Weighted Essentially Non-oscillatory Limiters Research Articles

A high order positivity-preserving polynomial projection remapping method

In this paper, we develop a high-order positivity-preserving polynomial projection remapping method based on the L2 projection for the discontinuous Galerkin (DG) scheme. Combined with the Lagrangian type DG scheme and the rezoning strategies, we present an indirect arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method. By clipping precisely the intersections between the old distorted mesh and the new rezoned mesh, our remapping method is high-order accurate and has no limitation for the mesh movements, so it is suitable for the large deformable problems. A positivity-preserving limiter is also added for the physical variables in computational fluid dynamics without losing the original high-order accuracy and conservation. A multi-resolution weighted essentially non-oscillatory (WENO) limiter is adopted to overcome numerical oscillations and it can keep the original high-order accuracy in the smooth region. This WENO limiter combines several different degrees of polynomials which are the local L2 projections of the original polynomial with nonlinear weights calculated by their smoothness, therefore, it is highly parallel efficient. The properties of positivity-preserving, non-oscillation and high-order accuracy of the remapping method will be shown by a variety of numerical experiments on one, two and three dimensional unstructured meshes. The performance of the ALE-DG scheme with rezoning and remapping is also tested for the Euler system in one and two dimensions.

High-Order CFD Solvers on Three-Dimensional Unstructured Meshes: Parallel Implementation of RKDG Method with WENO Limiter and Momentum Sources

In aerospace engineering, high-order computational fluid dynamics (CFD) solvers suitable for three-dimensional unstructured meshes are less developed than expected. The Runge–Kutta discontinuous Galerkin (RKDG) finite element method with compact weighted essentially non-oscillatory (WENO) limiters is one of the candidates, which might give high-order solutions on unstructured meshes. In this article, we provide an efficient parallel implementation of this method for simulating inviscid compressible flows. The implemented solvers are tested on lower-dimensional model problems and real three-dimensional engineering problems. Results of lower-dimensional problems validate the correctness and accuracy of these solvers. The capability of capturing complex flow structures even on coarse meshes is shown in the results of three-dimensional applications. For solving problems containing rotary wings, an unsteady momentum source model is incorporated into the solvers. Such a finite element/momentum source hybrid method eliminates the reliance on advanced mesh techniques, which might provide an efficient tool for studying rotorcraft aerodynamics.

High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters on triangular meshes

In this paper, high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are designed for solving hyperbolic conservation laws on triangular meshes. These multi-resolution WENO limiters are new extensions of the associated multi-resolution WENO finite volume schemes [49,50] which serve as limiters for RKDG methods from structured meshes [47] to triangular meshes. Such new WENO limiters use information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [24], to build a sequence of hierarchical L2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, and fourth-order RKDG methods with associated multi-resolution WENO limiters are developed as examples, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong shocks or contact discontinuities by gradually degrading from the highest order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on triangular meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on triangular meshes. Extensive one-dimensional (run as two-dimensional problems on triangular meshes) and two-dimensional tests are performed to demonstrate the effectiveness of these RKDG methods with the new multi-resolution WENO limiters.

A note on non‐negativity correction for a multimoment finite‐volume transport model with WENO limiter

AbstractAn oscillation‐less multimoment global transport model was proposed by Tang et al. in 2018 by introducing a slope limiter based on the weighted essentially non‐oscillatory (WENO) concept. The spurious oscillations around the discontinuities and strong gradients can be effectively removed and the model preserves the fourth‐order accuracy in spherical geometry for smooth solutions. However, the WENO limiter does not strictly guarantee the non‐negativity of numerical solution and the resulting model will violate the physical principle due to the existence of the nonphysical negative undershoots. As the numerical fluxes, which determine the time tendency of the volume‐integrated average, are evaluated by the point values (PVs) defined along the cell boundaries in the multimoment finite‐volume schemes, the non‐negativity preserving model can be accomplished by modifying those PVs to implement the corrections on the numerical fluxes proposed in the positive definite flux‐corrected transport (FCT) method by Smolarkiewicz in 1989. In this study, a non‐negativity correction algorithm is proposed for a global transport model using the MM‐FVM_WENO scheme and verified by simulating the widely used benchmark tests.

