Weighted Essentially Non-oscillatory Research Articles

Characteristic boundary conditions for magnetohydrodynamic equations

In the present study, a characteristic-based boundary condition scheme is developed for the compressible magnetohydrodynamic (MHD) equations in the general curvilinear coordinate system, which is an extension of the characteristic boundary scheme for the Navier–Stokes equations. The eigenstructure and the complete set of characteristic waves are derived for the ideal MHD equations in general curvilinear coordinates (ξ,η,ζ). The characteristic boundary conditions are derived and implemented in a high-order MHD solver where the sixth-order compact scheme is used for the spatial discretization. The fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme is also employed for the spatial discretization of problems with discontinuities. In our MHD solver, the fourth-order Runge–Kutta scheme is utilized for time integration. The characteristic boundary scheme is first verified for the non-magnetic (i.e., B=0) Sod shock tube problem. Then, various in-house test cases are designed to examine the derived MHD characteristic boundary scheme for three different types of boundaries: non-reflecting inlet and outlet, solid wall, and single characteristic wave injection. The numerical examples demonstrate the accuracy and robustness of the MHD characteristic boundary scheme.

Improved high-order high-resolution energy stable weighted essentially non-oscillatory plus scheme for shock/vortex problems

In the present study, a high-order high-resolution energy stable weighted essentially non-oscillatory plus (ESWENO-P) scheme was developed by improving the weighting function in the energy stable weighted essentially non-oscillatory (ESWENO) scheme. First, the weighting function was modified by introducing a new user-defined parameter. Then, the fine-tuning term in the weighted essentially non-oscillatory-Z plus (WENO-ZP) scheme was adopted to the modified weighting function. A parametric study was conducted to determine the value of the user-defined parameter included in the fine-tuning term. To validate the accuracy and the resolution of the present ESWENO-P scheme, well known one- and two-dimensional benchmark flow problems involving shocks and vortices were tested. It was found that, compared to the ESWENO, the present ESWENO-P scheme is more stable near strong discontinuities. It was also observed that, compared to the WENO-ZP, the present ESWENO-P scheme was less sensitive to the user-defined parameter included in the fine-tuning term. Overall, compared to the existing weighted essentially non-oscillatory (WENO) family schemes, the present ESWENO-P scheme consistently resolves flows with high resolution without much computational overhead.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.

Multi-level WENO schemes with an adaptive characteristic-wise reconstruction for system of Euler equations

The Weighted Essentially Non-Oscillatory (WENO) scheme is an accurate and robust reconstruction procedure to simulate compressible flows, especially in the presence of discontinuities in the solution. The recently introduced multi-level WENO schemes like WENO-ZQ (Zhu & Qiu, 2016), WENO-AO (Balsara et al. 2016), WENO-ZS (Zhu & Shu 2019) provide better resolution of the solution near discontinuities and accuracy for smooth problems than classical WENO scheme. Despite having better accuracy, a multi-level WENO scheme is not computationally efficient as it involves the computation of an extra smoothness indicator as compared to the classical WENO scheme. In this article, we have proposed two new algorithms of multi-level schemes for system of Euler equations, which retains the accuracy and resolution of multi-level schemes while being less costly. The first algorithm involves identifying the troubled-cells and employing the characteristic-wise WENO reconstruction in the troubled-cells to avoid spurious oscillations and component-wise WENO reconstruction in smooth regions. While in the second algorithm, the latter part is replaced by the component-wise finite difference upwind reconstruction. The important aspect of the new algorithm is the troubled-cell indicator, which has been developed utilizing the information of the extra smoothness indicator of multi-level schemes. The present algorithms are independent of order of multi-level WENO schemes. To demonstrate the new idea, we have considered fifth order WENO-AO(5,3), WENO-ZQ, and WENO-ZS schemes. Several 1D & 2D numerical experiments are performed to demonstrate the efficiency of proposed algorithms and its performance is compared with multi-level schemes. The proposed algorithms are efficient and take 15%–75% less computational time than multi-level schemes while retaining the advantageous features of multi-level schemes.

Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws

We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge–Kutta time discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to recognize undershoots/overshoots, as well as smooth solutions, is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D. In this context, we enforce global bounds and prove preservation of accuracy for the linear advection equation.

