Monotonicity Preserving Research Articles

Numerical simulation of vapor explosion bubbles in the presence of a non-condensable gas and a phase change

The complex dynamics of two flow problems of vapor explosion bubbles near a free surface and between two parallel plates are simulated by a compressible three-phase flow code. Firstly, a coupled high-order monotonicity-preserving (MP) interface capturing scheme is developed for the numerical computation of compressible three-phase flows, which is followed by validation with previous numerical and experimental data. Phase changes are taken into consideration and the empirical condensation coefficient is calibrated with the experimental data. For both two flow problems, good agreement between the numerical results and reference data is obtained. Most physical phenomena, including the free surface spike, the liquid jet before the impact and the complex interface deformations during multiple expanding and collapsing oscillations, are reasonably well reproduced. Afterwards, the effects of the initial vapor pressure and the non-condensable gas volume fraction in the surrounding water on the behavior of vapor bubbles are analyzed and discussed. It is found that both the initial vapor pressure and the non-condensable gas volume fraction have a substantial influence on the behavior of the bubble. Higher initial vapor pressure does indeed tend to result in larger bubbles. The presence of non-condensable gases tends to hinder the transfer of heat, slowing down condensation and potentially resulting in larger bubbles. When the initial vapor pressure is in the range of 10 to 20 atmospheres and the gas volume fraction is in the range of 0 to 0.2, the maximum vapor bubble volume is found to increase linearly with an increase of the amount of non-condensable gas.

Gradient based reconstruction: Inviscid and viscous flux discretizations, shock capturing, and its application to single and multicomponent flows

This paper presents a gradient-based reconstruction approach for simulations of compressible single and multi-species Navier–Stokes equations. The novel feature of the proposed algorithm is the efficient reconstruction via derivative sharing between the inviscid and viscous schemes: highly accurate explicit and implicit gradients are used for the solution reconstruction expressed in terms of derivatives. The higher-order accurate gradients of the velocity components are reused to compute the viscous fluxes for efficiency and significantly improve the solution and gradient quality, as demonstrated by several viscous-flow test cases. The viscous schemes are fourth-order accurate and carefully designed with a high-frequency damping property, which has been identified as a critically important property for stable compressible-flow simulations with shock waves (Chamarthi et al., 2022). Shocks and material discontinuities are captured using a monotonicity-preserving (MP) scheme, which is also improved by reusing the gradients. For inviscid test cases, The proposed scheme is fourth-order for linear and second-order accurate for non-linear problems. Several numerical results obtained for simulations of complex viscous flows are presented to demonstrate the accuracy and robustness of the proposed methodology.

Application of a high-order MP scheme to computation of multi-phase flows with heat and mass transfer

Reliable prediction of unsteady multiphase flows with mass changes, large density ratios, and complex flow features is very challenging due to the need to accurately resolve both complex interface deformations and steep gradients associated with material interfaces. This paper describes an application of our recently developed high-order monotonicity preserving (MP) scheme – a reliable and robust scheme for long-term computations of flows contain complex smooth structures interspersed with discontinuities (Ha and Lee, 2020) − to the solution of unsteady incompressible two-phase flows in the presence of heat and mass changes. The flow is governed by the multiphase Navier–Stokes equations, which comprises of the mixture continuity equation, the mixture momentum equations, the mixture energy equation, and the continuity equation for the individual phase. The numerical discretization is implemented on generalized curvilinear grids for facilitating flow simulations of the complex geometries. Our high-order MP scheme is applied to the calculations of the convective and surface tension terms. To obtain an effectively simultaneous solution of all equations, a novel dual-time stepping approach in conjunction with an implicit Runge–Kutta three-stage scheme is proposed. The reliability of the proposed numerical algorithm is then examined for a 1D Stefan condensation problem and two condensing flows including single bubble and steam-jet condensation problems. Particularly, for the single bubble condensation test, the bubble characteristics at different time instants are evaluated both qualitatively and quantitatively. For the steam-jet condensation test, the capability for capturing the evolution of interface deformations under different thermal and hydrodynamic flow conditions is highlighted. Finally, the effects of subcooling degree and steam mass flow rate on heat and mass transfer characteristics are investigated and discussed.

