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Showing papers in "Computers & Mathematics With Applications in 2014"


Journal ArticleDOI
TL;DR: In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant.
Abstract: This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators.Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R -linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.

234 citations


Journal ArticleDOI
TL;DR: The electric field potential, electric field and magnetic field in the form of traveling wave solutions for the two-dimensional ZK equation are found by applying the extended direct algebraic method and the efficiency of the method can be demonstrated.
Abstract: The Zakharov-Kuznetsov (ZK) equation is an isotropic nonlinear evolution equation, first derived for weakly nonlinear ion-acoustic waves in a strongly magnetized lossless plasma in two dimensions. In the present study, by applying the extended direct algebraic method, we found the electric field potential, electric field and magnetic field in the form of traveling wave solutions for the two-dimensional ZK equation. The solutions for the ZK equation are obtained precisely and the efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves are analyzed by making graphs of the exact solutions.

209 citations


Journal ArticleDOI
TL;DR: The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations.
Abstract: In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we use Kansa's approach to approximate the spatial derivatives. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. Also, we show that convergence order of the time discrete scheme is O(@t). We solve the two-dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular, triangular and circular domains and also the three-dimensional case is solved on the cubical and spherical domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations. Also, several test problems including the three-dimensional case are given. Numerical examples confirm the efficiency of the proposed scheme.

131 citations


Journal ArticleDOI
TL;DR: This work demonstrates that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme, and proves convergence of the approximate solution to the weak solution.
Abstract: In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann-Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic problems, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature.

127 citations


Journal ArticleDOI
TL;DR: This paper proposes new ways of deriving second-order in time approximations of the potential term of the Allen–Cahn and Cahn–Hilliard equations and proposes a new adaptive time-stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step.
Abstract: In this paper, we focus on efficient second-order in time approximations of the Allen–Cahn and Cahn–Hilliard equations. First of all, we present the equations, generic second-order schemes (based on a mid-point approximation of the diffusion term) and some schemes already introduced in the literature. Then, we propose new ways of deriving second-order in time approximations of the potential term (starting from the main schemes introduced in Guillen-Gonzalez and Tierra (2013)), yielding to new second-order schemes. For these schemes and other second-order schemes previously introduced in the literature, we study the constraints on the physical and discrete parameters that can appear to assure the energy-stability, unique solvability and, in the case of nonlinear schemes, the convergence of Newton’s method to the nonlinear schemes. Moreover, in order to save computational cost we have developed a new adaptive time-stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, we compare the behaviour of the schemes and the effectiveness of the adaptive time-stepping algorithm through several computational experiments.

112 citations


Journal ArticleDOI
TL;DR: A three-dimensional model of the generalized thermoelasticity without energy dissipation under temperature-dependent mechanical properties is established and the modulus of elasticity is taken as a linear function of the reference temperature.
Abstract: A three-dimensional model of the generalized thermoelasticity without energy dissipation under temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of the reference temperature. The resulting formulation in the context of Green and Naghdi model II is applied to a half-space subjected to a time-dependent heat source and traction free surface. The normal mode analysis and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled equations. Numerical results for the field quantities are given in the physical domain and illustrated graphically. The results are also compared to results obtained in the case of temperature-independent modulus of elasticity.

93 citations


Journal ArticleDOI
Regina Ammer, Matthias Markl, Ulric Ljungblad1, Carolin Körner, Ulrich Rüde 
TL;DR: The paper provides a detailed explanation of the modeling of the electron beam gun properties, such as the absorption rate and the energy dissipation, and an algorithm for the construction of a realistic powder bed is discussed.
Abstract: This paper introduces a three dimensional (3D) thermal lattice Boltzmann method for the simulation of electron beam melting processes. The multi-distribution approach incorporates a state-of-the-art volume of fluid free surface method to handle the complex interaction between gas, liquid, and solid phases. The paper provides a detailed explanation of the modeling of the electron beam gun properties, such as the absorption rate and the energy dissipation. Additionally, an algorithm for the construction of a realistic powder bed is discussed. Special emphasis is placed to a parallel, optimized implementation due to the high computational costs of 3D simulations. Finally, a thorough validation of the heat equation and the solid-liquid phase transition demonstrates the capability of the approach to considerably improve the electron beam melting process.

90 citations


Journal ArticleDOI
TL;DR: A new inflow boundary condition is proposed which regularizes the adjoint problem, allowing the use of a stronger test norm, and the robustness of the method is proven.
Abstract: We introduce a DPG method for convection-dominated diffusion problems. The choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter (Demkowicz and Heuer 2011) [18]. We propose a new inflow boundary condition which regularizes the adjoint problem, allowing the use of a stronger test norm. The robustness of the method is proven, and numerical experiments demonstrate the method's robust behavior.

