Weighted Essentially Non-oscillatory Research Articles

A systematic analysis of three-dimensional Riemann problems for verification of compressible-flow solvers

Numerical simulation is a well-established way of analyzing compressible flows. Due to high computational demands of solvers for such flow problems, their verification is typically limited to two-dimensional (2D) cases. However, 2D simulations suppress fundamental three-dimensional (3D) aspects of the flow evolution in (compressible) turbulent flows or shock-bubble and shock-drop interactions. With increase of computational power, 3D simulations become more feasible for routine analyses. The verification of 3D simulation frameworks is often limited to transformations of lower dimensional test cases in the 3D space. There is a lack of strictly 3D reference test cases for gas dynamics. In this work, we present a set of genuine 3D Riemann problems in order to validate and verify numerical solvers for compressible flows. The problems are designed such that each octant’s constant initial data connects two neighboring states by an elementary wave only. The problem design is inspired by well-established 2D Riemann problems most prominently posted by Lax and Liu (1998). In contrast to the twenty published 2D cases, more than 300 distinct combinations can be found in 3D. We provide example solutions for the particularly interesting ones of these case combinations and show how the cases help to expose shortcomings of numerical solvers. We provide reference data from computations with an open-source compressible multiresolution flow solver. For the reference solutions, we employ the Harten-Lax-van Leer contact (HLLC) Riemann solver and a weighted essentially non-oscillatory (WENO) reconstruction stencil of fifth order. The reference solutions use an effective resolution of one billion cells. We additionally make the full compute pipeline of this work publicly available, so interested researchers may reproduce and extend the current work.

A general multi-objective Bayesian optimization framework for the design of hybrid schemes towards adaptive complex flow simulations

Achieving accuracy with underresolved simulation of complex compressible flows with multiscale flow structures is a challenge. Either the numerical dissipation or the resolution and thereby the numerical cost is impractically high. Also, in the design of numerical solvers, the application of a solver for specific flow classes is balanced by robustness allowing the study of a broad range of flows. In this study, we propose a hybrid fifth-order targeted essentially non-oscillatory (TENO5)-based scheme tailored to optimally simulate compressible flows with underresolved dilatational and vortical multiscale structures. For optimal design, three data-driven objectives are defined. A novel objective that derives from the numerical dissipation rate analysis is a key element to deal with underresolved complex flows in practical applications. The optimization process employs a multi-objective Bayesian optimization framework with an expected hypervolume improvement and three flow configurations representative for a broad range of two- and three-dimensional flows with genuine and non-genuine subgrid scales. The optimized hybrid scheme is validated by comparing with shock-capturing schemes of the weighted essentially non-oscillatory (WENO)- and TENO- families with flows of complex shock interactions, Kelvin-Helmholtz instabilities, shock-vortex interactions, vortical and turbulent flows

A hybrid lattice Boltzmann/immersed boundary method/finite-difference model for thermal fluid-solid interactions

Particulate flows are frequently observed in both industrial and natural environments, and their simulation becomes particularly challenging when heat transfer is involved. This study introduces a hybrid numerical approach, combining the lattice Boltzmann Method (LBM), Immersed Boundary Method (IBM), and Finite Difference (FD) energy solver to alleviate the limitations observed in double distributed lattice Boltzmann methods and classical Computational Fluid Dynamics (CFD). The model incorporates an enhanced Hermite polynomial-based central moments collision operator. This enhancement boosts stability and overcomes the Galilean-invariant issues observed in classical LBM. For heat transfer, a Finite-Difference technique is employed for resolving the energy equation, discretizing the advection term through a Weighted Essentially Non-Oscillatory (WENO) approach. The Boussinesq approximation handles buoyancy forces, while the direct-forcing IBM manages fluid-particle interaction. Validation across various test cases confirms the model's accuracy, showcasing its ability to account for variations in thermal properties like specific heat capacity Cp over a wide range. The model is further utilized to investigate a reactor containing 300 catalyst particles. In summary, this hybrid methodology is found to be a promising tool for comprehensively studying and designing particle-laden systems with thermal effects.

KFVM-WENO: A High-order Accurate Kernel-based Finite Volume Method for Compressible Hydrodynamics

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided.

High-order genuinely multidimensional finite volume methods via kernel-based WENO

In this paper a family of fully multidimensional kernel-based reconstruction schemes for use in finite volume methods (FVMs) will be presented. These methods are intended for use in shock dominated problems, and stability is achieved through a suitable adaptation of the Adaptive Order Weighted Essentially Non-Oscillatory (WENO-AO) method to the proposed kernel-based reconstruction schemes. There are a number of key difficulties in the design of high-order finite volume schemes which will be discussed and addressed. High (4th and 6th) order convergence will be demonstrated on smooth exact solutions of the ideal MHD equations. The very same scheme will then be applied to extremely stringent astrophysical benchmark problems.