Research on Computational Method of Supersonic Inlet/Isolator Internal Flow

In this research, a CFD solver is developed for solving the 2D/3D compressible flow problem: the finite volume method based on multi-block structural grids is used to solve the compressible Reynolds averaged Navier–Stokes equations (RANS). Included in the methodology are multiple high-order reconstruction schemes, such as the 3rd-order MUSCL (Monotone Upstreamcentered Schemes for Conservation Laws), 5th-order WENO (Weight Essentially Non-Oscillatory), and 5th-order MP (Monotonicity-Preserving) schemes. Of the variety of turbulence models that are embedded, this solver is mainly based on the shear stress transport model (SST), which is compatible with OpenMP/MPI parallel algorithms. This research uses the CFD solver to conduct steady-state flow simulation for a two-dimensional supersonic inlet/isolator, incorporating these high-precision reconstruction schemes to accurately capture the shock wave/expansion wave interaction and shock wave/turbulent boundary layer interaction (SWTBLI), among other effects. By comparing the 2D/3D computation results of the same inlet configuration, it is found that the 3D effects of the side wall cannot be ignored due to the existing strong lateral flow near the corner. To obtain a more refined turbulence simulation, the commercial software ANSYS Fluent 18.0 is used to carry out the detached eddy simulation (DES) and the large eddy simulation (LES) of the same supersonic inlet, so as to reveal the flow details near the separation area and boundary layers.

Highly accurate monotonicity-preserving Semi-Lagrangian scheme for Vlasov-Poisson simulations

In this paper, we study a highly accurate monotonicity-preserving (MP) Semi-Lagrangian scheme for Vlasov-Poisson simulations. The classical Semi-Lagrangian scheme is known to be highly accurate and free from CFL condition, but it does not satisfy local maximum principle. To remedy this drawback, using the conservative form of the Semi-Lagrangian scheme, we recast existing MP schemes for the numerical flux in a common framework, and then substitute the local extremum by some “better” guess, in order to avoid as much as possible loss of accuracy and clipping near extrema, while keeping the monotonicity on monotone portions. With the limiter, on the one hand, the scheme keeps the good properties of the unlimited scheme: it is conservative, free from CFL condition and highly accurate. On the other hand, for locally monotonic data, the monotonicity of the solution is preserved. Numerical tests are made on free transport equation and Vlasov-Poisson system illustrating that the limited scheme is more diffusive compared to cubic splines but has better L1 conservation, which is primarily an advantage for problems with sharp gradients.

Monotonicity-preserving splitting schemes for solving balance laws

In this paper, some monotonicity-preserving (MP) and positivity-preserving (PP) splitting methods for solving the balance laws of the reaction and diffusion source terms are investigated. To capture the solution with high accuracy and resolution, the original equation with reaction source termis separated through the splitting method into two sub-problems including the homogeneous conservation law and a simple ordinary differential equation (ODE). The resulting splitting methods preserve monotonicity and positivity property for a normal CFL condition. A trenchant numerical analysis made it clear that the computing time of the proposed methods decreases when the so-called MP process for the homogeneous conservation law is imposed. Moreover, the proposed methods are successful in recapturing the solution of the problem with high-resolution in the case of both smooth and non-smooth initial profiles. To show the efficiency of proposed methods and to verify the order of convergence and capability of these methods, several numerical experiments are performed through some prototype examples.

A low-dissipation shock-capturing framework with flexible nonlinear dissipation control

In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially non-oscillatory (TENO) scheme (Fu et al. 2016 [9]). Rather than being discarded directly as with TENO schemes, the nonsmooth candidates are filtered by an extra nonlinear limiter, such as a monotonicity-preserving (MP) limiter or a total variation diminishing (TVD) limiter. Consequently, high-order reconstruction is achieved by assembling candidate fluxes with optimal linear weights since they are either smooth reconstructions or filtered ones which feature good non-oscillation property. A weight renormalization procedure as with the standard TENO paradigm is not necessary. This new framework concatenates the concepts of TENO, WENO and other nonlinear limiters for shock-capturing, and provides a new insight to designing low-dissipation nonlinear schemes. Through the adaptation of nonlinear limiters, nonlinear dissipation in the newly proposed framework can be controlled separately without affecting the performance in smooth regions. Based on the proposed framework, a family of new six- and eight-point nonlinear schemes with controllable dissipation is proposed. A set of critical benchmark cases involving strong discontinuities and broadband fluctuations is simulated. Numerical results reveal that the proposed new schemes capture discontinuities sharply and resolve the high-wavenumber fluctuations with low dissipation, while maintaining the desired accuracy order in smooth regions.

Numerical viscosity in simulations of the two-dimensional Kelvin-Helmholtz instability

The Kelvin-Helmholtz instability serves as a simple, well-defined setup for assessing the accuracy of different numerical methods for solving the equations of hydrodynamics. We use it to extend our previous analysis of the convergence and the numerical dissipation in models of the propagation of waves and in the tearing-mode instability in magnetohydrodynamic models. To this end, we perform two-dimensional simulations with and without explicit physical viscosity at different resolutions. A comparison of the growth of the modes excited by our initial perturbations allows us to estimate the effective numerical viscosity of two spatial reconstruction schemes (fifth-order monotonicity preserving and second-order piecewise linear schemes).