A finite difference Hermite RBF‐WENO scheme for hyperbolic conservation laws

AbstractOne of the best ideas for controlling the Gibbs oscillations in hyperbolic conservation laws is to apply weighted essentially non‐oscillatory (WENO) schemes. The traditional WENO and Hermite WENO (HWENO) schemes are based on the polynomial interpolation. In this research, a non‐polynomial HENO and HWENO finite difference scheme in order to enhance the local accuracy and convergence is proposed. The idea of the reconstruction in HWENO schemes is derived from traditional WENO schemes, with the difference that in traditional WENO schemes only function values are evolved and used, while in HWENO schemes both the values of the function and the first derivative of the function are evolved in time and used. The Hermite radial basis functions (RBFs) interpolation offers the flexibility to control the local error by optimizing the free parameter. The numerical results demonstrate that the developed Hermite RBF‐ENO and RBF‐WENO schemes enhance the local accuracy and give sharper solution profile than the Hermite ENO and WENO schemes based on the polynomial interpolation.

A neural network enhanced weighted essentially non-oscillatory method for nonlinear degenerate parabolic equations

In this paper, a new modification of the weighted essentially non-oscillatory (WENO) method for solving nonlinear degenerate parabolic equations is developed using deep learning techniques. To this end, the smoothing indicators of an existing WENO algorithm, which are responsible for measuring the discontinuity of a numerical solution, are modified. This is done in such a way that the consistency and convergence of our new WENO-DS (deep smoothness) method is preserved and can be theoretically proved. A convolutional neural network (CNN) is used and a novel and effective training procedure is presented. Furthermore, it is shown that the WENO-DS method can be easily applied to additional dimensions without the need to retrain the CNN. Our results are presented using benchmark examples of nonlinear degenerate parabolic equations, such as the equation of a porous medium with the Barenblatt solution, the Buckley–Leverett equation, and their extensions in two-dimensional space. It is shown that in our experiments, the new method outperforms the standard WENO method, reliably handles sharp interfaces, and provides good resolution of discontinuities.

Simulation of three-dimensional forced compressible isotropic turbulence by a redesigned discrete unified gas kinetic scheme

In this paper, we implemented the Boltzmann-equation-based mesoscopic model, developed recently by Chen et al. [“Inverse design of mesoscopic models for compressible flow using the Chapman–Enskog analysis,” Adv. Aerodyn. 3, 5 (2021)], to simulate three-dimensional (3D) forced compressible isotropic turbulence. In this model, both the Prandtl number and the ratio of bulk to shear viscosity can be arbitrary prescribed. The statistically stationary turbulent flow is driven by a large-scale momentum forcing in the Fourier space, with the internal heating due to the viscous dissipation at small scales being removed by a thermal cooling function. Under the framework of discrete unified gas kinetic scheme (DUGKS), a 3D direct numerical simulation code has been developed, incorporating a generalized Strang-splitting scheme. The weighted essentially non-oscillatory (WENO) scheme is used to increase local spatial accuracy in the reconstruction of particle distribution functions at the cell interface. A 3D discrete particle velocity model with a ninth-order Gauss–Hermite quadrature accuracy is used to ensure accurate evaluation of viscous stress and heat flux in the continuum regime. We simulate forced compressible isotropic turbulence at both low and high turbulent Mach numbers. A direct comparison is performed with the results obtained from a hybrid compact finite difference-WENO scheme solving directly the Navier–Stokes–Fourier system. The comparison validates our DUGKS code and indicates that DUGKS is a reliable and promising tool for simulating forced compressible isotropic turbulence. The work represents a first study to directly simulate forced compressible turbulence by a mesoscopic method based on the Boltzmann equation.

A high‐order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations

AbstractIn this research the numerical solution of nonlinear degenerate parabolic equations is investigated by a new sixth‐order finite difference weighted essentially nonoscillatory (WENO) based on exponential polynomials. In smooth regions, the new scheme, named as EPWENO6, can achieve the maximal approximation order while in critical points it does not lose its accuracy. In order to better approximation near steep gradients without spurious oscillations, the EPWENO6 scheme is designed by the exponential polynomials that are incorporated into the WENO reconstruction. In design of nonlinear weights that play a very important role in WENO reconstructions, a global smoothness indicator using generalized undivided differences is introduced. To analyze the convergence order of the EPWENO6 method in full detail, the Lagrange‐type exponential functions have been used. By comparing the EPWENO6 scheme and other existing WENO schemes for solving degenerate parabolic equations, it can be seen that the new scheme offers better results while its computational cost is less. To illustrate the effectiveness of the proposed scheme, a number of numerical experiments are considered.

