WENO Scheme Research Articles

LES of autoignition process in a turbulent droplet-laden mixing layer – impact of an evaporation model and discretisation method

This paper focuses on the numerical modelling of the autoignition process in an ethanol-air mixture within a turbulent droplet-laden temporally evolving mixing layer, utilising the Large Eddy Simulation (LES) method. We compare results obtained by implementing two notably distinct discretisation methods (second-order Total Variation Diminishing – TVD, and fifth-order Weighted Essentially Non-Oscillatory – WENO) in the species and energy transport equations. Additionally, we examine three widely recognised evaporation models (the ‘ D 2 ’ model, the Abramzon–Sirignano model, and a non-equilibrium model based on the Langmuir-Knudsen law). For modelling the combustion process, we employ the Eulerian Stochastic Field method and a laminar chemistry model. The simulations encompass two droplet diameter distributions, one with low and the other with significant diameter non-uniformity. Furthermore, nine distinct initial conditions are considered. Our findings indicate that adjusting the discretisation scheme or the evaporation model can yield simulation outcomes differing to a substantial degree, similar to the impact of modifying the actual properties of the spray droplets or their initial distribution. This is particularly pronounced in the case of sprays featuring a narrowed droplet diameter distribution. We found that the Abramzon–Sirignano model results in the most rapid autoignition. The TVD scheme contributes to a more even spatial distribution of the evaporated fuel, consequently facilitating quicker evaporation and faster autoignition. Using the WENO scheme generates higher fuel vapor gradients near the droplets, leading to less intense evaporation and slower fuel transport in space.

WENO schemes using optimized third order fuzzy weight limiter functions

Abstract This work presents an improved version of non-linear weight limiters to obtain third order non-oscillatory WENO scheme. The construction of this modified weight limiter is based on fuzzy inference system, which is a knowledge based rule system. The linear combination of overlapped basis functions is used to achieve the optimized weight limiters by exploring the linguistics hedges operator on the basis functions. The WENO scheme using optimized weight limiter achieves third order of accuracy and gives higher resolution to discontinuities compared to other established third order WENO schemes.

Local Characteristic Decomposition–Free High-Order Finite Difference WENO Schemes for Hyperbolic Systems Endowed with a Coordinate System of Riemann Invariants

Local Characteristic Decomposition–Free High-Order Finite Difference WENO Schemes for Hyperbolic Systems Endowed with a Coordinate System of Riemann Invariants

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

High-order finite volume multi-resolution WENO schemes with adaptive linear weights on triangular meshes

This paper presents high-order finite volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights to solve hyperbolic conservation laws on triangular meshes. They are abbreviated as the ALW-MR-WENO schemes. The novel third-order, fourth-order, and fifth-order ALW-MR-WENO schemes are designed by applying two unequal-sized hierarchical central stencils in comparison to the classical WENO schemes which utilize many equal-sized upwind biased/central stencils. With the application of one simple condition, only two linear weights are automatically adjusted to be positive values on condition that their summation is one. The novel finite volume ALW-MR-WENO schemes could maintain the designed order of accuracy in smooth areas and reduce to the first-order accuracy so as to keep essentially non-oscillatory properties around strong discontinuities. So it is the first time that any high-order WENO schemes with the application of only two unequal-sized stencils are obtained on triangular meshes. And the major benefits are their efficiency, compactness, and simplicity in large scale engineering applications on unstructured meshes. Finally, several tests are used to indicate the effectiveness of these new finite volume WENO schemes.

WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modeled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modeling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (2018) [3] to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.

New Finite Volume Mapped Unequal-Sized WENO Scheme for Hyperbolic Conservation Laws

New Finite Volume Mapped Unequal-Sized WENO Scheme for Hyperbolic Conservation Laws

A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations

A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations

Compact high-order gas-kinetic scheme for direct numerical simulation of compressible turbulent flows

In this paper, the three-dimensional fully compact high-order gas-kinetic scheme (HGKS) is proposed for the direct numerical simulation of compressible turbulent flows. Because of the high-order gas evolution model, the numerical fluxes as well as the point-wise conservative variables can be evaluated from the time-accurate gas distribution function at the cell interface. As a result, both the cell-averaged variables and their cell-averaged gradients can be updated inside each cell. Based on the cell averaged values and their gradients, the compact Hermite weighted essentially non-oscillatory (HWENO) scheme is developed, in which the dimension-by-dimension reconstruction is used for three-dimensional turbulences. In both normal and tangential directions, the fifth-order HWENO reconstruction is adopted. Compared with the classical WENO scheme, the stencil for the HWENO scheme only contains 33 cells for each cell. To achieve the temporal accuracy, the two-stage fourth-order temporal discretization is used. For the evaluation of point-wise variables, the simplified third-order gas-kinetic solver is used. Several classical benchmark problems are simulated, which validate the accuracy, resolution, and robustness of compact HGKS. As a comparison, the numerical results of HGKS using non-compact WENO reconstruction are also provided. Due to the compact stencil, the compact HGKS has a favorable performance for turbulence simulation in resolving the multi-scale structures.

