High Order Weighted Essentially Non-oscillatory Schemes Research Articles

A new family of downwind-limited, scale-invariant WENO schemes with optimal accuracy

Despite improvement over decades, there are still at least two key issues not fully solved in Weighted Essentially Non-oscillatory (WENO) schemes. Firstly, most WENO schemes hardly satisfy the scale-invariant property, which undermines their robustness and accuracy when simulating flows involving different scales such as reactive multi-component flows and plasma. Secondly, most of existing Essentially Non-oscillatory (ENO) constraints are not sufficient and necessary conditions to achieve optimal accuracy. Therefore, in this work, we design a new family of WENO schemes based on a three-cell stencil to address these two issues. The new WENO non-linear weights are designed as bounded asymptotic functions which are dependent on cell Normalised Volume Integrated Average (NVIA) values and a shape parameter θ. Thus, the new WENO weighting functions are scale-independent. The shape parameter θ controls the errors between the nonlinear weight and the ideal weight. As the value of θ increases, the new weighting functions become more accurate but also become more likely to produce numerical oscillations. Thus, we propose a downwind limiting method as a supplementary constraint of ENO property. The maximum value of θ which corresponds to the optimal accuracy of a weighting function can then be determined. The accuracy analysis via the standard Taylor expansion at critical points reveals that the proposed weighting functions outperform classical WENO schemes. The accuracy, scale-invariant property and ENO property of proposed schemes are further validated through benchmark tests, which shows the superior performance of the new schemes in comparison with existing three-cell-based WENO schemes. Thus, this work proposes a new direction to construct scale-invariant WENO weighting functions with optimal accuracy, which can potentially be extended to higher-order WENO schemes.

Efficient implementation of high-order WENO schemes with sharing function for solving Euler equations

Due to the high-order accuracy and essentially non-oscillatory (ENO) property, the weighted ENO (WENO) schemes have a wide range of successful applications. The component-wise reconstruction WENO (CP WENO) scheme for fluxes or variables has simple formulations but it may produce numerical oscillations near discontinuities when solving the Euler equations. Although the characteristic-wise reconstruction WENO (CH WENO) scheme can reduce such oscillations, it involves too many characteristic projection operations. In this paper, first, we introduced a sharing function to indicate the discontinuities in the Euler equations and then constructed new adaptive characteristic-wise WENO (Ada-WENO) scheme and common-weights WENO (Co-WENO) scheme with this function. Several one and two dimensional problems are used to test the performances of Ada-WENO and Co-WENO. Numerical results show that, Ada-WENO can reduce the computational cost of CH WENO while maintaining its oscillation-free property since it only switches from CP WENO to CH WENO near discontinuities, and Co-WENO can reduce the cost and oscillations of CP WENO, but it may still generate few oscillations.

A modified high-order symmetrical WENO scheme for hyperbolic conservation laws

Abstract This paper designs a modified weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws. Using the switching principle based on inflection points, the new scheme automatically adapts between linear upwind and WENO schemes. If there is at least one inflection point in the largest stencil available for reconstruction, a symmetrical WENO (SWENO) scheme is considered for the reconstruction of the numerical flux; otherwise the numerical flux is directly approximated by the reconstruction polynomial. By comparing the new scheme introduced in this paper with the classical WENO scheme and another improved scheme that has been proposed recently D. Chai, G. Xi, Z. Sun, Z. Wangand Z. Huang,Comput. Fluids, vol. 170, pp. 176–186, 2018), we can point out the robustness and better efficiency of this scheme. To examine and explain the features of the new scheme, a number of examples such as Euler equations have been considered.