A Semi-Lagrangian Computation of Front Speeds of G-equation in ABC and Kolmogorov Flows with Estimation via Ballistic Orbits

The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three dimensional periodic divergence free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. We give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as admissible trajectories for front speed estimates. To study the tightness of the estimates, we compute the front speeds of G-equation based on a semi-Lagrangian (SL) scheme with Strang splitting and weighted essentially non-oscillatory (WENO) interpolation. Time step size is chosen so that the Courant number grows sublinearly with the flow intensity. Numerical results show that the front speed growth rate in terms of the flow intensity may approach the analytical bounds from the ballistic orbits.

High order semi-implicit weighted compact nonlinear scheme for the full compressible Euler system at all Mach numbers

A high order semi-implicit weighted compact nonlinear scheme (WCNS) is presented to solve the full compressible Euler equations at all Mach numbers. To avoid the stringent acoustic CFL restriction for an explicit time discretization method, the pressure splitting methodology is introduced to split the Euler equations into nonstiff and stiff terms. The nonstiff and stiff terms are treated explicitly and implicitly, respectively, based on the high order IMEX Runge-Kutta time discretization method that is asymptotic preserving and asymptotically accurate in the zero Mach number limit. A fifth-order WCNS and fourth-order centered finite difference schemes with zero numerical viscosity are used for the spatial discretization. Numerical tests in one and two space dimensions are displayed to demonstrate the performance of the high order semi-implicit WCNS in compressible and incompressible regimes. Comparisons with the third-order explicit weighted essentially non-oscillatory (WENO) scheme are also made in order to better assess the proposed scheme.

An Improved 3rd-Order WENO Scheme Based on a New Reference Smoothness Indicator

In order to meet the requirement of high accuracy and high resolution in computational fluid dynamics (CFD), a new reference smoothness indicator was proposed to reduce the numerical dissipation of the classical 3rd-order weighted essentially non-oscillatory (WENO) scheme. The construction method is different from the classical WENO-Z scheme. It is obtained through the <i>L</i>²-norm approximation of the derivatives of the reconstruction polynomials of the whole global stencil, and the linear combination of the derivatives of the reconstruction polynomials on the candidate sub-stencils. With this calculation method, higher-order reference smoothness indicators can be obtained than the WENO-Z scheme. In addition, different reference smoothness indicators can be obtained by change of the value of free parameter $ \varphi$. A series of numerical examples prove the effectiveness of the reference smoothness indicator.

Modeling epidemic flow with fluid dynamics.

In this paper, a new mathematical model based on partial differential equations is proposed to study the spatial spread of infectious diseases. The model incorporates fluid dynamics theory and represents the epidemic spread as a fluid motion generated through the interaction between the susceptible and infected hosts. At the macroscopic level, the spread of the infection is modeled as an inviscid flow described by the Euler equation. Nontrivial numerical methods from computational fluid dynamics (CFD) are applied to investigate the model. In particular, a fifth-order weighted essentially non-oscillatory (WENO) scheme is employed for the spatial discretization. As an application, this mathematical and computational framework is used in a simulation study for the COVID-19 outbreak in Wuhan, China. The simulation results match the reported data for the cumulative cases with high accuracy and generate new insight into the complex spatial dynamics of COVID-19.

