Numerical Scheme Research Articles

Multiple solutions of nonlinear coupled constitutive relation model and its rectification in non-equilibrium flow computation

In this study, the multiple solutions of Nonlinear Coupled Constitutive Relation (NCCR) model are observed and a way for identifying the physical solution is proposed. The NCCR model proposed by Myong is constructed from the generalized hydrodynamic equations of Eu, and aims to describe rarefied flows. In the non-equilibrium regime, the NCCR equations are more reliable than the Navier-Stokes equations. And the NCCR equations have an advantage of efficiency over the discrete velocity methods and stochastic particle methods. However, the NCCR model is a complicated nonlinear system. Many assumptions have been used in the schemes for solving the NCCR equations. The corresponding numerical methods may be associated with unphysical solution and instability. At the same time, it is hard to analyze the physical accuracy and stability of NCCR model due to the uncertainties in the numerical discretization. In this study, a new numerical method for solving NCCR equations is proposed and used to analyze the properties of NCCR equations. More specifically, the nonlinear equations are converted into the solutions of an objective function of a single variable. Under this formulation, the multiple solutions of the NCCR system are identified and the criteria for picking up the physical solution are proposed. Therefore, a numerical scheme for solving NCCR equations is constructed. A series of flow problems in the near continuum and low transition regimes with a large variation of Mach numbers are conducted to validate the numerical performance of proposed method and the physical accuracy of NCCR model.

Automatically Differentiable Higher-Order Parabolic Equation for Real-Time Underwater Sound Speed Profile Sensing

This paper is dedicated to the acoustic inversion of the vertical sound speed profiles (SSPs) in the underwater marine environment. The method of automatic differentiation is applied for the first time in this context. Representing the finite-difference Padé approximation of the propagation operator as a computational graph allows for the analytical computation of the gradient with respect to the SSP directly within the numerical scheme. The availability of the gradient, along with the high computational efficiency of the numerical method used, enables rapid inversion of the SSP based on acoustic measurements from a hydrophone array. It is demonstrated that local optimization methods can be effectively used for real-time sound speed inversion. Comparative analysis with existing methods shows the significant superiority of the proposed method in terms of computation speed.

CFD Simulation of Stirling Engines: A Review

Stirling engines (SEs) have long attracted the attention of renewable energy researchers due to their external combustion design and flexibility in operating with various heat sources. The mathematical analysis of these devices is conducted by using a broad range of models ranging from basic zero-order to highly detailed fourth-order models, which are implemented through Computational Fluid Dynamics (CFD) simulations. The unique features of this last approach, combined with the increase in computing power, have promoted the use of CFD as a tool for analyzing SEs in recent years, significantly reducing the costs associated with prototype construction. However, Stirling CFD simulations are sophisticated due to the variety of physical phenomena involved, such as volume change, conjugated heat transfer, turbulent compressible fluid dynamics, and flow through porous media in the regenerator. Furthermore, there is currently no comprehensive review of CFD simulations of SEs in the literature; therefore, this contribution aims to fill that gap. Emphasis has been placed on identifying the type of engine, the physical phenomena modeled, the simplifying assumptions, and specific numerical aspects, such as mesh type, spatial and temporal discretization, and the order of the numerical schemes used. As a result, it has been found that in many cases, CFD numerical reports lack sufficient detail to ensure the reproducibility of the simulations. This work proposes guidelines for reporting CFD studies on Stirling engines to address this issue. Additionally, the need for a sufficiently detailed experimental benchmark database to validate future CFD studies is stressed. Finally, the use of Large Eddy Simulations on coupled key engine components—such as compression and expansion spaces, pistons, displacer, and regenerator—is suggested to provide further insights into the specific flow and heat transfer characteristics in Stirling engines.

Shannon Wavelet‐Based Approximation Scheme for Information Entropy Integrals in Confined Domain

ABSTRACTIn this work, the author attempted to develop a Shannon wavelet‐based numerical scheme to approximate the information entropies in both configuration and momentum space corresponding to the ground and adjacent excited energy states of one‐dimensional Schrödinger equation appearing in non‐relativistic quantum mechanics. The development of this scheme is based on the judicious use of sinc scale functions as an approximation basis and a suitable numerical quadrature to approximate entropies in position and momentum spaces. Priori and posteriori errors appearing in the approximations of wave functions and entropy integrals have been discussed. The scheme (coded in Python) has been subsequently exercised for various exactly solvable and quasi‐exactly solvable non‐relativistic quantum mechanical models in confined domain.

