WENO Scheme Research Articles

A family of well-balanced WENO and TENO schemes for atmospheric flows

We herein present a novel methodology to construct very high order well-balanced schemes for the computation of the Euler equations with gravitational source term, with application to numerical weather prediction (NWP). The proposed method is based on augmented Riemann solvers, which allow preserving the exact equilibrium between fluxes and source terms at cell interfaces. In particular, the augmented HLL solver (HLLS) is considered. Different spatial reconstruction methods can be used to ensure a high order of accuracy in space (e.g. WENO, TENO, linear reconstruction), being the TENO reconstruction the preferred method in this work. To the knowledge of the authors, the TENO method has not been applied to NWP before, although it has been extensively used by the computational fluid dynamics community in recent years. Therefore, we offer a thorough assessment of the TENO method to evidence its suitability for NWP considering some benchmark cases which involve inertia and gravity waves as well as convective processes. The TENO method offers an enhanced behavior when dealing with turbulent flows and underresolved solutions, where the traditional WENO scheme proves to be more diffusive. The proposed methodology, based on the HLLS solver in combination with a very high-order discretization, allows carrying out the simulation of meso- and micro-scale atmospheric flows in an implicit Large Eddy Simulation manner. Due to the HLLS solver, the isothermal, adiabatic and constant Brunt-Väisälä frequency hydrostatic equilibrium states are preserved with machine accuracy.

Comparison of Upwind and Symmetric WENO Schemes in Large Eddy Simulation of Basic Turbulent Flows

The properties of modern WENO schemes are examined as applied to large eddy simulation (LES). The WENO-ZM5 scheme with modified smoothness indicators (“upwind”) and the WEN-O‑SYMBOO6 scheme on a symmetric stencil (“symmetric WENO scheme”) are chosen. The schemes are compared on one-dimensional test problems (advection, Hopf, and Burgers equations) with both smooth and discontinuous solutions. The decay of isotropic turbulence is modeled within LES and the results are discussed. The solutions produced by the new schemes are compared with those based on the central difference scheme, the classical WENO5 scheme, and a hybrid scheme. The level of dissipation of the schemes is compared by analyzing their energy spectra. A similar comparison is made between the LES computations of the temporal evolution of a mixing layer, where the profiles of mean velocity and Reynolds stresses are considered in addition to the energy spectrum.

Comparison of Upwind and Symmetric WENO Schemes in Large Eddy Simulation of Basic Turbulent Flows

Comparison of Upwind and Symmetric WENO Schemes in Large Eddy Simulation of Basic Turbulent Flows

Novel TENO schemes with improved accuracy order based on perturbed polynomial reconstruction

Firstly, an improved TENO scheme based on perturbed polynomial reconstruction (TENO-P) is proposed for hyperbolic conservation laws on the foundation of fifth-order TENO5 scheme. Two different free-parameters are introduced into the reconstruction coefficients to eliminate the third-order errors in the candidate stencils and the fifth-order error in the global stencil, respectively. One-order of accuracy improvement is achieved by adaptively adjusting the values of the free-parameters, resulting a maximum sixth-order accuracy in the smooth stencil. Secondly, to improve the accuracy order of the fifth-order TENO5-THINC scheme, a sixth-order TENO-THINC-P scheme is proposed by combining the proposed TENO-P scheme for smooth regions with the THINC reconstruction for non-smooth discontinuities. Compared to the TENO5 scheme and the TENO5-THINC scheme, the numerical results show the dissipations of the TENO-P scheme and the TENO-THINC-P scheme are reduced, and their resolutions are significantly improved in resolving complicated flow structures with multiple discontinuities. Taking the computational efficiency and the numerical resolution into consideration, the TENO-P scheme is suitable for resolving vortex structures of turbulence while the TENO-THINC-P scheme is more suitable for resolving vortex structures near strong discontinuities. The derivation processes of the TENO-P scheme and the TENO-THINC-P scheme also demonstrate that the perturbed polynomial reconstruction has simple principle and good portability and expandability, so the idea of the TENO-P scheme and the TENO-THINC-P scheme can be easily extended to other improved very-high-order TENO schemes or WENO schemes.

