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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 main purpose of this paper is to study the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with multi-step implicit-explicit (IMEX) time discretization schemes, for solving time-dependent incompressible fluid flows. We will give theoretical analysis for the Oseen equation, and assess the performance of the schemes for incompressible Navier-Stokes equations numerically. For the Oseen equation, using first order IMEX time discretization as an example, we show that the IMEXLDG scheme is unconditionally stable for Qk elements on cartesian meshes, in the sense that the time-step τ is only required to be bounded from above by a positive constant independent of the spatial mesh size h. Furthermore, by the aid of the Stokes projection and an elaborate energy analysis, we obtain the L∞(L2) optimal error estimates for both the velocity and the stress (gradient of velocity), in both space and time. By the inf-sup argument, we also obtain the L∞(L2) optimal error estimates for the pressure. Numerical experiments are given to validate our main results. [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]

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]

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]

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form min(ut + cux, u - g(x)) = 0, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These "low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes. [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 implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case [H.J. Wang, C.-W. Shu and Q. Zhang, SIAM J. Numer. Anal. 53 (2015) 206–227; H.J. Wang, C.-W. Shu and Q. Zhang, Appl. Math. Comput. 272 (2015) 237–258], i.e., the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step τ is only required to be upper-bounded by a positive constant independent of the spatial mesh size h. Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e., if piecewise polynomial of degree k is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order O(hk+1+τs) for the sth order IMEX LDG scheme (s = 1, 2, 3) under consideration. Numerical experiments are also given to verify our main results. [ABSTRACT FROM AUTHOR]

We analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws. In one dimension, optimal a priori error estimates of order k+1 are obtained for the semidiscrete scheme when piecewise polynomials of degree at most k (k ≥ 0) are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge--Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier-Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced in [Y. Guo and Ian Tice, Anal. PDE 6 (2013) 287-369; Y. Guo and Ian Tice, Arch. Ration. Mech. Anal. 207 (2013) 459-531; L. Wu, SIAM J. Math. Anal. 46 (2014) 2084-2135], and the natural energy estimate, which is rarely used in the numerical study of this system before. [ABSTRACT FROM AUTHOR]

In this paper we present an a priori error estimate of the Runge-Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge-Kutta method and the finite element space is made up of piecewise polynomials of degree k ≥ Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition τ ≤ γh, where h is the element length and τ is time step, and γ is a positive constant independent of h and τ. Optimal estimates are also considered when the upwind numerical flux is used. [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]

In this paper, we generalize the so-called inverse Lax-Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144-8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235-256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundström, Math. Comput. 26 (1972) 649-686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge-Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems. [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]

We investigate the angular distribution of Lyα photons scattering or emerging from an optically thick medium. Since the evolution of specific intensity I in frequency space and angular space are coupled with each other, we first develop the WENO numerical solver 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 that I is linearly dependent on the angular variable μ, yield similar frequency profiles of the photon flux as those without the Eddington approximation. However, the solutions of the μ distribution evolution are significantly different from those given by the Eddington approximation. First, the angular distribution of I is found to be substantially dependent on the frequency of the photons. For photons with the resonant frequency v0, I contains only a linear term of μ. For photons with frequencies at the double peaks of the flux, the μ-distribution is highly anisotropic; most photons are emitted radially forward. Moreover, either at v0 or at the double peaks, the μ distributions actually are independent of the initial μ distribution of photons of the source. This is because the photons with frequencies either at v0 or the double peaks undergo the process of forgetting their initial conditions due to 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 a very small opening angle. [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]