Using the Discrete Wavelet Transform for Lossy On‐the‐Fly Compression of GPU Fluid Simulations

ABSTRACTHigh‐performance computing in fluid dynamics frequently confronts substantial memory demands, especially in large‐scale applications. Data compression techniques can alleviate these memory constraints, but introduce new challenges. This paper introduces an innovative on‐the‐fly low‐overhead lossy compression technique tailored for GPU‐based fluid simulations, utilizing the discrete wavelet transform (DWT). The technique is applicable to any numerical scheme where the data is stored on a regular grid and the time step is computed using a stencil. Our approach significantly diminishes memory requirements, achieving up to a 10‐fold long‐term reduction on a D3Q27 simulation, while minimally impacting the simulation accuracy. The methodology is built around careful design choices to achieve a satisfactory compression ratio/speed trade‐off. It effectively maintains mass conservation and accurately preserves essential discontinuities in simulations. Extensive testing with a D3Q27 Lattice‐Boltzmann method (LBM) simulation on a single GPU has shown that large‐scale grids can be processed with minimal impact on the simulation accuracy and acceptable compression times. This compression technique demonstrates a robust capability to handle memory limitations in fluid simulations, opening the door to more complex and larger‐scale simulations.

Efficient numerical approximations for a nonconservative nonlinear Schrödinger equation appearing in wind‐forced ocean waves

AbstractWe consider a nonconservative nonlinear Schrödinger equation (NCNLS) with time‐dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper, we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic nonlinear Schrödinger equation. Both schemes are second‐order accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour–Fortin–Payre scheme, and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly nonconservative problems. We finally compare the two numerical schemes and discuss their performance.

A Nonlinear Approach in the Quantification of Numerical Uncertainty by High-Order Methods for Compressible Turbulence with Shocks

This is a comprehensive overview on our research work to link interdisciplinary modeling and simulation techniques to improve the predictability and reliability simulations (PARs) of compressible turbulence with shock waves for general audiences who are not familiar with our nonlinear approach. This focused nonlinear approach is to integrate our “nonlinear dynamical approach” with our “newly developed high order entropy-conserving, momentum-conserving and kinetic energy-preserving methods” in the quantification of numerical uncertainty in highly nonlinear flow simulations. The central issue is that the solution space of discrete genuinely nonlinear systems is much larger than that of the corresponding genuinely nonlinear continuous systems, thus obtaining numerical solutions that might not be solutions of the continuous systems. Traditional uncertainty quantification (UQ) approaches in numerical simulations commonly employ linearized analysis that might not provide the true behavior of genuinely nonlinear physical fluid flows. Due to the rapid development of high-performance computing, the last two decades have been an era when computation is ahead of analysis and when very large-scale practical computations are increasingly used in poorly understood multiscale data-limited complex nonlinear physical problems and non-traditional fields. This is compounded by the fact that the numerical schemes used in production computational fluid dynamics (CFD) computer codes often do not take into consideration the genuinely nonlinear behavior of numerical methods for more realistic modeling and simulations. Often, the numerical methods used might have been developed for weakly nonlinear flow or different flow types other than the flow being investigated. In addition, some of these methods are not discretely physics-preserving (structure-preserving); this includes but is not limited to entropy-conserving, momentum-conserving and kinetic energy-preserving methods. Employing theories of nonlinear dynamics to guide the construction of more appropriate, stable and accurate numerical methods could help, e.g., (a) delineate solutions of the discretized counterparts but not solutions of the governing equations; (b) prevent numerical chaos or numerical “turbulence” leading to FALSE predication of transition to turbulence; (c) provide more reliable numerical simulations of nonlinear fluid dynamical systems, especially by direct numerical simulations (DNS), large eddy simulations (LES) and implicit large eddy simulations (ILES) simulations; and (d) prevent incorrect computed shock speeds for problems containing stiff nonlinear source terms, if present. For computation intensive turbulent flows, the desirable methods should also be efficient and exhibit scalable parallelism for current high-performance computing. Selected numerical examples to illustrate the genuinely nonlinear behavior of numerical methods and our integrated approach to improve PARs are included.

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with (2−α)-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

Improved Delayed Detached-Eddy Simulation of Turbulent Vortex Shedding in Inert Flow over a Triangular Bluff Body

The Improved Delayed Detached-Eddy Simulation (IDDES) is a modification of the original Detached-Eddy Simulation (DES) design to incorporate Wall Modeled Large Eddy Simulation (WMLES) capabilities and to extend the class of flows suitable for this methodology. For thin attached boundary layers, typically seen in external aerodynamic flows, the DES branch of the model is active, whereas with thick boundary layers, typically seen in internal flows and also wake flows, the WMLES branch is active, thus providing a numeric method suited to handling most flow cases automatically. The flow over a triangular bluff body is used to validate the suitability of the IDDES model and compare the results with experimental, DDES, and LES data. The IDDES model is found to be relatively accurate when compared with the experimental results, with recirculation length, streamwise velocity, and Reynolds stresses all showing good agreement with the experimental data. However, when compared with the DDES model, there is a ~4% overprediction of the recirculation length using the same mesh and numerical scheme. The code, with its extra complexity, is also ~3% slower to solve. The IDDES model has also been tested against different meshes, and the results show that even for a coarse mesh, there is still good agreement with the experimental data.