A robust fifth order finite difference Hermite WENO scheme for compressible Euler equations

In this paper, we develop a robust fifth order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations following the HWENO with limiter (HWENO-L) scheme (Zhang and Zhao, 2023). The HWENO-L scheme reduced storage and increased efficiency by using restricted derivatives only for time discretizations. However, it cannot control spurious oscillations well when facing strong shocks since the derivatives are directly used in spatial discretizations without any restrictions. To address such an issue, our proposed HWENO scheme performs flux reconstructions in the finite difference framework without using the derivative value of a target cell, which can result in a simpler and more robust scheme. The resulting scheme is simpler while achieving fifth order accuracy, making it more efficient. Besides, numerically we find it is very robust for some extreme problems even without positivity-preserving limiters. The proposed scheme also inherits advantages of previous HWENO schemes, including arbitrary positive linear weights in flux reconstructions, compact reconstructed stencils, and high resolution. Extensive numerical tests are performed to demonstrate the fifth order accuracy, efficiency, robustness, and high resolution of the proposed HWENO scheme.

A smoothness indicators free third-order weighted ENO scheme for gas-dynamic Euler equations

To improve the shock-capturing capability of the third-order WENO scheme and enhance its computational efficiency, in this paper, we design a new WENO scheme independent of the local smoothing factor, WENO-SIF. The weight function of the WENO-SIF scheme is the segmentation function of the sub-stencil, which is guaranteed to achieve the desired accuracy at higher order critical points. WENO-SIF does not need to compute the smoothing factor during the computation, which effectively reduces the computational consumption. The present WENO-SIF is compared with WENO-JS and other WENO schemes for numerical experiments at one- and two-dimensional benchmark problems with a suitable choice of $$\lambda =0.13$$ . The results demonstrate that the WENO scheme can further improve the resolution of WENO-JS, achieve optimal accuracy at high-order critical points, and significantly reduce the computational consumption.

A sixth-order finite difference WENO scheme for non-linear degenerate parabolic equations: an alternative technique

PurposeThis paper aims to construct a sixth-order weighted essentially nonoscillatory scheme for simulating the nonlinear degenerate parabolic equations in a finite difference framework.Design/methodology/approachTo design this scheme, we approximate the second derivative in these equations in a different way, which of course is still in a conservative form. In this way, unlike the common practice of reconstruction, the approximation of the derivatives of odd order is needed to develop the numerical flux.FindingsThe results obtained by the new scheme produce less error compared to the results of other schemes in the literature that are recently developed for the nonlinear degenerate parabolic equations while requiring less computational times.Originality/valueThis research develops a new weighted essentially nonoscillatory scheme for solving the nonlinear degenerate parabolic equations in multidimensional space. Besides, any selection of the constants (sum equals one is the only requirement for them), named the linear weights, will obtain the desired accuracy.

Formation and Evolution of Cellular Structures in Wedge-Induced Oblique Detonations

Numerical simulation of an oblique detonation induced by a wedge is performed to investigate the formation and evolution of the oblique cellular detonation structure and the quantitative comparison of the cellular structure of a normal and an oblique detonation. The compressible reactive Euler equations are solved using the seventh-order WENO scheme on an adaptive mesh based on the open source program Adaptive Mesh Refinement in Object-oriented C++ (AMROC). The numerical results show that there are two sets of transverse waves, the left-running transverse waves (LRTW) and the right-running transverse waves (RRTW), which form the oblique cellular detonation structure. Although both sets of transverse waves are convected downstream, they propagate at almost the same relative velocity in the opposite direction. The LRTWs start in the transition zone because of the detonation instability. However, the RRTWs form due to the interaction between the deformed detonation front and the reflected shock wave from the wedge. For the same degree of overdrive, numerical simulations reveal that the characteristic cell size of an oblique detonation is almost the same as that of a normal detonation.

Enhancing the Resolution of Blade Tip Vortices in Hover with High-Order WENO Scheme and Hybrid RANS–LES Methods

The accurate prediction of blade tip vortices continues to be challenging for the investigation of rotor aerodynamics. In the current work, the blade tip vortex system of a Caradonna–Tung rotor in the hovering state is simulated on account of the framework of a high-order WENO scheme and hybrid RANS/LES method progressively. With the RANS method based on a fifth-order WENO–Roe scheme, the spatial resolution of wake age and capture the accuracy of tip vortices are improved significantly. However, the unsteadiness of the vortex system fails to be distinguished. Then, the DDES and IDDES methods, coupled with a fifth-order WENO–Roe scheme, are implemented to further enhance the resolution of the spatial turbulence. Compared with RANS, the improvement of spatial resolution is reflected in both the statistical and transient results with DDES. The identifiable vortex core distributions can be predicted at older wake ages. The asymmetrical characteristic of blade tip vortices is revealed along with the release of fluctuation while the predicted distribution of velocity profiles in the vortex core is enhanced. With the application of IDDES, the secondary turbulent structures are captured around the primary vortex core of blade tip vortices while the complex behaviors of the vortices are observed. However, the statistics of the averaged flow field are similar to DDES.

