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In this paper, we present a class of new Hermite weighted essentially non-oscillatory (HWENO) schemes based on finite volume framework to directly solve the Hamilton–Jacobi (HJ) equations. For HWENO reconstruction, both the cell average and the first moment of the solution are evolved, and for two dimensional case, HWENO reconstruction is based on a dimension-by-dimension strategy which is the first used in HWENO reconstruction. For spatial discretization, one of key points for directly solving HJ equation is the reconstruction of numerical fluxes. We follow the idea put forward by Cheng and Wang (2014) [3] to reconstruct the values of solution at Gauss–Lobatto quadrature points and numerical fluxes at the interfaces of cells, and for neither the convex nor concave Hamiltonian case, the monotone modification of numerical fluxes is added, which can guarantee the precision in the smooth region and converge to the entropy solution when derivative discontinuities come up. The third order TVD Runge–Kutta method is used for the time discretization. Extensive numerical experiments in one dimensional and two dimensional cases are performed to verify the efficiency of the methods. [ABSTRACT FROM AUTHOR]

Abstract: The local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection–diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as Runge–Kutta LDG (RKLDG) when TVD Runge–Kutta method is applied for time discretization. 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 non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper, we investigated RKLDG methods with WENO and Hermite WENO (HWENO) limiters for solving convection–diffusion equations on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition. Numerical results are provided to illustrate the behavior of these procedures. [Copyright &y& Elsevier]

Abstract: A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillation. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. In this paper, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong shocks. The idea is to identify discontinuity by an discontinuity indicator, then reconstruct numerical flux by WENO approximation in discontinuous regions and up-wind linear approximation in smooth regions. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu (J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially non-oscillatory limiters, SIAM Journal of Scientific Computing 27 (2005) 995–1013). The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost. Detail numerical studies in one- and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we systematically investigate adaptive Runge–Kutta discontinuous Galerkin (RKDG) methods for hyperbolic conservation laws with different indicators which were based on the troubled cell indicators studied by Qiu and Shu [J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin mehtods using weighted essentially non-osillatory limiters, SIAM J. Sci. Comput. 27 (2005) 995–1013]. The emphasis is on comparison of the performance of adaptive RKDG method using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance for adaptive computation to save computational cost. Both h-version and r-version adaptive methods are considered in the paper. The idea is to first identify “troubled cells” by different troubled-cell indicators, namely those cells where limiting might be needed and discontinuities might appear, then adopt an adaptive approach in these cells. A detailed numerical study in one-dimensional case is performed, addressing the issues of efficiency (less CPU cost and more accurate), non-oscillatory property, and resolution of discontinuities. [Copyright &y& Elsevier]

In this paper, a compact and high order ADER (Arbitrary high order using DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for hyperbolic conservation laws. The newly-developed method employs the Lax-Wendroff procedure to convert time derivatives to spatial derivatives, which provides the time evolution of the variables at the cell interfaces. This information is required for the simple HWENO reconstructions, which take advantages of the simple WENO and the classic HWENO. Compared with the original Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages. Firstly, RK-HWENO method must solve the additional equations for reconstructions, which is avoided for the new method. Secondly, the SHWENO reconstruction is performed once with one stencil and is different from the classic HWENO methods, in which both the function and its derivative values are reconstructed with two different stencils, respectively. Thus the new method is more efficient than the RK-HWENO method. Moreover, the new method is more compact than the existing ADER-WENO method. Besides, the new method makes the best use of the information in the ADER method. Thus, the time evolution of the cell averages of the derivatives is simpler than that developed in the work (Li et al. (2021) [17]). Numerical tests indicate that the new method can achieve high order for smooth solutions both in space and time, keep non-oscillatory at discontinuities. • A compact fifth-order ADER scheme is proposed for hyperbolic conservation laws. • A compact reconstruction technique is designed by better utilizing the information in time evolution. • The computational stencil is more compact than the original ADERWENO scheme. • It is more efficient than the original multi-stage RK-HWENO scheme. [ABSTRACT FROM AUTHOR]

In this paper, a class of high-order central finite volume schemes is proposed for solving one- and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax–Wendroff type discretizations or the natural continuous extension of Runge–Kutta methods in time. Differently from the central Hermite WENO methods we developed previously in Tao et al. (2015) [34] , the spatial reconstructions, a core ingredient of the methods, are based on the zeroth-order and the first-order moments of the solution, and are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. This leads to much simpler implementation of the methods in higher dimension and better cost efficiency. Meanwhile, the proposed methods have the attractive features of the general central Hermite WENO methods such as being compact in reconstruction and requiring neither flux splitting nor numerical fluxes, while being accurate and essentially non-oscillatory. A collection of one- and two-dimensional numerical examples is presented to demonstrate high resolution and robustness of the methods in capturing smooth and non-smooth solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we generalize a new type of limiters based on the weighted essentially non-oscillatory (WENO) finite volume methodology for the Runge–Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed in [32] for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the entire polynomials of the DG solutions from the troubled cell and its immediate neighboring cells, and then apply the classical WENO procedure to form a convex combination of these polynomials based on smoothness indicators and nonlinear weights, with suitable adjustments to guarantee conservation. The main advantage of this new limiter is its simplicity in implementation, especially for the unstructured meshes considered in this paper, as only information from immediate neighbors is needed and the usage of complicated geometric information of the meshes is largely avoided. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good performance of this procedure. [Copyright &y& Elsevier]