Fifth-order WENO Research Articles

An Improved Fifth-Order WENO Scheme for Solving Hyperbolic Conservation Laws Near Critical Points

An Improved Fifth-Order WENO Scheme for Solving Hyperbolic Conservation Laws Near Critical Points

Numerical study of bifurcation structures in reflected shock-wave/laminar-boundary-layer interaction within an end-wall tube

Abstract The primary aim of this study is to analyze the unsteady characteristics of the interaction between a reflected shock wave and a laminar boundary layer in an end-wall shock tube. Our direct numerical simulations at shock Mach numbers of Ms=1.9, 2.5, and 3.5 using a fifth-order WENO scheme and three-step Runge-Kutta time integration method revealed inhomogeneity and anisotropy in the shock bifurcation. Surprisingly, the upper and lower bifurcated structures maintain a notably asymmetric flow during the forward propagation of the reflected shock bifurcation. The inverse flow in the bifurcation resembles a crooked earthworm structure, exhibiting high-frequency oscillations indicative of instability. However, at higher shock intensities, the earthworm transforms into a stable strip-like configuration, facilitating the entrapment of inverse flow and leading to rapid bifurcation height growth and early convergence. Additionally, isolated islands with high density, temperature, and pressure emerge in the transitional region behind the bifurcated shocks, due to variations in wave propagation speed.

Advancing instantaneous detonation prediction of condensed energetic materials based on high-order compressible multiphase fluid dynamics

The instantaneous detonation model (IDM) is widely used in simulating underwater explosions due to its efficiency and ability to ignore the detonation reaction process. In this study, we propose a new IDM to predict the fluid structure in the detonation zone of an RS211 explosive charge. This model is based on high-order solutions provided by the detonation shock dynamics model, where the spatial term is discretized using fifth-order WENO reconstruction in characteristic space and Lax–Friedrichs’s splitting and the temporal terms are discretized using a third-order TVD Runge–Kutta scheme. The interface motion is captured using the level-set method combined with MGFM, and a programmed burn model is provided to describe the generation and propagation of the detonation wave. The self-similarity of detonation wave propagation is validated, and the quantitative calculation formula of the instantaneous detonation model is obtained by averaging or curve fitting the dimensionless results. Consequently, the IDM of the RS211 charge is established using high-order polynomial approximations of the Taylor rarefaction zone and a constant static zone for 1D planar, cylindrical, and spherical RS211 charges. The application of the IDM involves direct mapping from the radial direction to the spatial structured grid for 1D planar, 2D cylindrical, and 3D spherical charges. Numerical results demonstrate that the IDM proposed in this paper shows good accuracy and high computational efficiency.

A Study of Fifth-Order WENO Reconstruction for Genuinely Two-Dimensional Convection-Pressure Flux Split Riemann Solver

A Study of Fifth-Order WENO Reconstruction for Genuinely Two-Dimensional Convection-Pressure Flux Split Riemann Solver

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

Distinct-positivity-preserving methods for fifth-order finite volume ALW-MR-WENO schemes with a bigger sufficient CFL number for extreme problems

In this paper, the new distinct-positivity-preserving (DPP) methods are proposed for fifth-order finite volume multi-resolution WENO schemes with adaptive linear weights (which are termed as the ALW-MR-WENO schemes) [J. Comput. Phys., 493 (2023), 112471] for solving some extreme problems on structured meshes. Associated WENO spatial reconstruction procedures only require two hierarchical unequal-sized central spatial stencils for achieving fifth-order accuracy in smooth regions and keeping the essentially non-oscillatory property in non-smooth regions. One redefines five cell averages after performing such spatial reconstructions and then designs one quartic polynomial and two cubic reconstruction polynomials based on them in one dimension. After that, a new detective process is used to examine the positivity of the point values of such cubic reconstruction polynomials at some checking points inside the target cell. If the negativity happens, a new over-determined compression limiter is carried out to ensure that the compressed polynomials can achieve the positivity in the whole target cell instead of only at some discrete points inside it. The quartic polynomial could use its four quadrature point values by using a four-point Gauss-Lobatto quadrature formula when performing numerical integrations. Since it is easy to rewrite the point values of one quartic polynomial with the point values of two cubic polynomials timing with associated linear weights, both cubic polynomials could use a four-point Gauss-Lobatto quadrature formula or a three-point Simpson's quadrature formula without losing any algebra precisions. By doing this novelty switching methodology between two quadrature formulas, it is the first time to theoretically prove and increase a sufficient CFL number from 1/12 to 1/7.2 for the fifth-order WENO schemes in one dimension. This methodology can be easily extended to multi-dimensions. Finally, the novelty DPP methods for fifth-order ALW-MR-WENO schemes with a larger practical CFL number of 0.6, instead of the sufficient CFL number of 1/7.2, are also simple and robust enough for some extreme problems without timely halving the CFL number to avoid the appearance of negative density and negative pressure.

