WENO Limiters Research Articles

Efficient dimension-by-dimension adaptive TENO-based limiter and troubled-cell indicator for nodal-based high-order spectral difference method

A new targeted essentially non-oscillatory (TENO) limiter with adaptive dissipation has been developed for the 3rd- and 4th-order spectral difference method. This limiter can potentially be useful for other nodal-based discontinuous high-order methods as well. Unlike traditional WENO limiters for these methods which limit low-order moments one by one, the new limiter is based on a direct convex combination of reconstruction polynomial candidates. This strategy ensures the good computational efficiency for high order reconstructions. The reconstruction stencils only involve nodal values from the target cell and its neighbors sharing faces with it. The reconstruction employs Hermite polynomials, making the new limiter compact. It can be implemented in a dimension-by-dimension manner for multi-dimensional problems, which is consistent with the spectral difference method. To further enhance efficiency, a TENO-based troubled-cell indicator is also developed to only activate limiters in troubled cells. Extensive one-dimensional and two-dimensional numerical experiments validate the performance of the new TENO-based limiter and indicator. In summary, the limiter is much less dissipative than WENO limiters and can resolve richer small-scale flow structures on coarse meshes. The indicator precisely marks out troubled cells, slightly improving over the KXRCF indicator.

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes

In this paper, we improve the simple WENO limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method in [34] for solving two-dimensional hyperbolic systems on unstructured meshes. The major improvement is reducing the number of polynomials transformed to the characteristic fields for each direction. For the triangular mesh, this new simple WENO limiter transforms only two polynomials to the characteristic fields for each direction, while the original simple WENO limiter [34] uses four polynomials. Thus the improved simple WENO limiter reduces the computational cost and improves the efficiency. It provides a simpler and more practical and efficient way to the characteristic-wise limiting procedure, while still simultaneously maintaining uniform high-order accuracy in smooth regions and controlling spurious nonphysical oscillations near discontinuities. We also apply this improved simple WENO limiter to a high order method constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds [10], where polygonal meshes are constructed based on random points and the traditional DG method was adopted on the constructed polygonal mesh. For such a complex polygonal mesh, this limiter still transforms only two polynomials to the characteristic fields. Thus the simplicity and efficiency of this new limiter is more evident in this case. Numerical results are provided to illustrate the accuracy and effectiveness of this procedure. For some examples, the improved limiter has less smearing and higher resolution than the original one.

Simulating magnetized neutron stars with discontinuous Galerkin methods

Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard minmod/$\Lambda\Pi^N$ limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized TOV stars without any fine-tuning of parameters. Among the limiters explored here, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method.

A cell-centered Lagrangian discontinuous Galerkin method using WENO and HWENO limiter for compressible Euler equations in two dimensions

A cell-centered Lagrangian discontinuous Galerkin method using WENO and HWENO limiter for compressible Euler equations in two dimensions

A study of higher-order reconstruction methods for genuinely two-dimensional Riemann solver

Balsara's genuinely multidimensional Riemann solvers, which possess a simple closed-form and high efficiency in the low-order cases, realize their multidimensional effect successfully by introducing the vertex-based framework. However, traditional higher-order (third-order and above) reconstruction methods built for the structured grids become ambiguous when applied directly into these solvers. Nowadays, there is little research on the higher-order reconstruction methods for such Riemann solvers besides Balsara's work by adopting the ADER-WENO schemes. Based on Balsara's work, we conduct research on the multidimensional higher-order reconstruction procedure in this study. This paper investigates two third-order reconstruction methods with the WENO approach for the genuinely two-dimensional HLLE Riemann solver. The critical idea of the reconstruction methods is to utilize the solution or its derivative to construct the spatial two-dimensional interpolation polynomials with the third-order accuracy. The first reconstruction method denoted by TWENO constructs the polynomials by the Taylor expansion, while the second one denoted by HWENO adopts the Hermite interpolation polynomials. By combining the polynomials with the WENO limiter, these schemes can provide required reconstructed values at the midpoints and corner-points of the cell interfaces for the framework and avoid oscillations near discontinuities. A discontinuity-detector technology is adopted to improve the computational efficiency of these reconstruction methods in simulating complex flows. Several numerical test cases are conducted and show that these third-order methods achieve their design accuracy and capture the shock waves with no overshoots or oscillations. The multidimensional nature of the solutions is kept well by the TWENO/HWENO methods, and the computational time is reduced by about 30% with the discontinuity-detector technology. Moreover, it is found that the genuinely two-dimensional Riemann solver fits the third-order reconstruction methods naturally due to its unique application of a Simpson rule.