Development and prospect of high-order WENO schemes

Development and prospect of high-order WENO schemes

Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

The relativistic hydrodynamics (RHD) equations have three crucial intrinsic physical constraints on the primitive variables: positivity of pressure and density, and subluminal fluid velocity. However, numerical simulations can violate these constraints, leading to nonphysical results or even simulation failure. Designing genuinely physical-constraint-preserving (PCP) schemes is very difficult, as the primitive variables cannot be explicitly reformulated using conservative variables due to relativistic effects. In this paper, we propose three efficient Newton–Raphson (NR) methods for robustly recovering primitive variables from conservative variables. Importantly, we rigorously prove that these NR methods are always convergent and PCP, meaning they preserve the physical constraints throughout the NR iterations. The discovery of these robust NR methods and their PCP convergence analyses are highly nontrivial and technical. Our NR methods are versatile and can be seamlessly incorporated into any RHD schemes that require the recovery of primitive variables. As an application, we apply them to design PCP finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes incorporate high-order HWENO reconstruction, a PCP limiter, and strong-stability-preserving time discretization. We rigorously prove the PCP property of the fully discrete schemes using convex decomposition techniques. Moreover, we suggest the characteristic decomposition with rescaled eigenvectors and scale-invariant nonlinear weights to enhance the performance of the HWENO schemes in simulating large-scale RHD problems. Several demanding numerical tests are conducted to demonstrate the robustness, accuracy, and high resolution of the proposed PCP HWENO schemes and to validate the efficiency of our NR methods.

New simple local smoothness indicators for fifth-order WENO schemes simulating compressible flows

For the WENO-Z scheme, the structure of the local smoothness indicators has a significant impact on the computational performance of the weighted essentially non-oscillatory scheme with a given global smoothness indicator. In this paper, a new class of two-parameter local smoothness indicators is constructed by combining the classical fifth-order WENO-JS and WENO-UD5 schemes and locally appending the coefficients of the higher-order terms of the new local smoothness indicators. A new WENO scheme, WENO-NSLI, is constructed using the global smoothness indicators of WENO-UD5. Numerical experiments show that the new scheme can achieve optimal accuracy at the critical point when the parameter (λ,μ)=(0.5,10) is taken and has a higher resolution than WENO-JS, WENO-Z, and WENO-UD5.

Improving the third-order WENO schemes by using exponential polynomial space with a locally optimized shape parameter

Improving the third-order WENO schemes by using exponential polynomial space with a locally optimized shape parameter

WENO smoothness indicator based troubled‐cell indicator for hyperbolic conservation laws

SummaryHybrid algorithms are an efficient and popular choice for computing the solutions of hyperbolic conservation laws. In general, hybrid algorithms involve low‐cost high‐order accurate schemes in smooth regions and non‐oscillatory shock‐capturing schemes in the vicinity of discontinuities. Troubled‐cell indicators which measure the smoothness of the solution play a significant role in the efficiency of hybrid algorithms. This article proposes a new troubled‐cell indicator utilising the smoothness indicators of WENO schemes for hyperbolic conservation laws. The proposed troubled‐cell indicators are simple, efficient, effective, and are used to construct three new adaptive WENO algorithms of high‐order accuracy. The hybrid algorithms developed are independent of the order and type of the WENO reconstruction. For demonstration, we have considered the fifth and seventh order WENO‐Z reconstruction. The first two algorithms have comparable accuracy and resolution of the solution across discontinuities to that of the WENO‐Z scheme but at a less computational cost. The third algorithm ensures the convergence of the proposed scheme to the correct entropy solution when applied to a hyperbolic conservation law with non‐convex flux for which the WENO schemes fail. We have performed several 1D and 2D numerical experiments to demonstrate the efficiency of the proposed algorithms and their performance compared with the WENO‐Z schemes. The proposed algorithms are efficient and take 30%–75% less computational time than the WENO‐Z schemes while retaining the advantages of WENO‐Z schemes.

A new type of increasingly higher order finite difference and finite volume MR-WENO schemes with adaptive linear weights for hyperbolic conservation laws

In this paper, the new fifth-order, seventh-order, and ninth-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (MR-WENO) schemes with adaptive linear weights are presented for hyperbolic conservation laws on structured meshes. They are termed as high-order finite difference and finite volume ALW-WENO schemes. These ALW-WENO schemes only apply one small stencil and one large stencil in reconstruction processes, which could achieve the desired accuracy in the region of smoothness and non-oscillatory properties in the region of containing strong shocks. The linear weights that sum to one can be automatically adjusted to any positive numbers. This is the first time that arbitrary high-order finite difference and finite volume WENO schemes are designed by using only two unequal-sized central spatial stencils. The structure of these novel WENO schemes is simple, so it is easier for obtaining high-order accuracy and solving multi-dimensional problems in large scale engineering applications. Compared to traditional MR-WENO schemes with same order, the computational efficiency can be further improved. Some benchmark tests indicate that these new ALW-WENO schemes have good robustness and performance.

