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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]

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]

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]

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]

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]

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]

This is an extension of our earlier work [9] in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. An algorithm of building a suitable polygonal mesh based on the random points was given and the traditional discontinuous Galerkin (DG) method was adopted on the constructed polygonal mesh. Numerical results in [9] show that the current scheme will generate spurious numerical oscillations when dealing with solutions containing strong shocks. In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for DG schemes on two-dimensional unstructured triangular meshes [27], to our high order method on polygonal meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the original method in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt is particularly simple to implement and will not harm the conservativeness and compactness of the original method. Moreover, we also extend the maximum-principle-satisfying limiter for the scalar case and the positivity-preserving limiter for the Euler system to our method. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of these limiters. [ABSTRACT FROM AUTHOR]

H(div) conforming and discontinuous Galerkin (DG) methods are designed for incompressible Euler's equation in two and three dimension. Error estimates are proved for both the semidiscrete method and fully-discrete method using backward Euler time stepping. Numerical examples exhibiting the performance of the methods are given. [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 analyze implicit-explicit (IMEX) Runge-Kutta (RK) time discretization methods for solving linear convection-diffusion equations. The diffusion operator is treated implicitly via the embedded discontinuous Galerkin (EDG) method and the convection operator explicitly via the upwinding discontinuous Galerkin method. [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 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]

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]

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]

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]

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 provide a priori L2 error estimates for the semi-discrete discontinuous Galerkin method [3] and the local discontinuous Galerkin method [22] for one- and two-dimensional nonlinear Hamilton-Jacobi equations with smooth solutions. With a special Gauss-Radau projection, the optimal error estimates on rectangular meshes are obtained. [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 we investigate the superconvergence of local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time-dependent fourth-order problems. We prove that the error between the LDG solution and a particular projection of the exact solution, u, achieves (k + 3/2)th-order supercon-vergence when polynomials of degree k {k ≥ 1) are used. Numerical experiments with Pk polynomials, with 1 ≤ k ≤ 3, are displayed to demonstrate the theoretical results, which show that the error eu actually achieves (k + 2)th-order superconvergence, indicating that the error bound for u obtained in this paper is suboptimal. Initial boundary value problems, nonlinear equations and solutions having singularities, are numerically investigated to verify that the conclusions hold true for very general cases. [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]

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]

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 study the strong stability preserving (SSP) property of a class of deferred correction time discretization methods, for solving the method-of-fines schemes approximating hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

?We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis. [ABSTRACT FROM AUTHOR]

High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods are more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil It is particularly important to treat the points near the inflow boundary accurately, as the information would flow into the computational domain and would affect global accuracy. In the literature, the numerical solution at these boundary points are either fixed with the exact solution, which is not always feasible, or computed with a first order discretization, which could reduce the global accuracy. In this paper, we discuss two strategies to handle the inflow boundary conditions. One is based on the numerical solutions of a first order fast sweeping method with several different mesh sizes near the boundary and a Richardson extrapolation, the other is based on a Lax-Wendroff type procedure to repeatedly utilizing the PDE to write the normal spatial derivatives to the inflow boundary in terms of the tangential derivatives, thereby obtaining high order solution values at the grid points near the inflow boundary. We explore these two approaches using the fast sweeping high order WENO scheme in [18] for solving the static Eikonal equation as a representative example. Numerical examples are given to demonstrate the performance of these two approaches. [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]

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]

A numerical simulation of two-dimensional compressible Navier-Stokes equations using a high-order weighted essentially nonoscillatory finite difference shock capturing scheme is carried out in this paper, to study the effect of shock waves on the development of Rayleigh-Taylor instability. Shocks with different Mach numbers are introduced ahead or behind the Rayleigh-Taylor interface, and their effect on the transition to instability is demonstrated and compared. It is observed that shock waves can speed up the transition to instability for the Rayleigh-Taylor interface significantly. Stronger shocks are more effective in this speed-up process. [ABSTRACT FROM AUTHOR]

In this paper we further explore and apply our recent anti-diffusive flux corrected high order finite difference WENO schemes for conservation laws [18] to compute the Saint-Venant system of shallow water equations with pollutant propagation, which is described by a transport equation. The motivation is that the high order anti-diffusive WENO scheme for conservation laws produces sharp resolution of contact discontinuities while keeping high order accuracy for the approximation in the smooth region of the solution. The application of the anti-diffusive high order WENO scheme to the Saint-Venant system of shallow water equations with transport of pollutant achieves high resolution [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]

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]