Order Numerical Scheme Research Articles

Property preserving reformulation of constitutive laws for the conformation tensor

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg numbers. This study presents a new approach to preserve the symmetric positive definiteness of the conformation tensor and to bound the magnitude of its eigenvalues. The idea behind this transformation is lies with the matrix logarithm formulation. Under the logarithmic transformation, the eigenvalue spectrum of the new conformation tensor varies from infinite positive to infinite negative. But, reconstruction the classical formulation from unbounded eigenvalues doesn't achieve meaningful results. This enhanced formulation, hyperbolic tangent, prevails the previous numerical failure by bounding the magnitude of eigenvalues in a manner that positive definite is always satisfied. In order to evaluate the capability of the hyperbolic tangent formulation we performed a numerical simulation of FENE-P fluids in a rectangular channel in the context of the finite element method. Under this new transformation, the maximum attainable Weissenberg number increases 21.4% and 112.5% comparing the standard log-conformation and classical constitutive equation respectively.

Fast numerical simulation of a new time-space fractional option pricing model governing European call option

When the fluctuation of option price is regarded as a fractal transmission system and the stock price follows a Lévy distribution, a time-space fractional option pricing model (TSFOPM) is obtained. Then we discuss the numerical simulation of the TSFOPM. A discrete implicit numerical scheme with a second-order accuracy in space and a 2−γ order accuracy in time is constructed, where γ is a transmission exponent. The stability and convergence of the obtained numerical scheme are analyzed. Moreover, a fast bi-conjugate gradient stabilized method is proposed to solve the numerical scheme in order to reduce the storage space and computational cost. Then a numerical example with exact solution is presented to demonstrate the accuracy and effectiveness of the proposed numerical method. Finally, the TSFOPM and the above numerical technique are applied to price European call option. The characteristics of the fractional option pricing model are analyzed in comparison with the classical Black–Scholes (B-S) model.

A second order numerical scheme for the annealing of metal–intermetallic laminate composite: A ternary reaction system

Metal–Intermetallic Laminate (MIL) composites are laminate structures that are fabricated by optimizing unique benefits of constituent components to have attractive physical and mechanical features, such as high strength, stiffness, and toughness. The synthesis of MIL composites involves annealing reaction of multiple metallic elements at high temperatures. In this work, we propose, analyze, and implement a second-order semi-implicit scheme for the annealing of a ternary metallic reaction system. The robustness of such a numerical scheme is demonstrated by various numerical experiments, as well as the expected accuracy tests. At the theoretical level, we provide a detailed convergence analysis, which confirms the second-order accuracy in both the temporal and spatial discretization. Moreover, this numerical scheme is used to study the annealing process of the Al–Fe–Ni ternary system. The computation reproduces the formation of the Al-rich and Ni-rich layers in the annealing process. We present the computational results as diffusion paths in a ternary phase diagram, with a nice agreement with that of experimental data. We also study the growth kinetics of the two layers by calculating the rate constant and kinetic exponent of the reaction. The morphology of interfaces between different layers is thoroughly investigated as well. The computational results indicate that the numerical scheme is an effective, useful tool for predicting the microstructure evolution in the annealing process of a ternary reaction system.

A comparative study of limiting strategies in discontinuous Galerkin schemes for the M1 model of radiation transport

Abstract The M 1 minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation, and has many desirable properties. Among them are symmetric hyperbolicity, entropy decay, moment realizability, and correct behavior in the diffusion and free-streaming limits. However, numerical difficulties arise when approximating the solution of the M 1 model by high order numerical schemes; namely maintaining the realizability of the numerical solution and controlling spurious oscillations. In this paper, we extend a previously constructed one-dimensional realizability limiting strategy to 2D. In addition, we perform a numerical study of various combinations of the realizability limiter and the TVBM local slope limiter on a third order Discontinuous Galerkin (DG) scheme on both triangular and rectangular meshes. In several test cases, we demonstrate that in general, a combination of the realizability limiter and a TVBM limiter is necessary to obtain a robust and accurate numerical scheme. Our code is published so that all results can be reproduced by the reader.

A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection

In this paper we present a second order accurate, energy stable numerical scheme for the epitaxial thin film model without slope selection, with a mixed finite element approximation in space. In particular, an explicit treatment of the nonlinear term, $$\frac{\nabla u}{1+|\nabla u|^2}$$ , greatly simplifies the computational effort; only one linear equation with constant coefficients needs to be solved at each time step. Meanwhile, a second order Douglas–Dupont regularization term, $$A\tau \varDelta ^2 ( u^{n+1} - u^n)$$ , is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an $$\ell ^\infty (0,T; L^2)$$ convergence analysis for the proposed scheme, with an $$O (\tau ^2 + h^q)$$ error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using $$Q_q$$ finite elements, which shows that the spatial convergence order can be improved to $$q+1$$ on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme.