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.

IDDES Investigation of Rotor Blade Tip Vortices in Hovering State with Tip Slots

The slots on a blade tip can effectively suppress the tip vortices by diffusing the core structure of a vortex and thus mitigate the adverse impacts of vortex interactions. In this paper, the effect of a slotted tip on vortex diffusion in the hovering state is investigated numerically by implementing the IDDES method using the fifth-order WENO scheme. The resulting wake flow field show that the number of secondary turbulent structures of so-called vortex worms near the slotted tip reduces significantly, indicating the weakening of vortex core rotation. It was found that the slotted tip reduced the peak value of velocity profiles in the vicinity of the vortex core by about 50%. The numerical result is close to the experimental conclusion. Correspondingly, the diameter of the vortex core is approximately doubled at each wake age relative to the baseline blade because the local flow passing through the slots will disintegrate the stable structure of the core region during the early stages of tip vortex generation. This promotes the turbulence mixture, resulting in the reduction in the vorticity value and rotating effect near the vortex core. Based on the overall results, the study of the slot parameters demonstrated that reducing the slot number or diameter will weaken the diffusing effect, causing a decrease in the flow rate.

Modeling COVID-19 Using a Modified SVIR Compartmental Model and LSTM-Estimated Parameters

This article presents a modified version of the SVIR compartmental model for predicting the evolution of the COVID-19 pandemic, which incorporates vaccination and a saturated incidence rate, as well as piece-wise time-dependent parameters that enable self-regulation based on the epidemic trend. We have established the positivity of the ODE version of the model and explored its local stability. Artificial neural networks are used to estimate time-dependent parameters. Numerical simulations are conducted using a fourth-order Runge–Kutta numerical scheme, and the results are compared and validated against actual data from the Autonomous Communities of Spain. The modified model also includes explicit parameters to examine potential future scenarios. In addition, the modified SVIR model is transformed into a system of one-dimensional PDEs with diffusive terms, and solved using a finite volume framework with fifth-order WENO reconstruction in space and an RK3-TVD scheme for time integration. Overall, this work demonstrates the effectiveness of the modified SVIR model and its potential for improving our understanding of the COVID-19 pandemic and supporting decision-making in public health.

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.

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.

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.

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.

Fifth order AWENO finite difference scheme with adaptive numerical diffusion for Euler equations

In solving hyperbolic conservation laws using the fifth-order characteristic-wise alternative WENO finite-difference scheme (AWENO) with Z-type affine-invariant nonlinear Ai-weights, the classical local Lax–Friedrichs flux (LLF) is modified with an adaptive numerical diffusion (ND) coefficient to form an adaptive LLF flux (LLF-M). The adaptive ND coefficient depends nonlinearly on the local scale-independent smoothness measures on the pressure, density, dilation, and vorticity. The feature sensor combines the well-known Durcos’ sensor on the dilation and vorticity of the velocity field and Jameson’s sensor on the density and pressure. Based on the measure of the feature sensor, the ND coefficient of the LLF-M flux is designed to transit smoothly and quickly from a set minimum to the maximum, the local spectral radius of the eigenvalues of the Jacobian of the flux. Hence, the modified AWENO scheme improves the resolution of small-scale structures due to a substantial reduction of excessive dissipation while capturing discontinuities essentially non-oscillatory (ENO-property). The performance of the improved AWENO scheme is validated in one- and two-dimensional benchmark problems with discontinuities and smooth small-scale structures. The results show that the new adaptive LLF-M flux improves resolution, captures fine-scale structures, and is robust in a long-term simulation.

A fifth-order finite difference HWENO scheme combined with limiter for hyperbolic conservation laws

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two-dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the nonlinear HWENO reconstruction which is the combination of a quintic polynomial with two quadratic polynomials, where the linear weights can be artificial positive numbers only if the sum equals one. And other fluxes in the derivative equations are approximated by high-degree polynomials directly. For the purpose of controlling spurious oscillations, an HWENO limiter is applied to modify the derivatives. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in the modified HWENO scheme (Z. Zhao et al. (2020) [27]), we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme, the proposed HWENO scheme is simpler, more accurate, efficient, and has better resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and is more efficient than the same order finite difference WENO scheme of Jiang and Shu (1996) [11] and WENO scheme with artificial linear weights of Zhu and Qiu (2016) [29]. Various benchmark numerical examples are presented to show the fifth-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.

