Weighted Essentially Non-oscillatory Scheme Research Articles

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows.

A new fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow

A new fifth-order alternative finite difference multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve the hyperbolic conservation laws in this paper. With the application of the original finite difference multi-resolution WENO schemes (Zhu and Shu, 2018; Zhu and Shu, 2020), a series of unequal-sized central spatial stencils are adopted to perform the WENO procedures, but the difference is that the methodology adopted in this paper is directly based on the point values of the solution rather than on the flux values. The proposed new multi-resolution WENO scheme can inherit many advantages of the original high order schemes, such as the arbitrary choice of the linear weights, the maintenance of the essentially non-oscillatory property in the vicinity of strong discontinuities, the smaller L1 and L∞ truncation errors than that of the same order classical WENOJS schemes (Balsara et al., 2016; Jiang and Shu, 1996) in smooth regions, and better convergence properties of the residue when solving some steady-state problems. Compared with the original multi-resolution WENO schemes, the presented WENO scheme also has some advantages. Firstly, an arbitrary monotone flux can be used in this framework, while the original method is only suitable for smooth flux splitting technique. Secondly, it has smaller L1 and L∞ truncation errors than that of the original same order multi-resolution WENO scheme. And finally, it avoids some time-consuming intermediate processes in the original multi-resolution WENO reconstruction procedures, thus resulting in less calculation time under the same conditions. Some inviscid and viscous numerical examples are provided to verify the superior performance of the new fifth-order alternative finite difference multi-resolution WENO scheme.

A New Sixth-Order Finite Difference WENO Scheme for Fractional Differential Equations

In this paper, we propose a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. After splitting the Caputo fractional derivative of order $$\alpha $$ $$(1<\alpha \le 2)$$ into a weakly singular integral and a classical second derivative, the classical Gauss–Jacobi quadrature is used to solve the weakly singular integral and a new spatial WENO-type reconstruction methodology is proposed to approximate the second derivative. There are two advantages of the new WENO scheme: the first is that the linear weights can be any positive numbers on condition that their summation equals one, and the second is its simplicity in the engineering applications. The new WENO reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on three unequal-sized spatial stencils in a traditional WENO fashion. This new sixth-order WENO scheme uses smaller number of cell average information than that of the same order classical WENO schemes and could eliminate non-physical oscillations near strong discontinuities when solving the fractional differential equations. Some benchmark examples are given to demonstrate the efficiency, robustness, and good performance of this new finite difference WENO scheme.

A new type of WENO scheme in SPH for compressible flows with discontinuities

In the past few decades, the weighted essentially non-oscillatory (WENO) scheme has been widely used in mesh-based methods and has achieved great success, but its application to the smoothed particle hydrodynamics (SPH) is still limited. In this paper, a simple and accurate implementation of the WENO scheme to the SPH method is proposed for solving compressible flows with discontinuities and small-scale structures. In the proposed scheme, several equal spacing points along the line joining two interacting particles are selected to constitute candidate stencils. However, due to the Lagrangian characteristic of SPH, some points may be lost. To solve this problem, we first search the particles closest to these missing points, and then the first-order Taylor series expansion is used to obtain the corresponding primitive variables of these points. After that, through the WENO strategy, the left and right states at the interface between two interacting particles are reconstructed. Finally, the inter-particle interactions are determined by using Roe’s approximate Riemann solver. Several numerical tests show that the proposed WENO-SPH method is robust and able to accurately capture shockwaves, and benefiting from the low-dissipation property, it also has a good performance in resolving small-scale structures in flows.