RISK MEASURES DERIVED FROM A REGULATOR’S PERSPECTIVE ON THE REGULATORY CAPITAL REQUIREMENTS FOR INSURERS

AbstractIn this study, we propose new risk measures from a regulator’s perspective on the regulatory capital requirements. The proposed risk measures possess many desired properties, including monotonicity, translation-invariance, positive homogeneity, subadditivity, nonnegative loading, and stop-loss order preserving. The new risk measures not only generalize the existing, well-known risk measures in the literature, including the Dutch, tail value-at-risk (TVaR), and expectile measures, but also provide new approaches to generate feasible and practical coherent risk measures. As examples of the new risk measures, TVaR-type generalized expectiles are investigated in detail. In particular, we present the dual and Kusuoka representations of the TVaR-type generalized expectiles and discuss their robustness with respect to the Wasserstein distance.

Modified Monotonicity Preserving Constraints for High-Resolution Optimized Compact Scheme

The monotonicity-preserving (MP) scheme is an accurate shock-capturing scheme. However, its performance is still inefficient for resolving high-frequency waves. In this paper, to improve the resolution characteristics, an upwind compact interpolation is proposed as a substitute to the original one in the MP scheme, and the coefficients of that were analytically optimized to minimize the dispersion and dissipation errors. Moreover, it was found that the limiting part of the original MP scheme degenerates the accuracy in a high-wavenumber region due to unnecessarily activation. This limitation is improved by applying a new indicator and criterion. The results of the nonlinear wave (N-wave) propagation demonstrate that the proposed scheme guarantees the robustness at the sharp discontinuity. At the same time, the solutions of linear wave propagation prove the excellent resolution of the proposed scheme. We intensively evaluated the performance for the standard and long-time situations of the shock-entropy wave interaction problems. The results prove that the usefulness of proposed scheme is more pronounced in the flow fields involving both of shock and waves.

A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows

The monotonicity preserving (MP) polynomial-based reconstruction scheme with Runge-Kutta time stepping by Suresh and Huynh (1997) has been applied extensively in computational fluid dynamics for solving hyperbolic partial differential equations and other convection-dominated problems. However, despite of its algorithmic simplicity in achieving a desired high-order of accuracy, the MP scheme is shown to lose the accuracy and robustness for a long-term computation of flows containing complex smooth structures interspersed with discontinuities. In this paper, the original MP scheme is first reviewed with a focus on the potential causes that lead to its failure. We then propose an improved MP scheme that can overcome the weakness in the MP scheme. The proposed MP scheme is constructed so that it can bypass smooth extrema without loss of accuracy while preserving monotonicity and properly reducing the order of accuracy near discontinuities. Various challenging scalar advection test cases are considered to demonstrate the performance of the presented method. Numerical results reveal that it achieves the highly accurate solutions in smooth parts of the flow and yields a significant improvement in accuracy and shape-preserving property. Moreover, the successful application to several classes of two- and three-dimensional unsteady multi-phase flows highlights the potential of the present method for resolving both sharp and complex interface deformations.

Multidimensional Vlasov–Poisson Simulations with High-order Monotonicity- and Positivity-preserving Schemes

Abstract We develop new numerical schemes for Vlasov–Poisson equations with high-order accuracy. Our methods are based on a spatially monotonicity-preserving (MP) scheme and are modified suitably so that the positivity of the distribution function is also preserved. We adopt an efficient semi-Lagrangian time integration scheme that is more accurate and computationally less expensive than the three-stage TVD Runge–Kutta integration. We apply our spatially fifth- and seventh-order schemes to a suite of simulations of collisionless self-gravitating systems and electrostatic plasma simulations, including linear and nonlinear Landau damping in one dimension and Vlasov–Poisson simulations in a six-dimensional phase space. The high-order schemes achieve a significantly improved accuracy in comparison with the third-order positive-flux-conserved scheme adopted in our previous study. With the semi-Lagrangian time integration, the computational cost of our high-order schemes does not significantly increase, but remains roughly the same as that of the third-order scheme. Vlasov–Poisson simulations on mesh grids have been successfully performed on a massively parallel computer.

Monotonicity preserving transformations of MOT and SEP

Recently, Beiglböck and Juillet (2016) and Beiglböck et al. (2015) established that optimizers to the martingale optimal transport problem (MOT) are concentrated on c-monotone sets. In this article we characterize monotonicity preserving transformations revealing certain symmetries between optimizers of MOT for different cost functions. Due to the intimate connection of MOT and the Skorokhod embedding problem (SEP) these transformations are also monotonicity preserving and disclose symmetries for certain solutions to the optimal SEP. Furthermore, the SEP picture allows to easily understand the geometry of these transformations once we have established the SEP counterparts to the known solutions of MOT based on the monotonicity principle for SEP which in turn allows to directly read off the structure of the MOT optimizers.