Enhancing Accuracy in Numerical Simulations for High-Speed Flows: Integrating High-Order Corrections with Weighted Essentially Non-Oscillatory Flux

This study introduces a novel method to enhance numerical simulation accuracy for high-speed flows by refining the weighted essentially non-oscillatory (WENO) flux with higher-order corrections like the modified weighted compact scheme (MWCS). Numerical experiments demonstrate improved sharpness in capturing shock waves and stability in complex conditions like two interacting blast waves. Key highlights include simultaneous capture of small-scale smooth fluctuations and shock waves with precision surpassing the original WENO and MWCS methods. Despite the significantly improved accuracy, the extra computational cost brought by the new method is only marginally increased compared to the original WENO, and it outperforms MWCS in both accuracy and efficiency. Overall, this method enhances simulation fidelity and effectively balances accuracy and computational efficiency across various problems.

Shock wave induced by the collapse of a bubble cluster with each bubble distributed randomly

The phenomenon of bubble collapse is ubiquitous in both natural and industrial settings, often manifesting in the form of bubble clusters. Despite its prevalence, there has been limited research on the dynamics of bubble clusters. In order to gain a better understanding of shock wave emission, liquid jet propagation, and the resulting pressure loads during bubble cluster collapse, we have developed a high-fidelity numerical approach aimed at accurately capturing shock waves and phase interfaces. The interface compression technique is introduced into the compressible two-phase model to improve the sharpness of the phase interface. We solve the physical model using the BVD (Boundary Variation Diminishing) principle. Both the 5th order WENO (Weighted Essentially Non-Oscillatory) scheme and the interface capturing function THINC (Tangent of Hyperbola for INterface Capturing) are utilized to reconstruct primary variables at cell boundaries, effectively minimizing numerical dissipation. For further resolution improvement, we implement these numerical methods on the AMR (Adaptive Mesh Refinement) mesh. Our numerical results demonstrate that the distance between the wall and the bubble cluster, as well as the nearest bubble to the wall, significantly influence the wall peak pressure. However, they have a minimal impact on the overall evolution of the bubble cluster and the peak pressure induced in the flow field. Remarkably, the spatial distribution of bubbles and the total number of bubbles play a crucial role in shaping the dynamic behavior of the bubble cluster and the pressure loads exerted both on the solid wall and in the flow field. The crucial findings offer profound insights into a range of applications, including cavitation erosion, underwater explosions, and drug delivery.

FLEW: A DNS Solver for Compressible Flows in Generalized Curvilinear Coordinates

AbstractWe present FLEW, an in-house high-fidelity solver for direct numerical simulation (DNS) of turbulent compressible flows over arbitrary shaped geometries. FLEW solves the Navier–Stokes equations written in a generalized curvilinear coordinate system, in which the surface coordinates are non-orthogonal, whereas the third axis is normal to the surface. Spatial discretization relies on high-order finite-difference schemes. The convective terms are discretized using an hybrid approach, combining the near-zero numerical dissipation provided by central approximations with the robustness of weighted essentially non-oscillatory (WENO) schemes, required to capture shock waves. Central schemes are stabilized using a skew-symmetric-like splitting of convective derivatives, endowing the solver with the energy-preserving property in the inviscid limit. The maximum order of accuracy is eighth for central schemes (also used for viscous terms discretization) and seventh for WENO. The code is oriented to modern high-performance computing (HPC) platforms thanks to message passing interface (MPI) parallelization and the ability to run on graphics processing unit (GPU) architectures. Reliability, accuracy and robustness of the code are assessed in the low-subsonic, transonic and supersonic regimes. We present the results of several benchmarks, namely the inviscid Taylor–Green vortex, turbulent curved channel flow, transonic laminar flow over a NACA 0012 airfoil and turbulent supersonic ramp flow. The results for all configurations proved to be in excellent agreement with previous studies.

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes I: Steady flows

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization

The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated.

Development of a tokamak magnetohydrodynamic code with the discontinuous Galerkin and Weighted Essentially Non-Oscillatory methods

In this study, we present the development of a new initial-value magnetohydrodynamic (MHD) code for toroidal geometry using discontinuous Galerkin (DG) and Weighted Essentially Non-Oscillatory (WENO) methods. The code utilizes a triangular mesh based on the flux of the fixed boundary equilibrium in the poloidal plane, which is uniformly divided in the toroidal direction. By solving the conservative perturbed generalized Lagrange multiplier (GLM) MHD model, the code can simulate both ideal and resistive toroidal plasmas. We validate the code by performing the linear calculations of the internal kink mode and tearing mode, and our results show good agreement with calculations conducted using an eigen-value MHD code. Furthermore, nonlinear simulations of the resistive internal kink mode and tearing mode are also carried out. We discuss the challenges and limitations of the code, as well as the future efforts to improve its performance. The development of this new initial-value MHD code opens up new possibilities for studying and understanding the dynamics of toroidal plasmas, contributing to the advancement of fusion research.