Comparison Study of the Fifth-Order WCNS and WENO Scheme In the Truncation Error, Dissipation and Dispersion Terms

This paper makes a comparison study of the fifth-order weighted compact nonlinear scheme (WCNS) and weighted essentially non-oscillatory (WENO) scheme in the truncation error, dissipation and dispersion terms. Truncation error analysis on the WCNS shows that numerical discretization leads to dissipation and dispersion terms. The Fourier spectral analysis methods is used to analyze the dissipative and dispersive features of both schemes.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Fourth-order gas-kinetic scheme for turbulence simulation with multi-dimensional WENO reconstruction

In this paper, a fourth-order multi-dimensional weighted essentially non-oscillatory (WENO) reconstruction is developed, which is aiming at the complicated flows. In corporate with two-stage fourth-order temporal discretization, a high-order gas-kinetic scheme is proposed for the direct numerical simulation (DNS) and large eddy simulation (LES) of turbulence. In such WENO scheme, a simple strategy of selecting stencils is adopted and the topology independent linear weights are used. The fourth-order reconstruction improves the order of accuracy of the previous third-order WENO reconstruction [Computers and Fluids 198 (2020) 104401], which is too dissipative for the simulation of turbulent flows. Compared with the classical dimension-by-dimension WENO reconstruction, the order of accuracy can be also achieved for any non-uniform and curvilinear meshes in the finite volume framework. Numerical examples, including the classical turbulence cases, are provided to illustrate the good performance of such WENO scheme for turbulence simulation. More importantly, the fourth-order WENO scheme is robust and works well from the subsonic to hypersonic flows. In the future, the current scheme will be extended to the genuinely unstructured meshes for more complicated turbulent flows.

Application of approximate dispersion-diffusion analyses to under-resolved Burgers turbulence using high resolution WENO and UWC schemes

This paper presents a space-time approximate diffusion-dispersion analysis of high-order, finite volume Upwind Central (UWC) and Weighted Essentially Non-Oscillatory (WENO) schemes. We perform a thorough study of the numerical errors to find a-priori guidelines for the computation of under-resolved turbulent flows. In particular, we study the 3-rd, 5-th and 7-th order UWC and WENO reconstructions in space, and 3-rd and 4-th order Runge-Kutta time integrators. To do so, we use the approximate von Neumann analysis for non-linear schemes introduced by Pirozzoli. Moreover, we apply the “1% rule” for the dispersion-diffusion curves proposed by Moura et al. [41] to determine the range of wavenumbers that are accurately resolved by each scheme. The dispersion-diffusion errors estimated from these analyses agree with the numerical results for the forced Burgers' turbulence problem, which we use as a benchmark. The cut-off wavenumbers defined by the “1% rule” are evidenced to serve as a good estimator of the beginning of the dissipation region of the energy cascade and they are shown to be associated to a similar level of dissipation, with independence of the scheme.Finally, we show that WENO schemes are more diffusive than UWC schemes, leading to stable simulations at the price of more dissipative results. It is concluded both UWC and WENO schemes may be suitable schemes for iLES turbulence modeling, given their numerical dissipation level acting at the appropriate wavenumbers.

A New Hybrid WENO Scheme on a Four-Point Stencil for Euler Equations

To enhance the performance of the third-order weighted essentially non-oscillatory (WENO) scheme on the four-point stencil, a new hybrid central-type scheme is proposed in this paper. Developments are conducted in the following two aspects. First, a new fourth-order central WENO scheme, named as WENO4-LC, is devised for shock capturing, and a low dissipative third-order grid-centered upwind scheme is adopted for the smooth region. Second, a discontinuity indicator is extended to the four-point stencil version, which is simpler than its counterpart in our former article (Guo et al. in J Sci Comput 83, 28, 2020), for switching between the linear and nonlinear branch of the hybrid scheme. Numerical tests of the Euler and Navier–Stokes benchmarks show that the new indicator is good on detecting the discontinuities. Moreover, the performances of the central-type WENO4-LC and its corresponding hybrid scheme have obvious improvements compared with those of WENO3-JS and WENO3-L, and even behave slightly better than classical WENO5-JS on resolving the fluid structures. Meanwhile, the hybrid scheme is efficient which only costs 76% CPU time of WENO3-JS and about 58% of WENO5-JS.

A third-order WENO scheme based on exponential polynomials for Hamilton-Jacobi equations

In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.