Optimally accurate second-order numerical scheme on layer-resolving graded mesh for singularly-perturbed boundary turning point problems exhibiting power-of-type-1 and hybrid layers

The paper discusses singularly-perturbed problems with boundary turning points. It describes constructions of layer-resolving grids based on layer-eliminating coordinate transformations and analyses the convergence of numerical solutions obtained using the upwind scheme on the layer-resolving grids and Richardson extrapolations. Theoretical error estimates and numerical experiments confirm that the upwind scheme with Richardson extrapolation on the proposed layer-resolving grids is uniformly convergent of order two without any logarithmic factor.

An elasto-visco-plastic constitutive model for snow: Theory and finite element implementation

Snow exhibits unique mechanical behaviour due to its evolving properties influenced by temperature, stress conditions, and viscous effects. This paper introduces a nonlinear constitutive model for snow, featuring new formulations for the yield function and strain rate potential, and incorporating viscosity, sintering, and degradation effects. The model is numerically integrated into the Abaqus/Standard FEM software using a fully implicit backward Euler method for time integration and Powell’s hybrid method for linearizing the nonlinear system of equations. The robustness and stability of the numerical scheme ensure accurate simulation of snow behaviour under various loading conditions. The model is finally validated against experimental data available in the literature, demonstrating its effectiveness and reliability in capturing the complex mechanical response of snow.

Evolution of linear internal waves over large bottom topography in three-layer stratified fluids

Evolutions of internal waves of different modes, particularly mode 1 and mode 2, passing over variable bathymetry are investigated based on a new numerical scheme. The problem is idealized as interfacial waves propagating on two interfaces of a three-layer density stratified fluid system with large-amplitude bottom topography. The Dirichlet-to-Neumann operators are introduced to reduce the spatial dimension by one and to adapt the three-layer system and significant topographic effects. However, for simplicity, nonlinear interactions between interfaces are neglected. Numerical techniques such as the Galerkin approximation, proven effective in previous works, are applied to save computational costs. Shoaling of linear waves on an uneven bottom is first studied to validate the proposed formulation and the corresponding numerical scheme. Then, for two-dimensional numerical experiments, mode transition phenomena excited by locally confined bottom obstacles and quickly varying topographies, including the Bragg resonance, mode-2 excitation, wave homogenization, etc., are investigated. In three-dimensional simulations, internal wave refraction by a Luneberg lens is considered, and good agreement is found in comparison with the ray theory. Finally, in the limiting case, when the top layer can be negligible (for example, a gas layer of extremely small density), the problem is reduced to a two-and-a-half-layer fluid system, where an interface and a surface are unknown free boundaries. In this situation, the surface signature of an internal wave is simulated and verified by introducing the realistic bathymetry of the Strait of Gibraltar and qualitatively compared with the satellite image.

An explicit time integration method for Boussinesq approximation

AbstractThis paper presents a novel approach to the discretization of the Boussinesq equation using finite elements, coupled with an explicit pressure correction scheme. The entire numerical scheme is formulated exclusively in terms of matrix‐vector products, enabling a highly efficient numerical solution. The key advantage of this approach lies in its suitability for fast computations and seamless parallelization on graphics processing units (GPUs). We present the numerical scheme and discuss different computational examples.

Comparative study on predicting turbulent kinetic energy budget using high-order upwind scheme and non-dissipative central scheme

The accurate computation of different turbulent statistics poses different requirements on numerical methods. In this paper, we investigate the capabilities of two representative numerical schemes in predicting mean velocities, Reynolds stress and budget of turbulent kinetic energy (TKE) in low Mach number flows. With concerns on numerical order of accuracy, dissipation and dispersion properties, a high-order upwind scheme with relatively good dispersion and dissipation and a second-order non-dissipative central scheme with perfect dissipation but poor dispersion are adopted for this comparative study. By carrying out a series of numerical simulations including Taylor-Green vortex, turbulent channel flow at Reτ=180\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ au }=180$$\\end{document} and turbulent flow over periodic hill at Reb=10595\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{b}=10595$$\\end{document}, it can be obtained that although the high-order upwind scheme lacks perfection of dissipation in high wave number range, it still demonstrates superior predictive capability compared with the second-order non-dissipative central scheme, especially with relatively coarse grids. Finally, by taking the high-order upwind scheme as a suitable selection for turbulence simulation, the turbulent flow over a 30P30N multi-element airfoil is investigated as an application study. After briefly comparing the simulated profiles and spectrum with reference experimental results as validation, the budget of TKE is analyzed to locate the dominant flow structures and regions. It is found that the production and dissipation terms behave in a “monopole” pattern in the locations with strong shears and wakes. Whereas the advection and diffusion terms show an “inward” pattern and an “outward” pattern, which indicate the spatial transport of TKE between the center of the shear layer and nearby locations.