On high-order numerical schemes for viscous relativistic hydrodynamics through the Kelvin–Helmholtz instability

ABSTRACT This work assesses the dissipative properties of high-order numerical methods for relativistic hydrodynamics. A causal theory of physical dissipation is included within a finite volume high-resolution shock-capturing framework based on the Israel–Stewart theory to study high-order WENO (weighted-essentially non-oscillatory) schemes for simulating the relativistic Kelvin–Helmholtz instability. We provide an estimation of the numerical dissipation of high-order schemes based on results obtained both with and without physically resolved dissipation and determine an empirical relationship between the numerical dissipation and the grid resolution. We consider the appearance of secondary flow features within the evolution of the Kelvin–Helmholtz instability and determine that they are numerical artifacts — this is partly based on arguments presented in terms of a frame-dependent form of the relativistic Reynolds number. There is a potential advantage of using high-order schemes in terms of their accuracy and computational cost on coarser grid resolutions when directly compared to low-order schemes on a fine grid in the presence of physical viscosity. It is possible to find reasonable agreement between numerical results that employ lower-order schemes using a finer grid resolution and results that employ higher order schemes at a coarser grid resolution when sufficient viscosity is present. Overall, the present analysis gives an insight into the numerical dissipation of high-order shock-wave capturing schemes which can be relevant to computational studies of astrophysical phenomena in the relativistic regime. The results presented herein are problem and scheme-dependent and serve to highlight the different roles of numerical and physical dissipation.

Preconditioned Jacobian-free Newton–Krylov fully implicit high order WENO schemes and flux limiter methods for two-phase flow models

Motivated by the high-resolution properties of high-order Weighted Essentially Non-Oscillatory (WENO) and flux limiter (FL) for steep-gradient problems and the robust convergence of Jacobian-free Newton-Krylov (JFNK) methods for nonlinear systems, the preconditioned JFNK fully implicit high-order WENO and FL schemes are proposed to solve the transient two-phase two-fluid models. Specially, the second-order fully-implicit BDF2 is used for the temporal operator and then the third-order WENO schemes and various flux limiters can be adopted to discrete the spatial operator. For the sake of the generalization of the finite-difference-based preconditioning acceleration methods and the excellent convergence to solve the complicated and various operational conditions, the random vector instead of the initial condition is skillfully chosen as the solving variables to obtain better sparsity pattern or more positions of non-zero elements in this paper. Finally, the WENO_JFNK and FL_JFNK codes are developed and then the two-phase steep-gradient problem, phase appearance/disappearance problem, U-tube problem and linear advection problem are tested to analyze the convergence, computational cost and efficiency in detailed. Numerical results show that WENO_JFNK and FL_JFNK can significantly reduce numerical diffusion and obtain better solutions than traditional methods. WENO_JFNK gives more stable and accurate solutions than FL_JFNK for the test problems and the proposed finite-difference-based preconditioning acceleration methods based on the random vector can significantly improve the convergence speed and efficiency.

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.

Direct numerical simulation of supersonic flow and acoustics over a compression ramp

We present direct numerical simulations of the shock wave boundary layer interaction (SBLI) at Mach number 2.9 over a 24° ramp. We study both the numerical accuracy and flow physics. Two classes of spatial reconstruction schemes are employed: the monotonic upstream-centered scheme for conservation laws and the Weighted Essentially Non-Oscillatory (WENO) scheme, of accuracy ranging from 2nd- to 11th-order. Using the canonical Taylor–Green vortex test-case, a simple and computationally inexpensive rescaling of the candidate stencil values—within the context of the high-order WENO scheme—is proposed for reducing the numerical dissipation, particularly in under-resolved simulations. For the compression ramp case, higher-order schemes are shown to capture the size of the SBLI separation zone more accurately, a consequence of resolving much finer turbulence structures. For second- and fifth-order schemes, the energy of the unresolved small scale turbulence shifts the cascade of the turbulence kinetic energy (TKE) spectrum, thus resulting in more energetic large scale turbulent structures. Consequently, the λ-shock foot shifts further downstream, leading to a smaller separation bubble size. Nonetheless, other statistical quantities, such as the turbulence anisotropy invariant map and the turbulence kinetic energy budget terms, show little dependence on the type and order of the spatial reconstruction scheme. Finally, using the more accurate ninth-order WENO results, it is reasoned that the interaction of the λ-shock with the post-shock relaxation region drives the low-frequency oscillation of the λ-shock.