Very high order WENO schemes using efficient smoothness indicators

Very high order weighted essentially non-oscillatory (WENO) schemes can obtain more accurate results than those relatively low order WENO schemes for many problems. However, as the order increases, the candidate sub-stencils of WENO schemes will contain more node points. Then, the classical smoothness indicators become very complex and need much more floating point operations. Although they could be written in a more succinct form which is a sum of perfect squares, only a little improvement of the computational efficiency is obtained. Furthermore, for the power parameter p in the nonlinear weights of the previous (2r-1)th order WENO schemes, only the range of choices is given and the optimal value could be problem dependent. In this article, another class of very high order WENO schemes is constructed by using more efficient smoothness indicators. Compared with very high order WENO schemes using classical smoothness indicators, the new ones are more time efficient, and less error prone for programming. Furthermore, they have the same approximate dispersion relation (ADR) as their underlying linear schemes. And a more important advantage is that they have both good behavior near critical points and good stability near discontinuities. For Euler equations, a predetermined order reduction technique including two inequalities is proposed to deal with the problems caused by interactions between discontinuities of different characteristic fields. One inequality is constructed using this kind of smoothness indicators of pressure to suppress the non-physical oscillations. And the other is proposed according to the governing equation of the pressure to prevent the emergence of negative pressure. Several test problems are presented to demonstrate the good properties of the proposed WENO schemes.

超音速探测器-刚性盘-缝-带型降落伞系统的大涡模拟研究

The influences of the aerodynamic deceleration performance and the flow field structure characteristics of the capsule-rigid disk-gap-band parachute system at an initial Mach number of 2.0 and different block-structured adaptive mesh refinement were studied. In the unsteady compressible fluid, the hybrid WENO (weighted essentially non-oscillatory) and TCD (tuned center difference) schemes were used to simulate the shock wave and the smooth continuous flow field. The large-eddy simulation method with the stretched vortex subgrid model was used to deal with the turbulence. The results show that, at a low resolution of the block-structured adaptive mesh refinement, it is difficult to accurately simulate the important aerodynamic drag coefficient and capture the flow field characteristics of the parachute system. Subsequently, the convergence of the adaptive mesh refinement of the flow field was verified.

A fourth-order entropy condition scheme for systems of hyperbolic conservation laws

Based on the analysis of the entropy condition scheme formulation, the accuracy order comes from initial value interpolation and flux reconstruction. Following the limiters of the traditional second-order Total Variation Diminishing scheme, higher accuracy order and non-oscillatory nature are retained with a newly proposed smoothness threshold method. Then, the scheme using the solution formula method in Dong et al. [(2002). High-order discontinuities decomposition entropy condition schemes for Euler equations. CFD Journal, 10(4), 563–568] is generalized to fully-discrete fourth-order accuracy, which retains the advantages of former schemes, i.e. it is a fully-discrete, one-step scheme with no need to perform with a Runge–Kutta process in time; an entropy condition is satisfied with no need of an entropy fix with artificial numerical viscosity; and an exact solution is achieved for linear cases if CFL→1. Numerical experiments are given with a 1D scalar equation for a shock-tube problem, a blast-wave problem, and Shu–Osher problem, a 2D Riemann problem, a double Mach reflection problem and a transonic airfoil flow problem for NACA0012. All tests are compared with a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme. Numerical experiments and efficiency comparisons show that the efficiency of the new fourth-order scheme is superior to the fifth-order WENO scheme.

High Order Finite Difference Multi-resolution WENO Method for Nonlinear Degenerate Parabolic Equations

In this paper, we propose a new finite difference weighted essentially non-oscillatory (WENO) scheme for nonlinear degenerate parabolic equations which may contain non-smooth solutions. An alternative formulation is designed to approximate the second derivatives in a conservative form. In this formulation, the odd order derivatives at half points are used to construct the numerical flux, instead of the usual practice of reconstruction. Moreover, the multi-resolution WENO scheme is designed to circumvent the negative ideal weights and mapped nonlinear weights that appear when applying the standard WENO idea. We will describe the scheme formulation and present numerical tests for one- and two-dimensional, demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

A friction interface model for multi-material interactions in a Eulerian framework