A Quasi-Conservative Discontinuous Galerkin Method for Multi-component Flows Using the Non-oscillatory Kinetic Flux

In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Gr\"uneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution WENO limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1].

Runge–Kutta discontinuous Galerkin method for Baer–Nunziato model with ‘simple WENO’ limiting of conservative variables

Abstract The work is devoted to the application of Runge–Kutta discontinuous Galerkin (RKDG) method for solving Baer–Nunziato hyperbolic model for nonequilibrium two-phase flows. The approach is based on the application of the simple WENO limiter directly to the conservative variables. Mathematical model and the corresponding numerical algorithm are described. The results of numerical simulations for 1D and 2D tests are presented and discussed.

High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems

Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical L2 projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

A low-dissipation shock-capturing framework with flexible nonlinear dissipation control

In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially non-oscillatory (TENO) scheme (Fu et al. 2016 [9]). Rather than being discarded directly as with TENO schemes, the nonsmooth candidates are filtered by an extra nonlinear limiter, such as a monotonicity-preserving (MP) limiter or a total variation diminishing (TVD) limiter. Consequently, high-order reconstruction is achieved by assembling candidate fluxes with optimal linear weights since they are either smooth reconstructions or filtered ones which feature good non-oscillation property. A weight renormalization procedure as with the standard TENO paradigm is not necessary. This new framework concatenates the concepts of TENO, WENO and other nonlinear limiters for shock-capturing, and provides a new insight to designing low-dissipation nonlinear schemes. Through the adaptation of nonlinear limiters, nonlinear dissipation in the newly proposed framework can be controlled separately without affecting the performance in smooth regions. Based on the proposed framework, a family of new six- and eight-point nonlinear schemes with controllable dissipation is proposed. A set of critical benchmark cases involving strong discontinuities and broadband fluctuations is simulated. Numerical results reveal that the proposed new schemes capture discontinuities sharply and resolve the high-wavenumber fluctuations with low dissipation, while maintaining the desired accuracy order in smooth regions.

A compact subcell WENO limiting strategy using immediate neighbours for Runge-Kutta discontinuous Galerkin methods

A compact subcell weighted essentially non oscillatory (CSWENO) limiter is proposed for the solution of hyperbolic conservation laws with discontinuous Galerkin Method which uses only the immediate neighbours of a given cell. These neighbours are divided into the required stencil for WENO reconstruction and an existing WENO limiting strategy is used. Accuracy tests and results for one-dimensional and two-dimensional Burgers' equation and one-dimensional and two-dimensional Euler equations for Cartesian meshes are presented using this limiter. Comparisons with the parent WENO limiter are provided wherever appropriate and the performance of the current limiter is found to be slightly better than the parent WENO limiter for higher orders.

A p-weighted limiter for the discontinuous Galerkin method on one-dimensional and two-dimensional triangular grids

This paper presents an accuracy-preserving p-weighted limiter for discontinuous Galerkin methods on one-dimensional and two-dimensional triangular grids. The p-weighted limiter is the extension of the second-order WENO limiter by Li et al. (2018) [22] to high-order accuracy, with the following important improvements of the limiting procedure. First, the candidate polynomials of the p-weighted limiter are the p-hierarchical orthogonal polynomials of the current cell, and the linear polynomials constructed by minimizing the projection error on the face-neighboring cells. Second, the p-weighted procedure introduces a new smoothness indicator which has less numerical dissipation comparing with the classical WENO one. The smoothness indicator is efficiently computed through a quadrature-free approach that takes advantage of the orthogonal property of the basis functions. Third, the small positive number ϵ, which is introduced in the weights to avoid dividing by zero, is set as a function of the smoothness indicator to preserve accuracy near smooth extremas. Numerous benchmark problems are solved to test the p1, p3 and p5 discontinuous Galerkin schemes using the p-weighted limiter. Numerical results demonstrate that the p-weighted limiter is capable of capturing strong shocks while preserving accuracy in smooth regions.