High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations

A numerical framework of the generalized form of high order well-balanced finite difference weighted essentially non-oscillatory (WENO) interpolation-based schemes is proposed for the shallow water equations. It demonstrates more flexible construction process than the classical WENO reconstruction-based schemes. The weighted compact nonlinear schemes and finite difference alternative WENO schemes are two specific cases. To maintain the exact C-property, the splitting technique for the source term in the finite difference scheme [Xing and Shu, J. Comput. Phys. 208 (2005)] and the reconstruction technique in the finite volume WENO scheme [Xing and Shu, J. Comput. Phys. 214 (2006)] are adopted. The proposed scheme can be proved mathematically to maintain the exact C-property and demonstrates numerically that it is well-balanced by construction for the stationary water surface. Moreover, the local characteristic projections are employed to further mitigate the Gibbs oscillations. The proposed generic high order WENO schemes not only achieve high order accuracy but also capture the high gradients/shock waves essentially non-oscillatory. Meanwhile, the small perturbation problems can be resolved well on a coarse grid.

A new method towards high-order weno schemes on structured and unstructured grids

A new method is presented to develop high-order weighted essentially non-oscillatory (WENO) finite-volume schemes on structured and unstructured grids. The one-dimensional reconstruction used in the WENO-ZQ scheme is extended by introducing the concept of phantom points, which are constructed with the least-squares approximation to form a series of high-order WENO schemes. Numerical validations are performed on both regular structured grids and irregular unstructured grids. Analyses of the spectral property and numerical accuracy of the new schemes indicate attractive merits, including less dissipation and dispersion errors and low computational costs, which can be easily implemented into existing structured and unstructured grid CFD solvers.

A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes

In this continuing paper of Zhu and Shu (2018) [62], Zhu and Shu (2019) [63], we design a new third-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws on tetrahedral meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation. Comparing with classical third-order finite volume WENO schemes Zhang and Shu (2009) [56] on tetrahedral meshes, the crucial advantages of such new multi-resolution WENO schemes are their simplicity and compactness with the application of only three unequal-sized central stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, their easy choice of arbitrary positive linear weights without considering the topology of the tetrahedral meshes, their optimal order of accuracy in smooth regions, and their suppression of spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO scheme can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite volume WENO scheme on tetrahedral meshes. By performing such new spatial reconstruction procedures and adopting a third-order TVD Runge-Kutta method for time discretization, the occupied memory is decreased and the computing efficiency is increased. This new third-order finite volume multi-resolution WENO scheme is suitable for large scale engineering applications and could maintain good convergence property for steady-state problems on tetrahedral meshes. Benchmark examples are computed to demonstrate the robustness and good performance of these new finite volume WENO schemes.

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659–683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202–228, 1996) might suffer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain fifth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite-difference and finite-volume WENO schemes for solving steady-state problems. In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

Completed repeated Richardson extrapolation for compressible fluid flows

Abstract Richardson extrapolation is a powerful approach for reducing the spatial discretization errors and increasing, in this way, the accuracy of the computed solution obtained by using many numerical methods for solving different scientific and engineering problems. This approach has been used in a variety of computational fluid dynamics problems to reduce the numerical error, but it has been mainly restricted to the computation of incompressible fluid flows and on grids with coincident nodes. The purpose of this work is to present a completed repeated Richardson extrapolation (CRRE) procedure for a more generic type of grids not necessarily with coincident nodes, and test it on compressible fluid flows. Three tests are performed for one- and quasi-one-dimensional Euler equations, i.e. (i) the Rayleigh flow, (ii) the isentropic flow, and (iii) the adiabatic flow through a nozzle. The last test presents a normal shock wave. In order to build a simple CRRE solver, these problems are solved using a first-order upwind-type finite difference method as the base scheme. The normal shock wave problem is also solved using a high-order weighted essentially non-oscillatory (WENO) scheme to compare it with the CRRE procedure. The procedure we have proposed can increase the achieved accuracy and significantly decrease the magnitude of the spatial error in all three tests. Its performance can be best demonstrated in the Rayleigh flow test, where the spatial discretization error is reduced by seven orders of magnitude and the achieved accuracy is increased from 0.998 to 6.62 on the grid with 10240 nodes. Similar performance can be observed for the isentropic flow, for which the spatial discretization error is reduced by nine orders of magnitude and the achieved accuracy is again increased from 0.996 to 6.73 on the grid with 10240 nodes. Finally, in the adiabatic flow with a normal shock wave, the procedure can reduce the spatial discretization error both upstream and downstream of the shock. We do remark, however, that the more expensive high-order WENO scheme has errors of lower magnitude upstream of the shock and has a sharper shock transition for this shocked test case.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy

In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25,45], could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.

Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes

A new class of high order weighted essentially non-oscillatory (WENO) schemes (Zhu and Qiu, 2016, [50]) is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes (Jiang and Shu, 1996, [23]) might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in Jiang and Shu (1996) [23] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.

Computation of three-dimensional three-phase flow of carbon dioxide using a high-order WENO scheme

We have developed a high-order numerical method for the 3D simulation of viscous and inviscid multiphase flow described by a homogeneous equilibrium model and a general equation of state. Here we focus on single-phase, two-phase (gas–liquid or gas–solid) and three-phase (gas–liquid–solid) flow of CO2 whose thermodynamic properties are calculated using the Span–Wagner reference equation of state. The governing equations are spatially discretized on a uniform Cartesian grid using the finite-volume method with a fifth-order weighted essentially non-oscillatory (WENO) scheme and the robust first-order centered (FORCE) flux. The solution is integrated in time using a third-order strong-stability-preserving Runge–Kutta method. We demonstrate close to fifth-order convergence for advection–diffusion and for smooth single- and two-phase flows. Quantitative agreement with experimental data is obtained for a direct numerical simulation of an air jet flowing from a rectangular nozzle. Quantitative agreement is also obtained for the shape and dimensions of the barrel shock in two highly underexpanded CO2 jets.

Robust second-order scheme for multi-phase flow computations

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

Gravitational waveforms from binary neutron star mergers with high-order weighted-essentially-nonoscillatory schemes in numerical relativity

The theoretical modeling of gravitational waveforms from binary neutron star mergers requires precise numerical relativity simulations. Assessing convergence of the numerical data and building the error budget is currently challenging due to the low accuracy of general-relativistic hydrodynamics schemes and to the grid resolutions that can be employed in $(3+1)$-dimensional simulations. In this work, we explore the use of high-order weighted-essentially-nonoscillatory (WENO) schemes in neutron star merger simulations and investigate the accuracy of the waveforms obtained with such methods. We find that high-order WENO schemes can be robustly employed for simulating the inspiral-merger phase and they significantly improve the assessment of the waveform's error budget with respect to finite-volume methods. High-order WENO schemes can be thus efficiently used for high-quality waveforms production, also in future large-scale investigations of the binary parameter space.

Numerical simulation of wave transformation, breaking and runup by a contravariant fully non-linear Boussinesq equations model

In this paper we propose a new model based on a contravariant integral form of the fully non-linear Boussinesq equations (FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory (WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations (NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento (Italy).

A high-order symmetrical weighted hybrid ENO-flux limiter scheme for hyperbolic conservation laws

In this paper, we propose a new weighted essentially non-oscillatory (WENO) procedure for solving hyperbolic conservation laws, on uniform meshes. The new scheme combines essentially non-oscillatory (ENO) reconstructions together with monotone upwind schemes for scalar conservation laws’ interpolants. In a one-dimensional context, first, we obtain an optimum polynomial on a five-cells stencil. This optimum polynomial is fifth-order accurate in regions of smoothness. Next, we modify a third-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction–evolution method limiter. Then, we consider the optimum polynomial as a symmetric and convex combination of four polynomials with ideal weights. After that, following the methodology of the classic WENO procedure, we calculate non-oscillatory weights with the ideal weights. Also, the numerical solution is advanced in time by means of the linear multi-step total variation bounded (TV B0) technique. Numerical examples on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and yields sharp results when solving profiles with discontinuities. Comparing the new scheme with high-order WENO schemes shows that our method reduces smearing near shocks and corners, and in some cases it is more accurate near discontinuities. Finally, the new method is extended to multi-dimensional problems by a dimension-by-dimension approach. Several multi-dimensional examples are performed to show that our method remains non-oscillatory while giving good resolution of discontinuities.