A friction model for hyper-elastic solid materials was proposed for the simulation of complex shear impacts in a Eulerian framework. The interfacial status of multi-material interactions was obtained in Harten, Lax, and van Leer discontinuity (HLLD) Riemann solver. The inverse deformation gradient tensor was utilized in governing equations to describe the shape change of hyper-elastic solid materials. The level set method (LSM) was used to identify multiple materials in computational domain and the interface boundary condition was applied using the real ghost fluid method (RGFM). A uniform eigensystem of hyperbolic equations was derived using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO) scheme. Six numerical tests of one-dimensional problems were used to illustrate the robustness of the 5th order WENO scheme. The proposed friction model was evaluated for impact cases in both one and two dimensions with three-dimensional components. The redistribution of strain and kinetic energy during solid-solid interactions was represented using the proposed friction model, while both ‘slip’ and ‘stick’ interface showed unphysical approximations. We believe that the friction interface model completes the theory of multi-material interaction and provide a more appropriate way to model the complex dynamic behavior in Eulerian framework for solid materials.

Direct numerical simulations of turbulent reacting flows with shock waves and stiff chemistry using many-core/GPU acceleration

Compressible reacting flows may display sharp spatial variation related to shocks, contact discontinuities or reactive zones embedded within relatively smooth regions. The presence of such phenomena emphasizes the relevance of shock-capturing schemes such as the weighted essentially non-oscillatory (WENO) scheme as an essential ingredient of the numerical solver. However, these schemes are complex and have more computational cost than the simple high-order compact or non-compact schemes. In this paper, we present the implementation of a seventh-order, minimally-dissipative mapped WENO (WENO7M) scheme in a newly developed direct numerical simulation (DNS) code called KAUST Adaptive Reactive Flows Solver (KARFS). In order to make efficient use of the computer resources and reduce the solution time, without compromising the resolution requirement, the WENO routines are accelerated via graphics processing unit (GPU) computation. The performance characteristics and scalability of the code are studied using different grid sizes and block decomposition. The performance portability of KARFS is demonstrated on a variety of architectures including NVIDIA Tesla P100 GPUs and NVIDIA Kepler K20X GPUs. In addition, the capability and potential of the newly implemented WENO7M scheme in KARFS to perform DNS of compressible flows is also demonstrated with model problems involving shocks, isotropic turbulence, detonations and flame propagation into a stratified mixture with complex chemical kinetics.

A low dissipation finite difference nested multi-resolution WENO scheme for Euler/Navier-Stokes equations

In this paper, a low dissipation fifth-order finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) scheme is presented for solving Euler/Navier-Stokes equations in multi-dimensions on structured meshes. In the reconstruction procedures of this finite difference WENO, a series of nested unequal-sized central stencils are used to design spatial reconstruction polynomials based on the local orthogonal Legendre basis functions. So this new WENO scheme could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property by gradually degrading from the fifth-order to third-order, and ultimately to the first-order accuracy near strong discontinuities. Compared with the classical fifth-order finite difference WENO scheme [18], the results show that this new nested multi-resolution WENO scheme could obtain smaller L1 and L∞ errors for solving smooth problems based on the same mesh level. Moreover, the presented WENO scheme also has smaller numerical dissipation without introducing any dissipation preserving technique, and can capture more subtle flow structures for solving viscous flow problems. The proposed multi-resolution WENO scheme can be easily extended to finite volume framework owing to its linear weights can be set as any positive numbers with only one requirement that their summation equals to one. Therefore, the new WENO scheme is more suitable for solving viscous problems containing both discontinuities and complex smooth structures, and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark inviscid extreme and viscous examples are illustrated to verify the good performance of this nested multi-resolution WENO scheme.

An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary

In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f(u) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f′(u) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32]. The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

&lt;p style='text-indent:20px;'&gt;In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [&lt;xref ref-type="bibr" rid="b21"&gt;21&lt;/xref&gt;], with a high order RKEI method [&lt;xref ref-type="bibr" rid="b7"&gt;7&lt;/xref&gt;], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.&lt;/p&gt;

On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high‐speed compressible flows

SummaryHigh‐speed compressible turbulent flows typically contain discontinuities and have been widely modeled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1.5 and Reτ = 222 is conducted using OpenSBLI, a high‐order finite difference computational fluid dynamics framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against the WALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The sixth‐order adaptive TENO scheme is found to produce comparable results to those obtained with nondissipative fourth‐ and sixth‐order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model.