Development of a WENO scheme based on radial basis function with an improved convergence order

In this article, we present a novel RBF-WENO scheme improving the fifth-order WENO techniques for solving hyperbolic conservation laws. The numerical flux is implemented by incorporating radial basis function (RBF) interpolation to cell average data. To do this, the classical RBF interpolation is amended to be suitable for cell average data setting. With the aid of a locally fitting parameter in the RBF, the RBF-WENO reconstruction attains an additional one order of improvement, resulting in the sixth-order of accuracy. In addition, on the purpose of detecting small scale structures and steep gradients more accurately, we present new smoothness indicators by devising a method of generalized undivided differences with exponential vanishing moments. Several experimental results are performed to confirm the effectiveness of the proposed WENO method.

Parallel Accelerated Fifth-Order WENO Scheme-Based Pipeline Transient Flow Solution Model

The water hammer phenomenon is the main problem in long-distance pipeline networks. The MOC (Method of characteristics) and finite difference methods lead to severe constraints on the mesh and Courant number, while the finite volume method of the second-order Godunov scheme has limited intermittent capture capability. These methods will produce severe numerical dissipation, affecting the computational efficiency at low Courant numbers. Based on the lax-Friedrichs flux splitting method, combined with the upstream and downstream virtual grid boundary conditions, this paper uses the high-precision fifth-order WENO scheme to reconstruct the interface flux and establishes a finite volume numerical model for solving the transient flow in the pipeline. The model adopts the GPU parallel acceleration technology to improve the program’s computational efficiency. The results show that the model maintains the excellent performance of intermittent excitation capture without spurious oscillations even at a low Courant number. Simultaneously, the model has a high degree of flexibility in meshing due to the high insensitivity to the Courant number. The number of grids in the model can be significantly reduced and higher computational efficiency can be obtained compared with MOC and the second-order Godunov scheme. Furthermore, this paper analyzes the acceleration effect in different grids. Accordingly, the acceleration effect of the GPU technique increases significantly with the increase in the number of computational grids. This model can support efficient and accurate fast simulation and prediction of non-constant transient processes in long-distance water pipeline systems.

High order weighted essentially non-oscillatory WENO-ZN schemes for hyperbolic conservation laws

In Shen et al. (2020), the authors have proposed a novel weighting method to construct the fifth-order WENO-ZN scheme to improve the accuracy at the second-order critical point. Its basic idea is that, the square of the fourth-order undivided difference on the global five-point stencil used by the fifth-order WENO scheme is suggested as the global smoothness indicator. To keep the ENO property and enhance robustness for resolving shock waves, the constant 1 used to calculate the un-normalized weights in the original WENO-Z schemes is replaced by an adaptive function, which can approach a small value if the global stencil contains a discontinuity or approach a large value if the solution is smooth enough. The fifth-order WENO-ZN scheme can obtain fifth order accuracy at both the first- and second-order critical points. However, limited by the smoothness indicators, the scheme cannot improve the convergence rate at the third-order and above critical points. In this paper, we extend the idea of the fifth-order WENO-ZN scheme to construct higher-order WENO-ZN schemes and investigate their performance. Numerical experiments show that the (2r−1)th-order (r≥3) WENO-ZN schemes are robust for capturing shock waves and can improve the accuracy order in smooth regions including the maximum (2r−4)th-order critical points.

A study of multidimensional fifth-order WENO method for genuinely two-dimensional Riemann solver

In this paper we conduct a research on the multidimensional fifth-order reconstruction method for Balsara's genuinely two-dimensional (2D) HLLE Riemann solver. This solver possesses the multidimensional effect successfully by solving the 2D Riemann problem at the cell vertexes. However, it is difficult to be applied to higher-order reconstruction technology. In order to meet the requirement of the reconstruction of the variables at the vertexes, we construct the general WENO reconstruction formula to complete the reconstruction procedure along different spatial directions in turns. The Gaussian quadrature is employed to achieve the fifth-order accuracy of the numerical flux integration at the interface. Typical Simpson rule in Balsara's 2D HLLE solver is no longer used, and each Gaussian point is regarded as an independent 2D Riemann problem to obtain the 2D Riemann flux directly. Due to the inevitable numerical overshoots in the process of the fifth-order reconstruction, an accuracy-preserving limiter is introduced to enhance the computational stability, which could maintain the numerical accuracy in the smooth regions and suppress the overshoots near discontinuities. Solution accuracy tests show that the multidimensional WENO method could achieve the fifth-order accuracy. Several solution quality tests are also presented, which indicate that the method proposed in this study performs well in all test cases and can capture discontinuities and complex flow structures accurately and efficiently.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.