An efficient hybrid multi-resolution WCNS scheme for solving compressible flows

A hybrid finite difference fifth-order multi-resolution weighted compact nonlinear scheme (WCNS) is presented for simulating the inviscid and viscous compressible flows in this paper. This scheme borrows and improves the idea of the recently developed multi-resolution WENO scheme for node-to-midpoint nonlinear interpolation procedures of the WCNS scheme, which implements higher-order approximate interpolation at midpoints based on a series of nested stencils. This strategy can achieve the expected design accuracy under different linear weights, and can gradually decrease from high-order accuracy to first-order accuracy near the discontinuities. In addition, a hybridization strategy is proposed to further improve the computational efficiency of the presented multi-resolution WCNS scheme. The strategy does not depend on a specific problem to be solved, nor does it contain any problem-related artificial parameters. Finally, some benchmark examples are used to verify the performance of the presented scheme in terms of numerical accuracy, shock capture capability, robustness, computational efficiency, etc.

A robust common-weights WENO scheme based on the flux vector splitting for Euler equations

This paper proposes a common-weights weighted essentially non-oscillatory (Co-WENO) scheme for solving the Euler equations of gas dynamics. Different from the usual component-wise weighting methods, common-weights means that, on one global stencil, a set of weights is commonly shared by the split flux vector of Euler equations in one spatial dimension. The common-weights WENO scheme has two significant advantages. First, since only one set of weights is calculated and used for the split flux vector, the method has an improved computational efficiency. Second, for a stencil (or each cell on the stencil), the Co-WENO scheme keeps the same contribution on each component numerical flux in a hyperbolic system of equations. How to calculate the weights is one of the vital issues in developing this kind of Co-WENO schemes. In this paper, based on the flux vector split method, the product of density, pressure, and the split flux of energy equation(Γ±=ρpfE±) is proposed to calculate the common weights. This is based on the following considerations: (1) the density jumps at shocks and contact discontinuities; (2) the split energy flux contains the term of the third power of the velocity (for example, u3) and makes the resulting scheme has upwind characteristic; (3) the pressure always jumps at shocks, and it can help improve the stability in high speed flows, in which the kinetic energy is much larger than the internal energy. Numerical experiments also show that the proposed common-weights WENO scheme has good robustness and low numerical dissipation, and it can help suppress phase errors.

Fully discrete WENO with double entropy condition for hyperbolic conservation laws

ABSTRACTThis paper put forward a new fully discrete scheme construction method – double entropy condition solution formula method. With that, we turn the state-of-the-art semi-discrete WENO + RK scheme into a fully discrete scheme, which is named as Full-WENO. A major difficulty of this work is that we lack exact solution expressions for nonlinear equations in general cases. A feasible way we can go is to linearize equations and get quasi-exact solution formulas. The critical challenge is keeping both accuracy and efficiency in a scheme. Then, we get a class of new high-order schemes far better than traditional WENO schemes in the following aspects: (1) One-step to consistent high accuracy order in both space and time; (2) Resolution improves with the increasing CFL number; (3) Less CPU time and memory space, 1/s times of WENO with s-stage RK method in theory; (4) Excellent entropy condition satisfying property. Compared with our original work , the new method applies the more sophisticated WENO reconstruction and solves the resolution loss problems in multi-dimensional cases. The numerical tests show that the new scheme is equipped with the merits of high efficiency, high resolution and low dissipation, especially for long-time nonlinear problems.

Fifth-order well-balanced positivity-preserving finite difference AWENO scheme with hydrostatic reconstruction for hyperbolic chemotaxis models

We propose a fifth-order well-balanced positivity-preserving finite difference alternative WENO (AWENO) scheme with the affine-invariant WENO interpolation based on the Z-type nonlinear weights for the hyperbolic chemotaxis models. By using the techniques of source term reformulation, hydrostatic reconstruction of the interpolated conservative variables and modification of the numerical fluxes, the finite difference discretization is fifth-order and well-balanced. Moreover, the first-order interpolation with the Lax-Friedrichs (LF) flux and a reduced time step for the proposed discretized scheme has been shown to satisfy the density's positivity-preserving (PP) property. Thus, a simple positivity-preserving (PP) limiter conjugating the fifth-order hydrostatic reconstructed flux with the first-order positivity-preserving LF flux is introduced for extreme problems. Meanwhile, this improved approach strictly guarantees well-balanced property at the discrete level. Finally, one-, two-, and three-dimensional numerical examples are given to demonstrate the performance of the proposed AWENO scheme for this class of chemotaxis problems.