Implementation and assessment of wall friction models for LWR core analysis

The modeling of frictional pressure drop in the nuclear thermal hydraulics subchannel code, CTF, is improved through the addition of three new modeling options. Two of the new models allow the code to account for the effects of surface roughness and the third enables a user-supplied option. After the initial implementation, a variety of analyses are performed to test the software quality. First, a series of defect tests are designed for both single- and multi-channel configurations which compare simulated results to approximate solutions. The single-channel tests assess the friction model implementation; a suite of three-by-three bundle tests are used to ensure proper implementation of the roughness averaging scheme. The maximum relative error in the pressure drop over all defect tests is less than 0.15%. A solution verification test is performed to ensure that the first order numerical scheme in CTF is not significantly disrupted by the friction model. Finally, the wall friction model is validated using both separate and integral effects experimental data. Overall, the software quality, verification, and validation procedure ensures that the new model is coded correctly, that it properly interacts with the rest of CTF, and that it can be used to model real-world data for turbulent single-phase flow. The work completed herein provides a complete demonstration of modern coding practices. Future work could include a formal equation analysis of the numerical error in the friction model, as well as an analysis of validation data for one dimensional two-phase flow.

Contaminant Particle Motion in Lubricating Grease Flow: A Computational Fluid Dynamics Approach

In this paper, numerical simulations of particle migration in lubricating grease flow are presented. The rheology of three lithium greases with NLGI (National Lubricating Grease Institute) grades 00, 1 and 2 respectively are considered. The grease is modeled as a single-phase Herschel–Bulkley fluid, and the particle migration has been considered in two different grease pockets formed between two concentric cylinders where the inner cylinder is rotating and driving the flow. In the wide grease pocket, the width of the gap is much smaller compared to the axial length scale, enabling a one-dimensional flow. In the narrow pocket, the axial and radial length is of the same order, yielding a three-dimensional flow. It was found that the change in flow characteristics due to the influence of the pocket lateral boundaries when going from the wide to the narrow pocket leads to a significantly shorter migration time. Comparing the results with an existing migration model treating the radial component contribution, it was concluded that a solution to the flow in the whole domain is needed together with a higher order numerical scheme to obtain a full solution to the particle migration. This result is more pronounced in the narrow pocket due to gradients in the flow induced by the lateral boundaries.

Non-Fourier effects in the thermal protection against high-power ultra-fast laser pulses

In this paper the heat transfer process between laser beam and a solid body is investigated in the context of the protective materials, which can be used in the military, manufacturing or scientific fields. An ultra-fast high-power interaction regime is being considered. Non-Fourier effects are taken into account. To simulate effects, which can be of significance in the protective materials design, Cattaneo-Vernotte equation is being solved by means of MacCormack explicit second order numerical scheme. Selected results are discussed in detail.

Numerical modelling on pulsatile flow of Casson nanofluid through an inclined artery with stenosis and tapering under the influence of magnetic field and periodic body acceleration

The present study investigates the pulsatile flow of Casson nanofluid through an inclined and stenosed artery with tapering in the presence of magnetic field and periodic body acceleration. The iron oxide nanoparticles are allowed to flow along with it. The governing equations for the flow of Casson fluid when the artery is tapered slightly having mild stenosis are highly non-linear and the momentum equations for temperature and concentration are coupled and are solved using finite difference numerical schemes in order to find the solutions for velocity, temperature, concentration, wall shear stress, and resistance to blood flow. The aim of the present study is to analyze the effects of flow parameters on the flow of nanofluid through an inclined arterial stenosis with tapering. These effects are represented graphically and concluded that the wall shear stress profiles enhance with increase in yield stress, magnetic field, thermophoresis parameter and decreases with Brownian motion parameter, local temperature Grashof number, local nanoparticle Grashof number. The significance of the model is the existence of yield stress and it is examined that when the rheology of blood changes from Newtonian to Casson fluid, the percentage of decrease in the flow resistance is higher with respect to the increase in the parameters local temperature Grashof number, local nanoparticle Grashof number, Brownian motion parameter, and Prandtl number. It is pertinent to observe that increase in the Brownian motion parameter leads to increment in concentration and temperature profiles. It is observed that the concentration of nanoparticles decreases with increase in the value of thermophoresis parameter.