We investigate the effects of dust on Lyα photons emergent from an optically thick medium by solving the integro-differential equation of radiative transfer of resonant photons. To solve the differential equations numerically, we use the weighted essentially non-oscillatory method. Although the effects of dust on radiative transfer are well known, the resonant scattering of Lyα photons makes the problem non-trivial. For instance, if the medium has an optical depth of dust absorption and scattering of τa » 1, τ » 1, and τ » τa, the effective absorption optical depth in a random walk scenario would be equal to √τa(τa + τ). We show, however, that for a resonant scattering at frequency v0, the effective absorption optical depth would be even larger than τ(v0). If the cross section of dust scattering and absorption is frequency-independent, the double-peaked structure of the frequency profile given by the resonant scattering is basically dust-independent. That is, dust causes neither narrowing nor widening of the width of the double-peaked profile. One more result is that the timescales of the Lyα photon transfer in an optically thick halo are also basically independent of the dust scattering, even when the scattering is anisotropic. This is because those timescales are mainly determined by the transfer in the frequency space, while dust scattering, either isotropic or anisotropic, does not affect the behavior of the transfer in the frequency space when the cross section of scattering is wavelength-independent. This result does not support the speculation that dust will lead to the smoothing of the brightness distribution of a Lyα photon source with an optically thick halo. [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]

In this article, we present a high-order weighted essentially non-oscillatory (WENO) scheme, coupled with a high-order fast sweeping method, for solving a dynamic continuum model for bi-directional pedestrian flows. We first review the dynamic continuum model for bi-directional pedestrian flows. This model is composed of a coupled system of a conservation law and an Eikonal equation. Then we present the first-order Lax-Friedrichs difference scheme with first-order Euler forward time discretization, the third-order WENO scheme with third-order total variation diminishing (TVD) Runge-Kutta time discretization, and the fast sweeping method, and demonstrate how to apply them to the model under study. We present a comparison of the numerical results of the model from the first-order and high-order methods, and conclude that the high-order method is more efficient than the first-order one, and they both converge to the same solution of the physical model. [ABSTRACT FROM AUTHOR]

The analytical shock-fitting algorithm outperforms traditional numerical methods in solving the Lighthill-Whitham-Richards (LWR) traffic flow model for homogeneous highways. In this study, we extend the algorithm to an inhomogeneous highway in which two homogeneous sections with different fundamental diagrams are connected by a junction (interface). According to the entropy condition of the Riemann problem, the flow conditions at the interface can be categorized into four groups, and the density on both sides of the interface can be uniquely determined. Based on the physical conditions, we construct a fictitious element as an appropriate boundary condition for each homogeneous section. Consequently, the Riemann problem of an inhomogeneous highway can be transformed into a problem of equivalent homogeneous sections to which the shock-fitting algorithm can be applied. We apply this algorithm to some representative traffic flow cases and compare the results with numerical solutions obtained using the Weighted Essentially Non-Oscillatory (WENO) scheme. [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]

Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady Navier-Stokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ(x, t)=1/ρ(∂ρ/∂t+∂ρu/∂x), where x is the coordinate corresponding to the vortex axis. If ?>0, the sectional streamlines spiral inwards in the near region of the centre. If λ<0, the sectional streamlines spiral outwards in the same region. If λ=0, the sectional streamlines form a nonlinear centre. If λ changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown. [ABSTRACT FROM AUTHOR]

In this paper we explicitly construct the entropy solutions for the Lighthill-Whitham-Richards (LWR) traffic flow model with a flow-density relationship which is piecewise quadratic, concave, but not continuous at the junction points where two quadratic polynomials meet, and with piecewise linear initial condition and piecewise constant boundary conditions. The existence and uniqueness of entropy solutions for such conservation laws with discontinuous fluxes are not known mathematically. We have used the approach of explicitly constructing the entropy solutions to a sequence of approximate problems in which the flow-density relationship is continuous but tends to the discontinuous flux when a small parameter in this sequence tends to zero. The limit of the entropy solutions for this sequence is explicitly constructed and is considered to be the entropy solution associated with the discontinuous flux. We apply this entropy solution construction procedure to solve four representative traffic flow cases, compare them with numerical solutions obtained by a high order weighted essentially nonoscillatory (WENO) scheme, and discuss the results from traffic flow perspectives. [ABSTRACT FROM AUTHOR]