RKFD Scheme for Quantum Reflection Model of Bose-Einstein Condensates (BEC) from Silicon Surface

We applied the numerical combination of Runge-Kutta and Finite Difference (RKFD) scheme for a quantum reflection model of Bose-Einstein condensate (BEC) from a silicon surface. It is by the time-dependent Gross-Pitaevskii equation (GPE), a non-linear Schrödinger equation (NLSE) in the context of quantum mechanics. The role of cut-off potential δ and negative imaginary potential Vim is essential to estimating non-interacting BEC reflection models. Relying on these features, we performed a numerical simulation of the BEC quantum reflection model and calculated the effect of reflection probability R versus incident speed vx. The model is based on the three rapid potential variations: positive-step potential +Vstep, negative-step potential -Vstep, and Casimir-Polder potential VCP. As a result, the RKFP numerical scheme was successfully set up and applied to the quantum reflection model of BEC from the silicon surface. The numerical simulations results show that the reflection probability R decays exponentially to the incident speed vx.

POD and mesh-adaptivity based reduce order model: Application to solve PDEs and inviscid flows

Time efficiency is a critical concern in Computational Fluid Dynamics (CFD) simulations of industrial applications. Despite the extensive research to improve the underlying numerical schemes to reduce the processing time, many CFD applications still need to improve on this issue. Reduced-Order Models (ROM) appeared as a promising alternative for significantly enhancing cost-effectiveness. The dimension of the initial problem is reduced significantly while keeping an acceptable level of accuracy. Proper Orthogonal Decomposition (POD), a data analysis technique, is widely used to construct a ROM to solve Euler and Navier-Stokes equations in CFD. Many aspects, however, need to be improved. In this paper, first, a general upper-bound error formula is derived, and second, a mesh-adaptivity based algorithm for snapshot locations in parameter space is provided (a sensitive step for POD-based ROM) in the context of Partial Differential Equations (PDE). The method’s efficiency is demonstrated through numerical tests by comparison to exact solutions and numerical results of a finite volume scheme on inviscid flow over a NACA0012.

Optimal Control of a Domestic Frost-Free Refrigerator Inlet Freezer from the Perspective of Reducing Energy Consumption

This study is dedicated to the development of a mathematical model based on shape optimal control of the inlet domestic frost-free refrigerator in the context of energy consumption reduction. A three-dimensional thermofluid model is established for the numerical studies. As the physical system (with shelves and fruits) can be seen as a porous medium, the coupled state equations of the transport mechanism (for velocity and temperature fields) are governed by Navier–Stokes–Forchheimer/Fourier equations. Then, based on the finite element method, the numerical scheme computation is made with FreeFem++ software (Version 4.6). Numerical results are shown for three different cases: empty without shelves, empty with shelves, and loaded with foods. For each case, the velocity and temperature fields’ results are discussed for the optimal configurations. The characterization of these physical parameters would help engineers in domestic frost-free refrigerator design.

Numerical estimation of wave breaking forces on seawalls

ABSTRACT This paper presents a computationally efficient numerical scheme to estimate wave forces from plunging breakers on vertical rigid seawalls. The approximated velocity equation, an asymptotical solution of the Green-Naghdi equation in the Camassa-Holm scaling, is utilized to obtain the evolution of the free surface and the depth-averaged horizontal velocity over time, leading to the formation of a plunging breaker. The uniform dynamic pressure along the depth is obtained by utilizing the unsteady Bernoulli equation. The wave forces on seawalls are then determined using Froude-Krylov theory, with adjustments for diffraction effects. The results, while being comparable to some existing formulations, notably differ with some others. This highlights the limitations imposed by the inherent assumptions in the existing empirical formulations and underlines the importance of the proposed methodology that is founded on an accurate modelling of the wave breaking process to estimate wave breaking forces on seawalls.

New dynamics performances to the established dark solitons in Polariton Condensate

Abstract New diverse enormous soliton solutions to the Gross–Pitaevskii Equation, which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping, have been constructed for the first time by using two various Schemes. The two utilized Schemes are the Generalized Kudryashov Scheme, the (G'/G)-Expansion Scheme. Throughout these two suggested schemata we construct new diverse forms solutions that include dark, bright-shaped soliton solutions, combined bright-shaped, dark-shaped soliton solutions, hyperbolic function soliton solution, singular shaped soliton solution and other rational soliton solutions. The two 2D and 3D figures designs have been configured by using the Mathematica-program. In addition, the Haar Wavelet Numerical Scheme has been applied to construct the identical numerical behavior for all soliton solutions achieved by the two suggested Schemes to show existing similarity between the soliton solutions and numerical solutions.