A Hybrid WENO Scheme for Steady Euler Equations in Curved Geometries on Cartesian Grids

A Hybrid WENO Scheme for Steady Euler Equations in Curved Geometries on Cartesian Grids

High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations

High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balanced property, we adopt the similar idea of WENO-XS scheme (Xing and Shu (2005) [30]) to balance the flux gradients and the source terms. The fluxes in the original equations are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balanced property. Instead of using a five-point stencil in the fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction. Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.

Locally order-preserving mapping for WENO methods

In our previous studies (Li and Zhong, 2021a; Li and Zhong, 2021b), the commonly reported issue that most of the existing mapped WENO schemes suffer from either losing high resolutions or generating spurious oscillations in long-run simulations of hyperbolic problems has been successfully addressed, by devising the improved mapped WENO schemes, namely MOP-WENO-X, where “X” stands for the version of the existing mapped WENO scheme. However, all the MOP-WENO-X schemes bring about the serious deficiency that their resolutions in the region with smoothly varying high-frequency waves are dramatically decreased compared to their associated WENO-X schemes. The purpose of this paper is to overcome this drawback. We first present the locally order-preserving (LOP) mapping. Then, by using a new proposed posteriori adaptive technique, we apply this LOP property to obtain the new mappings from those of the WENO-X schemes. The essential idea of the posteriori adaptive technique is to identify the global stencil in which the existing mappings fail to preserve the LOP property, and then replace the mapped weights with the weights of the classic WENO-JS scheme to recover the LOP property. We build the resultant mapped WENO schemes and denote them as LOP-WENO-X. The numerical experiments demonstrate that the resolutions in the region with high-frequency but smooth waves of the LOP-WENO-X schemes are similar to or even better than those of their associated WENO-X schemes and naturally much higher than the MOP-WENO-X schemes. Furthermore, the LOP-WENO-X schemes gain all the great advantages of the MOP-WENO-X schemes, such as attaining high resolutions and in the meantime preventing spurious oscillations near discontinuities when solving the one-dimensional linear advection problems with long output times, and significantly reducing the post-shock oscillations in the simulations of the two-dimensional problems with shock waves.

New finite difference unequal-sized Hermite WENO scheme for Navier-Stokes equations

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32] , [33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32] , [33] . Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24] , [25] . So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme.

WENO schemes with adaptive order for Hamilton–Jacobi equations

In this work, a fifth-order weighted essentially nonoscillatory scheme based on Legendre polynomials is constructed for simulating Hamilton–Jacobi (HJ) equations in a finite difference framework. The new reconstruction is a convex combination of a fourth-degree polynomial and two quadratic polynomials in WENO-Z fashion. This reconstruction uses the same six-point information as the original fifth-order WENO scheme [G.-S. Jiang and D. Peng, SIAM J. Sci. Comput. 21, 2126 (2000)] and could obtain smaller absolute truncation errors and the same accuracy order in the smooth region, while it has less computational time. A detailed analysis of the approximation order of the designed WENO scheme is prepared. Some benchmark tests in one-dimensional and multi-dimensional space are considered to display the capability of the new proposed scheme.

A New Third-Order Finite Difference WENO Scheme to Improve Convergence Rate at Critical Points

In this work, a new, improved third-order finite difference weighted essentially non-oscillatory scheme is presented for one- and two-dimensional hyperbolic conservation laws and associated problems. The parameter p which is regulate dissipation is introduced in the nonlinear weights in the framework of the conventional WENO-Z scheme, and the higher-order global smoothness indicator is obtained by the idea of Wang [Wang, Y. H., Y. L. Du, K. L. Zhao and L. Yuan. 2020. ‘A Low-dissipation Third-order Weighted Essentially Nonoscillatory Scheme with a New Reference Smoothness Indicator’. International Journal for Numerical Methods in Fluids. 92 (9): 1212–1234.], the sufficient condition of nonlinear weights is proved by using Taylor expansions. Finally, the value range of parameter p is obtained. The proposed scheme is verified to achieve the optimal order near critical points by linear convection equations with different initial values, and the high-resolution characteristic of the present scheme is proved on a variety of one- and two- dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with the other third-order WENO schemes.