Electromagnetohydrodynamic transport of Al2O3 nanoparticles in ethylene glycol over a convectively heated stretching cylinder

In this study, the transport of Al2O3 nanoparticles in ethylene glycol conventional fluid over a linearly stretching cylinder is investigated. The current research employs a convective surface boundary condition for heat transfer exploration. Flux model proposed by Rosseland is employed to examine effect of thermal radiations. The governing flow problem comprises highly nonlinear ordinary differential equations. Similarity transformations are used to reduce the equations in similar forms, which are then solved by Runge–Kutta–Fehlberg fourth-fifth order numerical scheme with shooting algorithm in MATLAB software. In order to authenticate the accuracy of our results, we have contrasted results with those obtained by Ishak et al., Wang, and Pandey and Kumar and found that they are in better concord, as revealed in Table 2. The impact of numerous emerging parameters on velocity distribution and heat transfer distribution are argued in all aspects and depicted through graphs.

Combining Plane Wave Expansion and Variational Techniques for Fast Phononic Computations

In this paper the salient features of the Plane Wave Expansion (PWE) method and the mixed variational technique are combined for the fast eigenvalue computations of arbitrarily complex phononic unit cells. This is done by expanding the material properties in a Fourier expansion, as is the case with PWE. The required matrix elements in the variational scheme are identified as the discrete Fourier transform coefficients of material properties, thus obviating the need for any explicit integration. The process allows us to provide succinct and closed form expressions for all the matrices involved in the mixed variational method. The scheme proposed here preserves both the simplicity of expression which is inherent in the PWE method and the superior convergence properties of the mixed variational scheme. We present numerical results and comment upon the convergence and stability of the current method. We show that the current representation renders the results of the method stable over the entire range of the expansion terms as allowed by the spatial discretization. When compared with a zero order numerical integration scheme, the present method results in greater computational accuracy of all eigenvalues. A higher order numerical integration scheme comes close to the accuracy of the present method but only with significantly more computational expense.

P.3.d.032 - Deep brain stimulation in schizophrenia: a randomized pilot study testing anterior cingulated cortex vs. nucleus accumbens targets

The M1 minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation, and has many desirable properties. Among them are symmetric hyperbolicity, entropy decay, moment realizability, and correct behavior in the diffusion and free-streaming limits. However, numerical difficulties arise when approximating the solution of the M1 model by high order numerical schemes; namely maintaining the realizability of the numerical solution and controlling spurious oscillations. In this paper, we extend a previously constructed one-dimensional realizability limiting strategy to 2D. In addition, we perform a numerical study of various combinations of the realizability limiter and the TVBM local slope limiter on a third order Discontinuous Galerkin (DG) scheme on both triangular and rectangular meshes. In several test cases, we demonstrate that in general, a combination of the realizability limiter and a TVBM limiter is necessary to obtain a robust and accurate numerical scheme. Our code is published so that all results can be reproduced by the reader.

CFD Simulations of an Air-Water Bubble Column: Effect of Luo Coalescence Parameter and Breakup Kernels.

In this work, CFD simulations of an air-water bubbling column were performed and validated with experimental data. The superficial gas velocities used for the experiments were 0.019 and 0.038 m/s and were considered as an homogeneous regime. The former involves simpler physics when compared to a heterogeneous regime where the superficial velocities are higher. In order to simulate the system, a population balance model (PBM) was solved numerically using a discrete method and a closure kernels involving the Luo coalescence model as well as two different breakup models: Luo's and Lehr's. For the multi-phase calculations, an eulerian framework was selected and the interphase momentum transfer included drag, lift, wall lubrication, and turbulent dispersion terms. A sensitivity analysis was performed on a Luo coalescence kernel by changing the coalescence parameter (c0) from 1.1 to 0.1 and results showed that the radial profiles of gas holdup and axial liquid velocity were significantly affected by such parameter. From the simulation results, the main conclusions were: (a) A combination of the Luo coalescence and Luo breakup kernels (Luo-Luo) combined with a decreasing value of c0 improves the gas holdup profiles as compared to empirical values. However, at the lowest value of c0 investigated in this work, the axial liquid velocity deteriorates with regards to experimental data when using a superficial gas velocity of 0.019 m/s. (b) A combination of the Luo coalescence and Lehr breakup models (Luo-Lehr) was shown to improve the gas holdup values with experimental data when compared to the Luo-Luo kernels. However, as c0 decreases, the Luo-Lehr models underestimate the axial liquid velocity profiles with regards to empirical values. (c) A first and second order numerical schemes allowed predicting similar radial profiles of gas holdup and axial liquid velocity. (d) The mesh sensitivity results show that a 3 mm mesh size can be considered as reasonable for simulating experimental data. (e) The inclusion of wall lubrication parameter was found to be significant, although only when using finer meshing. In addition, it allows an improvement of the axial liquid velocity at the core of the bubble column.

