOR]

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 (k + 3/2)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 ≥ 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. [ABSTRACT FROM AUTHOR]

To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] to convection diffusion equations, we propose a nonconventional high order finite volume weighted essentially nonoscil-latory (WENO) scheme which can be proved maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported. [ABSTRACT FROM AUTHOR]

In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, J. Comput. Phys., 227 (2008) pp. 8191-8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L¹ norm and a very fast convergence speed, but only first order accuracy in the L∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples. [ABSTRACT FROM AUTHOR]

High order accurate weighted essentially nonoscillatory (WENO) schemes are usually designed to solve hyperbolic conservation laws or to discretize the first derivative convection terms in convection dominated partial differential equations. In this paper we discuss a high order WENO finite difference discretization for nonlinear degenerate parabolic equations which may contain discontinuous solutions. A porous medium equation (PME) is used as all example to demonstrate the algorithm structure and performance. By directly approximating the second derivative term using a conservative flux difference, the sixth order and eighth order finite difference WENO schemes are constructed. Numerical examples are provided to demonstrate the accuracy and nonoscillatory performance of these schemes. [ABSTRACT FROM AUTHOR]

We propose a new discontinuous Galerkin method based on [Y. Cheng and C.-W. Shu, J. Comput. Phys., 223 (2007), pp. 398-415] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with nonisotropic dynamics. Several numerical experiments show the relevance of our method, in particular, for front propagation. [ABSTRACT FROM AUTHOR]

It is well known that finite difference or finite volume total variation diminishing (TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy at smooth extrema [S. Osher and S. Chakravarthy, SIAM J. Numer. Anal., 21 (1984), pp. 955- 984], thus TVD schemes are at most second order accurate in the L1 norm for general smooth and nonmonotone solutions. However, Sanders [Math. Comp., 51 (1988), pp. 535-558] introduced a third order accurate finite volume scheme which is TVD, where the total variation is defined by measuring the variation of the reconstructed polynomials rather than the traditional way of measuring the variation of the grid values. By adopting the definition of the total variation for the numerical solutions as in [R. Sanders, Math. Comp., 51 (1988), pp. 535-558], it is possible to design genuinely high order accurate TVD schemes. In this paper, we construct a finite volume scheme which is TVD in this sense with high order accuracy (up to sixth order) in the L1 norm. Numerical tests for a fifth order accurate TVD scheme will be reported, which include test cases from traffic flow models. [ABSTRACT FROM AUTHOR]

We analyze a local discontinuous Galerkin method for fourth-order time-dependent problems. Optimal error estimates are obtained in one dimension and in multidimensions for Cartesian and triangular meshes. We extend the analysis to higher even-order equations and the linearized Cahn-Hilliard type equations. Numerical experiments are displayed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Hunter-Saxton (HS) equation and its zero-viscosity and zero-dispersion limits. The energy stability for general solutions are proved, and numerical simulation results for different types of solutions of the nonlinear HS equation are provided to illustrate the accuracy and capability of the LDG method. The zero-viscosity and zero-dispersion properties of the HS equation are studied in a numerical simulation. [ABSTRACT FROM AUTHOR]

High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the issue of convergence toward entropy solutions for the high order finite volume weighted essentially nonoscillatory (WENO) scheme and the discontinuous Galerkin (DG) finite element method approximating scalar nonconvex conservation laws. Although such high order nonlinearly stable schemes can usually converge to entropy solutions of convex conservation laws, convergence may fail for certain nonconvex conservation laws. We perform a detailed study to demonstrate such convergence issues for a few representative examples and suggest a modification of the high order schemes based on either first order monotone schemes or a second order entropic projection [Bouchut, Bourdarias, and Perthame, Math. Comp., 65 (1996), pp. 1438-1461] to achieve convergence toward entropy solutions while maintaining high order accuracy in smooth regions. [ABSTRACT FROM AUTHOR]

In this paper, we consider convergence of classical high-order Godunov-type schemes towards entropy solutions for scalar conservation laws. It is well known that sufficient conditions for such convergence include total variation boundedness of the reconstruction and cell or wavewise entropy inequalities. We prove that under large time steps, we only need total variation boundedness of the reconstruction to guarantee such convergence. We discuss high-order total variation bounded reconstructions to fulfill this sufficient condition and provide numerical examples on one-dimensional convex conservation laws to assess the performance of such large time step Godunov-type methods. To demonstrate the generality of this approach, we also prove convergence and give numerical examples for a large time step Godunov-like scheme involving Sanders' third-order total variation diminishing reconstruction using both cell averages and point values at cell boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Camassa-Holm equation which contains nonlinear high-order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for different types of solutions of the nonlinear Camassa-Holm equation to illustrate the accuracy and capability of the LDG method. [ABSTRACT FROM AUTHOR]

We present a multiscale model for numerical simulation of dynamics of crystalline solids. The method couples nonlinear elastodynamics as the continuum description and molecular dynamics as another component at the atomic scale. The governing equations on the macroscale are solved by the discontinuous Galerkin method, which is built up with an appropriate local curl-free space to produce a coherent displacement field. The constitutive data are based on the underlying atomistic model: it is either calibrated prior to the computation or obtained from molecular dynamics as the computation proceeds. The decision to use either the former or the latter is made locally for each cell based on suitable criteria. [ABSTRACT FROM AUTHOR]