Constructing a Limiter Based on Averaging the Solutions for the Discontinuous Galerkin Method

The Galerkin method with discontinuous basis functions has proved to be effect in solving hyperbolic systems of equations numerically. However, to ensure the solution yielded by this method is monotonic, a smoothing operator is required to be used, especially if the solution contains strong discontinuities. In this paper, a well-proven smoothing operator based on a WENO reconstruction and a smoothing operator of a new type based on averaging the solutions that takes into account the rate of change of the solution and the rate of change of its derivatives is considered. The effect of these limiters in solving a series of test problems is compared. The application of the proposed smoothing operator is shown to be as good as the action of a WENO limiter, in some cases even exceeding it in the accuracy of the resulting solution, which is confirmed by the numerical studies.

The discontinuous Galerkin spectral element methods for compressible flows on two-dimensional mixed grids

This paper presents a practical algorithm for constructing high order discontinuous Galerkin spectral element methods (DGSEM) on mixed triangular and quadrilateral grids. The traditional DGSEM belongs to the collocation-type nodal discontinuous Galerkin method which is computationally efficient on one-dimensional and tensor-product grids. This work generalizes DGSEM to triangular grids using symmetric quadrature points in sphere close packed arrangement. The dispersion and dissipation analysis shows that the scheme is stable for linear problems on triangles. A problem independent limiting procedure is proposed for capturing the shock waves. The main components are a modified KXRCF shock detector, a smoothness indicator and a high-resolution WENO limiter based on candidate polynomials constructed by newly developed projection approach. The implementation of the DGSEM on the mixed grids is also presented. Numerical examples on triangular and mixed grids verify the scheme's high order accuracy and robustness in computing inviscid and viscous compressible flows.

Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics

AbstractThis paper develops Runge-Kutta PK-based central discontinuous Galerkin (CDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1,2,3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that the approximate solutions on mutually dual meshes are coupled with each other, and the use of numerical flux will be avoided. The WENO limiter is adaptively implemented via two steps: the “troubled” cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree (2K+1) replacing the CDG solutions inside the “troubled” cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells. Because the WENO limiter is only employed for finite “troubled” cells, the computational cost can be as little as possible. The accuracy of the CDG without the numerical dissipation is analyzed and calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our Runge-Kutta CDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.

High order sub-cell finite volume schemes for solving hyperbolic conservation laws I: basic formulation and one-dimensional analysis

In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with $\mathrm{P_NP_M}$ technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.

Runge–Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics

This paper develops PK-based non-central and central Runge–Kutta discontinuous Galerkin (DG) methods with WENO limiter for the one- and two-dimensional special relativistic magnetohydrodynamical (RMHD) equations, K=1,2,3. The non-central DG methods are locally divergence-free, while the central DG are “exactly” divergence-free but have to find two approximate solutions defined on mutually dual meshes. The adaptive WENO limiter first identifies the “troubled” cells by using a modified TVB minmod function, and then uses the WENO technique and the cell average values of the DG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells to locally reconstruct a new polynomial of degree (2K+1) inside the “troubled” cells instead of the DG solution. The WENO limiting procedure does not destroy the locally or “exactly” divergence-free property of magnetic field and is only employed for finite “troubled” cells so that the computational cost can be as little as possible. Several test problems in one and two dimensions are solved by using our non-central and central Runge–Kutta DG methods with WENO limiter. The numerical results demonstrate that our methods are stable, accurate, and robust in resolving complex wave structures.

Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter on Unstructured Meshes

AbstractIn this paper we generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed in [38] for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion [11] while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.