Application of a high-order CFD harmonic balance method to nonlinear aeroelasticity

An Aeroelastic-Harmonic Balance (A-HB) formulation of the Euler flow equations using a high-order spatial discretization scheme coupled with structural dynamic equations is proposed. The main objective of this new approach is to dramatically reduce the computational cost required to predict unsteady, periodic problems such as limit cycle oscillations (LCO). To this end, a new solver based on the Monotonicity Preserving limiter together with the AUSM+-up flux function is developed for the harmonic balance equations. The use of high-order CFD schemes allows the reduction of the number of degrees of freedom required to achieve a given desired accuracy, with respect to lower order schemes. In this paper, the reduction in degrees of freedom of the fluid system is exploited in the context of a CFD based Harmonic-Balance framework using a frequency updating procedure to determine the limit cycle conditions. The standard A-HB methodology has shown over one order of magnitude speed-up over time-marching methods; by employing the proposed high-order scheme in conjunction with coarser grids, the LCO computational time is halved without compromising accuracy.

単調性保存スキームI : 定常移流・拡散方程式の局所厳密解に基づく有限可変差分法

単調性保存スキームI : 定常移流・拡散方程式の局所厳密解に基づく有限可変差分法

Monotonicity preserving representations of curves and surfaces

Abstract In this paper we revisit the problem of monotonicity preservation of curves and surfaces and we provide some new proofs and open problems. In particular, we prove a new formula for the derivation of rational Bézier curves. We also deal with the rational monotonicity preservation of rational Bézier surfaces and a related conjecture is presented.

Criteria for response monotonicity preserving in approximation of fractional order systems

In approximation of fractional order systems, a significant objective is to preserve the important properties of the original system. The monotonicity of time frequency responses is one of these properties whose preservation is of great importance in approximation process. Considering this importance, the issues of monotonicity preservation of the step response and monotonicity preservation of the magnitude-frequency response are independently investigated in this paper. In these investigations, some conditions on approximating filters of fractional operators are found to guarantee the preservation of step magnitude-frequency response monotonicity in approximation process. These conditions are also simplified in some special cases. In addition, numerical simulation results are presented to show the usefulness of the obtained conditions.

An improved accurate monotonicity-preserving scheme for the Euler equations

The accurate monotonicity-preserving (MP) scheme of Suresh and Huynh (1997) [5] is a high-order and high-resolution method for hyperbolic conservation laws. However, the robustness of the MP scheme is not very high. In this paper, a detailed analysis on this scheme is performed, and two potential causes which may account for the weak robustness are revealed. Furthermore, in order to enhance the robustness of the MP scheme, an improved version of the MP scheme is presented, in which a strict continuous total-variation-diminishing (TVD) numerical flux is used at a disturbed discontinuity so that oscillations cannot grow indefinitely without violating the TVD condition. Without destroying the very high resolution property, numerical tests show that the improved scheme shares a strong robustness in simulating extreme numerical tests.

Unsteady compressible flow computations using an adaptive multiresolution technique coupled with a high-order one-step shock-capturing scheme

This article deals with the development of adaptive multiresolution coupled with a one-step shock-capturing scheme for the numerical simulation of unsteady compressible flows in the transonic and supersonic regimes with high frequency oscillations. The discretization of the convective terms is based on a coupled time and space approach by using a one-step (OS) scheme, developed following the Lax–Wendroff approach by correcting the successive modified equations. A monotonicity preserving (MP) criterion is added in order to locally relax the TVD constraints for such schemes. The adaptive strategy relies on the Harten cell-average multiresolution analysis, with a dynamical data structure organized as a graded tree that dynamically evolves in time. We apply the method to several prototype test-cases of shock-wave propagation interaction. We validate this approach on 2D inviscid advection of a vortex. We then present 2D viscous test-cases of shock-shear layer interactions and a 3D spherical Riemann problem to demonstrate the capability of the present method. Results demonstrate that 7th order OSMP schemes coupled with adaptive grid refinement gives very accurate results in comparison with more classical schemes applied on a single grid. We then propose an appropriate MR threshold parameter value that ensures accurate results while achieving drastic gains on the CPU time and memory usage.

A monotonicity preserving, nonlinear, finite element upwind method for the transport equation

Abstract We propose a simple upwind finite element method that is monotonicity preserving and weakly consistent of order O ( h 3 2 ) . The scheme is nonlinear, but since an explicit time integration method is used the added cost due to the nonlinearity is not prohibitive. We prove the monotonicity preserving property for the forward Euler method and for a second order Runge–Kutta method. The convergence properties of the Runge–Kutta finite element method are verified on a numerical example.