87 citations


Journal ArticleDOI
TL;DR: An efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation is described and it will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function.
Abstract: In this article, we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log-normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials.

86 citations


Journal ArticleDOI
TL;DR: A momentum exchange-based immersed boundary-lattice Boltzmann method, which is used to solve the fluid-flexible-structure-interaction problem, uses a concept of momentum exchange on the boundary to calculate the interaction force.
Abstract: A momentum exchange-based immersed boundary-lattice Boltzmann method, which is used to solve the fluid-flexible-structure-interaction problem, is introduced in this paper. The present method, overcoming the drawback of the conventional penalty method employing a user-defined spring parameter for calculating the interaction force induced by the immersed boundary, uses a concept of momentum exchange on the boundary to calculate the interaction force. Numerical examples, including a laminar flow past a circular cylinder, a filament flapping in the wake of the cylinder, a single filament with the upstream end fixed flapping in a uniform flow field and the interaction of two filaments flapping in the flow, are provided to validate the present method and to illustrate its capability of dealing with the fluid-flexible-structure-interaction problem. Particularly, with considering the filament mass effects, a single filament with a fixed centre point undergoing a bending transition in the flow is firstly studied in the present paper. Our numerical results compare qualitatively well to experimental results. For a single filament with a fixed centre point, it is found that the flexure modulus has a significant effect on the final state of the filament: for a larger flexure modulus, the filament reaches the 'quasi-steady' state finally; for a small flexure modulus, the filaments will be flapping like two filaments.

85 citations


Journal ArticleDOI
TL;DR: Weak Galerkin (WG) finite element methods were used in this paper to approximate weak partial derivatives and their approximations for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra.
Abstract: This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual H 2 -semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an H 2 -equivalent norm for the WG finite element solutions. Error estimates in the usual L 2 norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

Journal ArticleDOI
TL;DR: It is shown that the present method has an edge over existing methods, particularly when applied to large systems of equations.
Abstract: In this paper, we present a three-step iterative method of convergence order five for solving systems of nonlinear equations. The methodology is based on the two-step Homeier's method with cubic convergence (Homeier, 2004). Computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present method has an edge over existing methods, particularly when applied to large systems of equations.

Journal ArticleDOI
TL;DR: By establishing a variational structure and using the critical point theory, the existence of multiple solutions for a class of fractional advection-dispersion equations arising from a symmetric transition of the mass flux is investigated.
Abstract: By establishing a variational structure and using the critical point theory, we investigate the existence of multiple solutions for a class of fractional advection-dispersion equations arising from a symmetric transition of the mass flux. Several criteria for the existence of multiple nonzero solutions are established under certain assumptions.

Journal ArticleDOI
TL;DR: Comparison results reveal that the proposed algorithm can cope with the highly non-linear problems and outperforms many algorithms which exist in the literature.
Abstract: This study proposes a novel chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations. Different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm. Several benchmark studies are carried out. Logistic map gives the best results and is utilized in solving nonlinear equation sets. Nine well known problems are solved with our algorithm and results are compared with Quantum Behaved Particle Swarm Optimization, Intelligent Tuned Harmony Search, Gravitational Search Algorithm and literature studies. Comparison results reveal that the proposed algorithm can cope with the highly non-linear problems and outperforms many algorithms which exist in the literature.

Journal ArticleDOI
TL;DR: The main aim is to generalize the Legendre operational matrices of derivatives and integrals to the three dimensional case and reduce the corresponding fractional order partial differential equations to a system of easily solvable algebraic equations.
Abstract: In this paper, we develop a new scheme for numerical solutions of the fractional two-dimensional heat conduction equation on a rectangular plane. Our main aim is to generalize the Legendre operational matrices of derivatives and integrals to the three dimensional case. By the use of these operational matrices, we reduce the corresponding fractional order partial differential equations to a system of easily solvable algebraic equations. The method is applied to solve several problems. The results we obtain are compared with the exact solutions and we find that the error is negligible.

Journal ArticleDOI
TL;DR: The cascaded LBM is found to be a considerably improved approach to the simulation of multiphase flow over the LBGK, significantly increasing the stability range of both density ratio and Reynolds number.
Abstract: To improve the stability of the lattice Boltzmann method (LBM) at high Reynolds number the cascaded LBM has recently been introduced. As in the multiple relaxation time (MRT) method the cascaded LBM introduces additional relaxation times into the collision operator, but does so in a co-moving reference frame. This has been shown to significantly increase stability at low viscosity in the single phase case. Here the cascaded LBM is further developed to include multiphase flow. For this the force term is calculated by the interaction potential method, and introduced into the collision operator via the exact difference method (EDM). Comparisons are made with the lattice Bhatnagar-Gross-Krook (LBGK) method, and an MRT implementation. Both the cascaded and MRT methods are shown to significantly reduce spurious velocities over the LBGK method. For the particular case of the Shan-Chen interparticle force term calculation with the EDM, the cascaded LBM is successfully combined with a multiphase method, and shown to perform as well as the more established MRT method. The cascaded LBM is found to be a considerably improved approach to the simulation of multiphase flow over the LBGK, significantly increasing the stability range of both density ratio and Reynolds number. Additionally the importance of including third order velocity terms in the equilibria of both the cascaded and MRT methods is discussed.