A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Süli [Proceedings of the International Symposium on Discontinuous Galerkin Methods, Springer, New York, pp. 291-300; Math. Comput., 72 (2003), pp. 577-606]. The postprocessor allows improvement in accuracy of the discontinuous Galerkin method for time-dependent linear hyperbolic equations from order k+1 to order 2k+1 over a uniform mesh. Assumptions on the convolution kernel along with uniformity in mesh size give a local translation invariant postprocessor that allows for simple implementation using small matrix-vector multiplications. In this paper, we present two alternatives for extending this postprocessing technique to include smoothly varying meshes. The first method uses a simple local L²-projection of the smoothly varying mesh to a locally uniform mesh and uses this projected solution to compute the postprocessed solution. By using this local L²-projection, recalculating the convolution kernel for every element can be avoided, and 2k+1 order accuracy of the postprocessed solution can be achieved. The second method uses the idea of characteristic length based upon the largest element size for the scaling of the postprocessing kernel. These two methods, local projection and characteristic length, are also applied to approximations over a mesh with elements that vary in size randomly. We discuss the computational issues in using these two techniques and demonstrate numerically that we obtain the 2k+1 order of accuracy for the smoothly varying meshes, and that although the 2k+1 order of accuracy is not fully realized for random meshes, there is significant improvement in the L²-errors. [ABSTRACT FROM AUTHOR]

The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408-463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241-282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82-104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199-224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255-283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks. [ABSTRACT FROM AUTHOR]

In this paper we discuss two explicit finite difference schemes, namely a first order upwind scheme and a second order high resolution scheme, for solving a hierarchical size-structured population model with nonlinear growth, mortality, and reproduction rates. We prove stability and convergence for both schemes and provide numerical examples to demonstrate their capability in solving smooth and discontinuous solutions. [ABSTRACT FROM AUTHOR]

In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge-Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second-order explicit TVD (total variation diminishing) Runge-Kutta method, and the ℙκ (piecewise polynomial) finite element is used. When κ = 1 (piecewise linear finite element), the error estimate is obtained under the usual CFL condition τ ≤ βh for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for ℙκ finite elements with κ > 1 are obtained under a more restrictive CFL condition. [ABSTRACT FROM AUTHOR]

In [SIAM J. Sci. Comput., 26 (2005), pp. 907--929], we initiated the study of using WENO (weighted essentially nonoscillatory) methodology as limiters for the RKDG (Runge--Kutta discontinuous Galerkin) methods. The idea is to first identify "troubled cells," namely, those cells where limiting might be needed, then to abandon all moments in those cells except the cell averages and reconstruct those moments from the information of neighboring cells using a WENO methodology. This technique works quite well in our one- and two-dimensional test problems [SIAM J. Sci. Comput., 26 (2005), pp. 907--929] and in the follow-up work where more compact Hermite WENO methodology is used in the troubled cells. In these works we used the classical minmod-type TVB (total variation bounded) limiters to identify the troubled cells; that is, whenever the minmod limiter attempts to change the slope, the cell is declared to be a troubled cell. This troubled-cell indicator has a TVB parameter $M$ to tune and may identify more cells than necessary as troubled cells when $M$ is not chosen adequately, making the method costlier than necessary. In this paper we systematically investigate and compare a few different limiter strategies as troubled-cell indicators with an objective of obtaining the most efficient and reliable troubled-cell indicators to save computational cost. [ABSTRACT FROM AUTHOR]

Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first- and second-order well-balanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a high-order finite difference weighted essentially nonoscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme to exactly preserve steady states and to retain high-order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. [ABSTRACT FROM AUTHOR]

In this paper we develop a discontinuous Galerkin (DG) method, within the framework of the heterogeneous multiscale method (HMM), for solving hyperbolic and parabolic multiscale problems. Hyperbolic scalar equations and systems, as well as parabolic scalar problems, are considered. Error estimates are given for the linear equations, and numerical results are provided for the linear and nonlinear problems to demonstrate the capability of the method. [ABSTRACT FROM AUTHOR]

The Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws. It uses ideas from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, total variation diminishing (TVD) Runge-Kutta time discretizations, and limiters. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation, and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially nonoscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper we investigate using WENO finite volume methodology as limiters for RKDG methods, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. The traditional finite volume WENO framework based on cell averages is used to reconstruct point values of the solution at Gaussian-type points in those cells where limiting is deemed necessary, and the polynomial solutions in those cells are then rebuilt through numerical integration using these Gaussian points. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure. [ABSTRACT FROM AUTHOR]

In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94-111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577-606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of sup erconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions. [ABSTRACT FROM AUTHOR]

In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge-Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge-Kutta method. Error estimates for the P¹ (piecewise linear) elements are obtained under the usual CFL condition τ ≤ γh for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and τ are the maximum element lengths and time steps, respectively, and the positive constant yγ is independent of h and τ. However, error estimates for higher order Pk (k ≥ 2) elements need a more restrictive time step τ ≤ γh4/3. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition τ ≤ γh for the Pk elements of degree k ≥ 2. Error estimates of O(hk+½ + τ²) are obtained for general monotone numerical fluxes, and optimal error estimates of O(hk+1 + τ²) are obtained for upwind numerical fluxes. [ABSTRACT FROM AUTHOR]

In this paper we develop a Lax–Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge–Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax–Wendroff time discretization procedure is more cost effective than the Runge–Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics. [ABSTRACT FROM AUTHOR]

In this paper we construct high-order weighted essentially nonoscillatory (WENO) schemes for solving the nonlinear Hamilton-Jacobi equations on two-dimensional unstructured meshes. The main ideas are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy to choose diversified smaller stencils to make up the bigger stencil in the WENO procedure. Both third-order and fourth-order WENO schemes using combinations of second-order approximations with nonlinear weights are constructed. Extensive numerical experiments are performed to demonstrate the stability and accuracy of the methods. High-order accuracy in smooth regions, good resolution of derivative singularities, and convergence to viscosity solutions are observed. [ABSTRACT FROM AUTHOR]