Computational Investigation of Supersonic Jet Screech in Circular Jets Using Higher-order Weno Scheme

Computational investigation of supersonic jet screech in imperfectly expanded circular jets is conducted using higher order Weighted Essentially Non-Oscillatory (WENO) scheme based solution of the axisymmetric Navier Stokes equations. The jet screech phenomenon is numerically simulated for an over expanded and an under expanded circular supersonic jet. Sound pressure levels, screech amplitude and frequency are calculated from the pressure history in the flow field and compared with values in literature. Shock cell spacings in the high speed jet are also extracted from numerical simulations and compared with that in literature. Important aspects of a screeching jet observed experimentally, like the interaction of instability waves with the shock train and the contribution of downstream propagating hydrodynamic fluctuations to the creation of the upstream moving acoustic wave in the acoustic feedback mechanism, are also captured in the present computations. The latter phenomenon has been observed experimentally but not reported previously in numerical simulations.

An efficient scaling of WENO-JS weights for accuracy preserving and higher resolution schemes

In this short paper, we present efficient scaling of WENO-JS weights which results in robust and formal accuracy preserving high resolution WENO-JS schemes. The proposed scaling makes the WENO-JS schemes robust in the sense that schemes works well irrespective of the choice for the parameter ϵ used in classical WENO-JS weights to prevent division by zero. In smooth solution region the proposed scaling keeps the scaled WENO weights in the vicinity of critical weights thus resulting scheme retains formal accuracy. Numerical experiments show that the WENO schemes with scaled weights provide higher resolution for discontinuities as well for smooth solution with high gradients.

An improved hybridization strategy for the fifth-order unequal-sized weighted essentially non-oscillatory scheme

In this paper, a hybrid unequal-sized weighted essentially non-oscillatory (US-WENO) scheme is developed to reduce the computational cost caused by the computation of nonlinear weights and local characteristic decompositions in its procedures. First, an improved discontinuous sensor is designed based on the mathematical properties of the fourth-degree reconstruction polynomial in the US-WENO scheme. The proposed hybridization strategy can automatically and efficiently identify the troubled cells, and does not contain artificial parameters related to the problems. Second, the efficient linear scheme and the existing US-WENO scheme are combined to simulate the inviscid and viscous compressible flow problems. Finally, some benchmark numerical examples are used to verify the performance of the presented hybrid WENO scheme in various aspects, such as computational efficiency, low dissipation characteristics, shock capture ability, etc. The numerical experiments show that the proposed hybrid method can inherit all the features of the existing US-WENO scheme while improving its computational efficiency.

An improved fifth-order WENO scheme with symmetry-preserving smoothness indicators for hyperbolic conservation laws

Ha et al. [1] have constructed a novel set of smoothness indicators by using the L1-norm measure and introducing a user-tunable parameter to propose a new WENO scheme, dubbed WENO-NS. Kim et al. [2] have improved WENO-NS with another user-tunable parameter to balance the contribution of substencils, leading to the WENO-P scheme. Recently, Rathan et al. [3] have modified WENO-P by devising a higher-order global smoothness indicator. The associated MWENO-P scheme aimed to fix the issue of WENO-P and WENO-NS that they cannot achieve the optimal convergence orders at the critical points where the first and second derivatives are zero. The above-said schemes outperform the existing many WENO schemes in some senses. However, their smoothness indicators violate the symmetry-preserving property preserved by those of many other well-established WENO schemes and rely on one or two user-tunable parameters. In the present study, a new set of smoothness indicators satisfying the symmetry-preserving property without using any user-tunable parameters is constructed. The resultant scheme is called the improved MWENO-P scheme and abbreviated as IMWENO-P. We provide detailed theoretical analysis and numerical examples to verify that the new scheme can attain optimal convergence orders even in the presence of second-order critical points. Euler equations are simulated to demonstrate the enhancements of the new scheme, such as the lower dissipation and better resolution.

Improved third-order WENO scheme with a new reference smoothness indicator

In this article, a new reference smoothness indicator for the third-order WENO scheme is proposed. The main construction idea is to analyze the first derivative of the reconstruction polynomial of the candidate sub-stencil and the first derivative of the reconstruction polynomial of the global stencil. Then calculate the normalized square sum of the L2-norm approximation of the linear convex combination of the first derivative of the reconstruction polynomial of all candidate sub-stencils and the first derivative of the reconstruction polynomial of the global stencil. The new reference smoothness indicator obtained based on the above strategy is denoted as τRe. The newly developed solution is called the WENO-Re solution. A series of one-dimensional and two-dimensional numerical examples show that the new scheme has higher resolution and smaller dissipation compared to several recently improved third-order WENO schemes.