High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters

In this paper, a new type of multi-resolution weighted essentially non-oscillatory (WENO) limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods is designed. This type of multi-resolution WENO limiters is an extension of the multi-resolution WENO finite volume and finite difference schemes developed in [43]. Such new limiters use information of the DG solution essentially only within the troubled cell itself, to build a sequence of hierarchical L2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, fourth-order, and fifth-order RKDG methods with these multi-resolution WENO limiters have been developed as examples, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original DG methods. Benchmark examples are given to demonstrate the good performance of these RKDG methods with the associated multi-resolution WENO limiters.

A simple, high-order and compact WENO limiter for RKDG method

In this paper, a new limiter using weighted essentially non-oscillatory (WENO) methodology is investigated for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving hyperbolic conservation laws. The idea is to use the high-order DG solution polynomial itself in the target cell and the linear polynomials which are reconstructed by the cell averages of solution in the target cell and its neighboring cells to reconstruct a new high-order polynomial in a manner of WENO methodology. Since only the linear polynomials need to be prepared for reconstruction, this limiter is very simple and compact with a stencil including only the target cell and its immediate neighboring cells. Numerical examples of various problems show that the new limiting procedure can simultaneously achieve uniform high-order accuracy and sharp, non-oscillatory shock transitions.

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

SummaryNumerical oscillation has been an open problem for high‐order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub‐cell information available in the local high‐order reconstructions. This paper presents a novel algorithm that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax–Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.

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

AbstractIn this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [29]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. The modification in this paper improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter.

A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

AbstractThis paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

An efficient WENO limiter for discontinuous Galerkin transport scheme on the cubed sphere

SummaryThe discontinuous Galerkin (DG) transport scheme is becoming increasingly popular in the atmospheric modeling due to its distinguished features, such as high‐order accuracy and high‐parallel efficiency. Despite the great advantages, DG schemes may produce unphysical oscillations in approximating transport equations with discontinuous solution structures including strong shocks or sharp gradients. Nonlinear limiters need to be applied to suppress the undesirable oscillations and enhance the numerical stability. It is usually very difficult to design limiters to achieve both high‐order accuracy and non‐oscillatory properties and even more challenging for the cubed‐sphere geometry. In this paper, a simple and efficient limiter based on the weighted essentially non‐oscillatory (WENO) methodology is incorporated in the DG transport framework on the cubed sphere. The uniform high‐order accuracy of the resulting scheme is maintained because of the high‐order nature of WENO procedures. Unlike the classic WENO limiter, for which the wide halo region may significantly impede parallel efficiency, the simple limiter requires only the information from the nearest neighboring elements without degrading the inherent high‐parallel efficiency of the DG scheme. A bound‐preserving filter can be further coupled in the scheme that guarantees the highly desirable positivity‐preserving property for the numerical solution. The resulting scheme is high‐order accurate, non‐oscillatory, and positivity preserving for solving transport equations based on the cubed‐sphere geometry. Extensive numerical results for several benchmark spherical transport problems are provided to demonstrate good results, both in accuracy and in non‐oscillatory performance. Copyright © 2015 John Wiley & Sons, Ltd.

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework on unstructured meshes

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36,33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes [45], to the correction procedure via reconstruction (CPR) framework for solving conservation laws. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper is particularly simple to implement and will not harm the conservativeness of the CPR framework. Also, it uses information only from the target cell and its immediate neighbors, and thus can maintain the compactness of the CPR framework. Since the CPR framework with the WENO limiter does not in general preserve positivity of the solution, we also extend the positivity-preserving limiters in [43,44,42] to the CPR framework. Numerical results in one and two dimensions are provided to illustrate the good behavior of this procedure.

Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes

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.

RKDG methods with WENO limiters for unsteady cavitating flow

In this paper, we develop the Runge–Kutta discontinuous Galerkin (RKDG) methods with the finite volume weighted essentially non-oscillatory (WENO) reconstruction as limiters to solve for the unsteady cavitating flow under the employment of the isentropic one-fluid model. To treat the cavitating flow and suppress the possible spurious oscillations in the vicinity of the cavitation boundary, the TVB limiter is used as an indicator to detect the “troubled cells” and hence take the advantage of utilizing the WENO reconstruction for the freedoms of the RKDG methods. Numerical results are provided to illustrate the viability of these procedures.

Local DG method using WENO type limiters for convection–diffusion problems

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.

Runge–Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes

In [J. Qiu, C.-W. Shu, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907–929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, 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. Numerical results are provided to illustrate the behavior of this procedure.