Analysis of single-mode Richtmyer–Meshkov instability using high-order incompressible vorticity—streamfunction and shock-capturing simulations

Two- and three-dimensional simulation results obtained using a new high-order incompressible, variable-density vorticity–streamfunction (VS) method and data from previous ninth-order weighted essentially nonoscillatory (WENO) shock-capturing simulations [M. Latini and O. Schilling, “A comparison of two- and three-dimensional single-mode reshocked Richtmyer-Meshkov instability growth,” Physica D 401, 132201 (2020)] are used to investigate the nonlinear dynamics of single-mode Richtmyer–Meshkov instability using a model of a Mach 1.3 air(acetone)/SF6 shock tube experiment [J. W. Jacobs and V. V. Krivets, “Experiments on the late-time development of single-mode Richtmyer–Meshkov instability,” Phys. Fluids 17, 034105 (2005)]. A comparison of the density fields from both simulations with the experimental images demonstrates very good agreement in the large-scale structure with both methods but differences in the small-scale structure. The WENO method captures the small-scale disordered structure observed in the experiment, while the VS method partially captures such structure and yields a strong rotating core. The perturbation amplitude growth from the simulations generally agrees well with the experiment. The simulation bubble and spike amplitudes agree well at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. The predictions of nonlinear single-mode instability growth models are shown to agree with the simulation amplitudes at early-to-intermediate times but underpredict the amplitudes at later times in the nonlinear regime. Visualizations of the mass fraction and enstrophy isosurfaces, velocity and vorticity fields, and baroclinic vorticity production and vortex stretching terms from the three-dimensional simulations indicate that, with the exception of the small-scale structure within the rollups, the VS and WENO results are in good agreement.

Numerical simulation of water jet load induced by spherical bubble collapse under underwater explosion

In this study, a numerical method combining the HLLC solver and FV-WENO scheme is proposed to accurately simulate the behavior of multiphase compressible fluids and investigate the formation process of water jets. The HLLC solver approximates the conservation equations of the fluid by using numerical fluxes from the approximate Riemann problem, while the FV-WENO scheme improves the accuracy of the numerical solution by employing the weighted essentially non-oscillatory (WENO) reconstruction method. The combination of these methods allows for a more accurate simulation of the behavior of multiphase compressible fluids, leading to an accurate depiction of the formation process of water jets. By simulating the interaction between underwater shock waves from explosions and single or multiple spherical bubbles, we study the generation, development, and collapse of water jets during the bubble collapse process. By revealing the characteristics of the load during the formation of water jets and the interaction mechanisms between bubbles, our study provides support for further research on water jet phenomena. The formation process of water jets is of great significance for understanding the propagation of underwater shock waves and structural damage. Through this study, we delve into the mechanism of water jet formation, providing a new perspective on the propagation of underwater shock waves and structural damage. Our research results demonstrate that the application of the HLLC solver and FVWENO scheme in water jet simulation effectively improves the accuracy and precision of the numerical solution.

Analysis of spectral properties of WENO schemes in smooth regions

Weighted essentially non-oscillatory (WENO) schemes are a class of numerical schemes designed for solving problems with discontinuities. The spectral properties of them are hard to analyze because of their nonlinearities. A method of analyzing the average scheme in effect is proposed, by which we found that WENO schemes with a special kind of smoothness indicators can even obtain better spectral properties than their underlying linear schemes. This facilitates the design of WENO schemes with good spectral properties. The formulas of these smoothness indicators on sub-stencils have the same form except for different subscripts, which makes the nonlinear weights slightly close to the mean value. With this weight distribution, the resultant schemes behave like optimized upwind schemes with low dissipation and optimized dispersion properties. However, these WENO schemes may suffer from anti-dissipation and accuracy order degradation. Hence, they are suggested to be combined with the mapped WENO or the Z-type WENO methodology to address these problems. Some WENO schemes are taken as examples to illustrate the proposed methodology, and the results are in good agreement with those from the approximated dispersion relation (ADR) method.

A new third-order EXP-WENO scheme for Hamilton–Jacobi equations

The authors of this research paper have introduced a new method for solving Hamilton–Jacobi (HJ) equations called third-order weighted essentially nonoscillatory (WENO) method. This method uses exponential polynomials to construct numerical fluxes and smoothness indicators, which helps distinguish between singular and smooth regions more efficiently. The smoothness indicators are created using a finite difference operator that eliminates exponential polynomials. The numerical flux is constructed using an interpolation method based on exponential polynomials, which results in better outcomes around steep gradients. The new method maintains a high level of accuracy (i.e. three) in smooth regions, even near critical points. The authors have presented some numerical results to demonstrate the effectiveness of the new method and compared it with other WENO methods.

