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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]

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]

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]

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

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

The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes; however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric exibility, suitability for h-and p-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. In this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivity-preserving limiter in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 8918-8934] and a new rotational positivity-preserving limiter. This combined limiter is simple to implement and we prove the properties of positivity-preserving and high order accuracy rigorously. One- and two-dimensional numerical results are provided to verify the good properties of the positivity-preserving DG schemes. [ABSTRACT FROM AUTHOR]

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

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

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

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

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]

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

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]

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

In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-­Jacobi equations. This method is based on the Runge-­Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high-order accuracy with a local, compact stencil, and is suited for efficient parallel implementation. One- and two- dimensional numerical examples are given to illustrate the capability of the method. At least kth order of accuracy is observed for smooth problems when kth degree polynomials are used, and derivative singularities are resolved well without oscillations, even without limiters. [ABSTRACT FROM AUTHOR]