Improving Numerical Accuracy for the Viscous-Plastic Formulation of Sea Ice

Accurate modeling of sea ice dynamics is critical for predicting environmental variables and is important in applications such as navigating ice breaker ships. Research for both modeling and simulating sea ice dynamics is ongoing, with the most widely accepted model based on the viscous-plastic (VP) formulation introduced by Hibler in 1979. Due to its highly nonlinear features, this model is intrinsically challenging for computational solvers. In particular, sea ice simulations often significantly differ from satellite observations. This study therefore focuses on improving the numerical accuracy of the VP sea ice model. Since the poor convergence observed in existing numerical simulations stems from the nonlinear nature of the VP formulation, this investigation proposes using the celebrated weighted essentially non-oscillatory (WENO) scheme -- as opposed to the frequently employed centered difference (CD) scheme -- for the spatial derivatives in the VP sea ice model. We then proceed to numerically demonstrate that WENO yields higher-order convergence for smooth solutions, and that furthermore it is able to resolve the discontinuities in the sharp features of sea ice covers -- something that is not possible using CD methods. Finally, our proposed framework integrates a potential function method that utilizes the phase field method to naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations, resulting in a modified transport equations which includes additional forcing terms. Our method does not require post-processing, thereby avoiding the possible introduction of discontinuities and corresponding negative impact on the solution behavior. Numerical experiments are provided to demonstrate the efficacy of our new methodology.

A new fifth order finite difference WENO scheme to improve convergence rate at critical points

In the present work, we construct a new, improved version of fifth-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme with less dissipation to approximate solutions for one- and two-dimensional hyperbolic conservation laws and associated problems. The higher-order information presented for the classical WENO scheme is applied to construct a new global smoothness indicator, a composition of first derivatives of local smoothness indicators with their respective interpolation polynomials. The sufficient condition of nonlinear weights is satisfied by using Taylor expansions. We verify the proposed scheme through several benchmark test cases for scalar and system of hyperbolic conservation laws. We observe from the numerical simulations that the designed WENO scheme demonstrates high resolutions, fifth-order accuracy, and robustness at critical points as compared with other modern WENO schemes.

Study on the flow characteristics in the non-detonation reaction zones of wedge-induced oblique detonation transitions

Wedge-induced oblique detonation transitions are simulated numerically to investigate the non-detonation reaction zone structures that include induction and deflagration regions. Compressible reactive Euler equations are solved using a seventh-order Weighted Essentially Non-Oscillatory (WENO) scheme on an adaptive mesh. Quasi-one-dimensional primitive variables along the streamline of oblique detonation are used to study the flow characteristics inside the non-detonation reaction zones of different types of initiation structures. For a smooth transition, the fluid along the streamline of the entire non-detonation zone is in an expansion state. The gradual increase in the oblique shock angle during the pressure wave's upward movement is the primary reason for the reaction length reduction along the y-direction. For a weak abrupt-type transition, the fluid near the wedge surface is in a state of expansion. Along the y-direction, the gradual increase in the oblique shock angle and the compressibility of the fluid ahead of the main reaction zone are the two main reasons for the reduction in reaction length. For a strong-abrupt transition, the fluid near the wedge surface is still in a state of expansion. Along the y-direction, the sharp increase in the degree of compressibility ahead of the main reaction zone is the main reason for the reduction in reaction length. Comparing the reaction length of the theoretical analysis with the simulation results, it is found that the hypothesis of constant pressure combustion (CPC) along the inclined wedge surface is more appropriate than the hypothesis of constant volume combustion (CVC).

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Development of a WENO-type numerical solver for two-phase two-fluid six-equation model

Advanced discretization methods are pursued to improve accuracy of numerical solver for two-phase two-fluid six-equation model. The Weighted Essentially-Non-Oscillatory (WENO) method was a popular high-order discretization one and was successfully applied to many applications. However, development of the WENO-type numerical solver for two-phase two-fluid six-equation model was limited, which was partly due to the lack of analytical eigenvalues and eigenvectors. In previous work, the author derived an approximate analytical eigenvalues and eigenvectors for the two-phase flow model. The analytical eigenvalues and eigenvectors were formulated in a compact and structured way and were valid for arbitrary form of equation of state. The analytical eigenvalues and eigenvectors enable the development of a new WENO-type numerical solver for the two-phase flow model. In this work, a new WENO-type numerical solver is developed. Numerical tests show that the newly developed WENO-type solver works very well and is capable of capturing all the characteristic waves and sharp discontinuities. The order of accuracy study shows that the accuracy of WENO is at least third-order for a smooth solution.

Adaptive interpolation with maximum order close to discontinuities

Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to this method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique but the design of the weights in this case is more simple. Also, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.