Two-step multi-resolution reconstruction-based compact gas-kinetic scheme on tetrahedral mesh

This paper presents the development of a third-order compact gas-kinetic scheme (GKS) for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme utilizes a time-dependent gas distribution function at a cell interface to not only calculate the fluxes needed for updating the cell-averaged flow variables but also to evaluate the flow variables at the cell interface. This leads to the evolution of cell-averaged gradients of flow variables. The success of this scheme heavily relies on the initial data reconstruction techniques, with an emphasis on their application to the tetrahedral mesh. Employing a conventional second-order unlimited least-square reconstruction directly on the cell-averaged flow variables of von Neumann neighbouring cells can introduce linear instability into the scheme. However, by using the updated cell-averaged gradients, the GKS with a third-order compact smooth reconstruction remains linearly stable under a large CFL number when applied to a tetrahedral mesh. To enhance the robustness of the high-order compact GKS for capturing a discontinuous solution, we propose a novel two-step multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. This innovative approach overcomes the stability issues associated with a second-order compact reconstruction by incorporating a pre-reconstruction step. Additionally, it simplifies the third-order non-linear reconstruction process by adding a single large stencil to those used in the second-order one. A high-order wall boundary condition is achieved by fusing the constrained least-square technique with the WENO procedure, where a quadratic element is used in the reconstruction for cells with a curved boundary. Numerical tests involving both the second-order and third-order compact GKS are presented, encompassing both inviscid and viscous flows at both low and high speeds. The results demonstrate that the proposed third-order compact scheme possesses robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.

Flux Vector Splitting Method of Weakly Compressible Water Navier-Stokes Equation and Its Application

Water is a weakly compressible fluid medium. Due to its low compressibility, it is usually assumed that water is an incompressible fluid. However, if there are high-pressure pulse waves in water, the compressibility of the water medium needs to be considered. Typical engineering applications include water hammer protection and pulse fracturing, both of which involve the problem of discontinuous pulse waves. Traditional calculation and simulation often use first-order or second-order precision finite difference methods, such as the MacCormark method. However, these methods have serious numerical dissipation or numerical dispersion, which hinders the accurate evaluation of the pulse peak pressure. In view of this, starting from the weakly compressible Navier–Stokes (N-S) equation, this paper establishes the control equations in the form of flux, derives the expressions of eigenvalues, eigenvectors, and flux vectors, and gives a new flux vector splitting (FVS) formula by considering the water equation of state. On this basis, the above flux vector formula is solved using the fifth-order weighted essentially non-oscillatory (WENO) method. Finally, the proposed FVS formula is verified by combining the typical engineering examples of water hammer and pulse fracturing. Compared with the traditional methods, it is proved that the FVS formula proposed in this paper is reliable and robust. As far as we know, the original work in this paper extends the flux vector splitting method commonly used in aerodynamics to hydrodynamics, and the developed model equation and method are expected to play a positive role in the simulation field of water hammer protection, pulse fracturing, and underwater explosion.

Exponential approximation space reconstruction weighted essentially nonoscillatory scheme for dispersive partial differential equations

In this work, we construct a fifth‐order weighted essentially non‐oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third‐order derivative formulation is developed directly using WENO spatial reconstruction procedure, and third‐order TVD Runge–Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the characteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the construction of conservative flux approximation, smoothness indicators, nonlinear weights, and verified that the proposed scheme provides the required fifth convergence order. One‐ and two‐dimensional numerical examples are presented to support the theoretical claims.

Hybrid conservative central/WENO finite difference scheme for two-dimensional detonation problems

In the present work, a hybrid finite difference scheme is proposed and tested for the study of two-dimensional detonation problems. The hybrid scheme consists of a fifth-order weighted essentially nonoscillatory (WENO) method for discontinuous regions and a fourth or sixth-order robust conservative central finite difference scheme for the remaining domain. An arbitrary shock sensor is used to identify the discontinuities. The proposed shock sensor is based on the absolute value of the density gradient with an arbitrary threshold computed at each time step using a fast image segmentation technique. The sensor is complemented with a Ducros sensor to avoid the selection of vortices as discontinuities. The hybrid scheme is tested for several benchmark examples, including a two-dimensional transverse detonation wave, detonation wave diffraction, and a multiple obstacles test case. The obtained results show that the proposed sensor detects discontinuities adequately and the hybrid schemes give the same results as the fifth-order WENO in faster computational time.