Numerical algorithms for the time-space tempered fractional Fokker-Planck equation

This paper aims to provide the high order numerical schemes for the time-space tempered fractional Fokker-Planck equation in a finite domain. The high order difference operators, called the tempered and weighted and shifted Lubich difference operators, are used to approximate the time tempered fractional derivative. The spatial operators are discretized by the central difference methods. We apply the central difference methods to the spatial operators and obtain that the numerical schemes are convergent with orders O(tau^{q} + h^{2}) (q = 1,2,3,4,5). The stability and convergence of the first order numerical scheme are rigorously analyzed. And the effectiveness of the presented schemes is testified with several numerical experiments. Additionally, some physical properties of this diffusion system are simulated.

Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in Hu et al. (2009), with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.

Marching schemes for inverse scattering problems in waveguides with curved boundaries

A marching scheme is developed for inverse scattering problems of the Helmholtz equation in waveguides with curved boundaries. We implement a local orthogonal transform to transform the irregular waveguide in physical plane into a regular rectangle in computing plane. Then the modified Helmholtz system in computational domain is piecewise solved through a second order numerical marching scheme, and we propose a spectral projector based on the truncated local propagating eigenfunction expansion to regularize the marching scheme. In the end, the marching scheme is verified by extensive numerical experiments, and it is shown that the scheme is efficient, stable and accurate in rapidly varying waveguides with curved boundaries, even when the number of propagating modes in the main propagation direction is variable. • A marching method is developed for Cauchy problems of Helmholtz equation. • The method deals easily backscattering in waveguides with curved boundaries. • We present a proof for the principles of a marching algorithm. • The method is extensively verified in waveguides with strong reflections.

Numerical Investigation of Focused Waves and Their Interaction With a Vertical Cylinder Using REEF3D

For the stability of offshore structures, such as offshore wind foundations, extreme wave conditions need to be taken into account. Waves from extreme events are critical from the design perspective. In a numerical wave tank, extreme waves can be modeled using focused waves. Here, linear waves are generated from a wave spectrum. The wave crests of the generated waves coincide at a preselected location and time. Focused wave generation is implemented in the numerical wave tank module of REEF3D, which has been extensively and successfully tested for various wave hydrodynamics and wave–structure interaction problems in particular and for free surface flows in general. The open-source computational fluid dynamics (CFD) code REEF3D solves the three-dimensional Navier–Stokes equations on a staggered Cartesian grid. Higher order numerical schemes are used for time and spatial discretization. For the interface capturing, the level set method is selected. In order to test the generated waves, the time series of the free surface elevation are compared with experimental benchmark cases. The numerically simulated free surface elevation shows good agreement with experimental data. In further computations, the impact of the focused waves on a vertical circular cylinder is investigated. A breaking focused wave is simulated and the associated kinematics is investigated. Free surface flow features during the interaction of nonbreaking focused waves with a cylinder and during the breaking process of a focused wave are also investigated along with the numerically captured free surface.

High Order Numerical Schemes for Second-Order FBSDEs with Applications to Stochastic Optimal Control

AbstractThis is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve the second-order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discretize the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple (Yt,Zt,At,Γt) of the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effectiveness of the proposed numerical schemes. Applications of our numerical schemes to stochastic optimal control problems are also presented.

A new numerical method for variable order fractional functional differential equations

In this letter, a high order numerical scheme is proposed for solving variable order fractional functional differential equations. Firstly, the problem is approximated by an integer order functional differential equation. The integer order differential equation is then solved by the reproducing kernel method. Numerical examples are given to demonstrate the theoretical analysis and verify the efficiency of the proposed method.

DG SOLVER FOR THE SIMULATION OF SIMPLIFIED ELASTIC WAVES IN TWO-DIMENSIONAL PIECEWISE HOMOGENEOUS MEDIA

The theory of elasticity is a very important discipline which has a lot of applications in science and engineering. In this paper we are interested in elastic materials with different properties between interfaces implicated the discontinu- ous coefficients in the governing elasticity equations. The main aim is to develop a practical numerical scheme for modeling the behaviour of a simplified piecewise homogeneous medium subjected to an external action in 2D domains. Therefore, the discontinuous Galerkin method is used for the simulation of elastic waves in such elastic materials. The special attention is also paid to treatment of boundary and interface conditions. For the treatment of the time dependency the implicit Euler method is employed. Moreover, the limiting procedure is incorporated in the resulting numerical scheme in order to overcome nonphysical spurious overshoots and undershoots in the vicinity of discontinuities in discrete solutions. Finally, we present computational results for two-component material, representing a planar elastic body subjected to a mechanical hit or mechanical loading.