Journal ArticleDOI
A. Mohsen1
TL;DR: A brief survey of the properties and different treatments of the one-dimensional (1D) and (2D) Bratu problems is presented and nonstandard finite-difference methods with an appropriate amplitude are recommended.
Abstract: A brief survey of the properties and different treatments of the one-dimensional (1D) and (2D) Bratu problems is presented. Different iterative treatments of the resulting nonlinear system of equations are discussed. The finite-difference treatment of the problem is considered. Nonstandard finite-difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended. Bounds on the amplitude for yielding both lower and upper solutions are given.

Journal ArticleDOI
TL;DR: In this article, a quasi-optimal version of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients was proposed and proved to have a sub-exponential convergence rate.
Abstract: In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane C^N. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates.

Journal ArticleDOI
TL;DR: Normal forms associated with codimension-two Hopf–Turing bifurcation are derived, which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the Hopf-Turing pitting point.
Abstract: Spatiotemporal dynamics in a ratio-dependent predator–prey model with diffusion is studied by analytical methods. Normal forms associated with codimension-two Hopf–Turing bifurcation are derived, which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the Hopf–Turing bifurcation point. In the vicinity of this degenerate point, a wealth of complex spatiotemporal dynamics are observed. Our theoretical results are confirmed by numerical simulations.

Journal ArticleDOI
TL;DR: A new numerical method for calculating the consolidation behavior of the stratified, transversely isotropic and poroelastic material is presented by combining the extended precise integration algorithm with the integral transformation techniques.
Abstract: A new numerical method for calculating the consolidation behavior of the stratified, transversely isotropic and poroelastic material is presented by combining the extended precise integration algorithm with the integral transformation techniques. Starting with the governing partial differential equations of a saturated medium with transversely isotropic skeleton and compressible fluid constituents, an ordinary differential matrix equation is deduced with the aid of a Laplace-Hankel transform. An extended precise integration method for internal loading situations is proposed to solve the ordinary differential matrix equation in the transformed domain, and the actual solution is recovered by a numerical inverse transformation. Numerical examples are also provided to prove the feasibility of this method.

Journal ArticleDOI
TL;DR: The implementation of an optimized 3D real-time thermal and turbulent fluid flow solver with a performance of half a billion lattice node updates per second is described in detail.
Abstract: Real-time fluid simulation is an active field of research in computer graphics, but they usually focus on visual impact rather than physical accuracy However, by combining a lattice Boltzmann model with the parallel computing power of a graphics processing unit, both real-time compute capability and satisfactory physical accuracy are now achievable The implementation of an optimized 3D real-time thermal and turbulent fluid flow solver with a performance of half a billion lattice node updates per second is described in detail The effects of the hardware error checking code and the competition between appropriate boundary conditions and performance capabilities are discussed

Journal ArticleDOI
TL;DR: Numerical results show that the proposed immersed boundary-lattice Boltzmann method has second-order accuracy and is not affected by the distribution of Lagrangian points, and shows that the non-slip boundary condition is satisfied on the boundary.
Abstract: A novel immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for incompressible viscous flows in complex geometries. Based on the momentum exchanged-based IB-LBM, an iterative technique is introduced to enforce the non-slip boundary condition at the boundary points. Moreover, the proposed IB-LBM overcomes the drawback that the numerical results of the previous work (Wu and Shu, 2009) which is affected by the distribution of Lagrangian points. A simple theoretical analysis is developed to obtain the optimal iteration parameters. Numerical results show that the present scheme has second-order accuracy and is not affected by the distribution of Lagrangian points. It also shows that the non-slip boundary condition is satisfied on the boundary. This verifies that our IB-LBM is capable of simulating flow problems with complex boundaries.

Journal ArticleDOI
TL;DR: A first and a second order semi-analytical Fourier spectral (SAFS) methods for solving the Allen-Cahn equation, which decompose the original equation into linear and nonlinear subequations, which have closed-form solutions in the Fourier and physical spaces, respectively.
Abstract: In recent years, Fourier spectral methods have been widely used as a powerful tool for solving phase-field equations. To improve its effectiveness, many researchers have employed stabilized semi-implicit Fourier spectral (SIFS) methods which allow a much larger time step than a usual explicit scheme. Our mathematical analysis and numerical experiments, however, suggest that an effective time step is smaller than a time step specified in the SIFS schemes. In consequence, the SIFS scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes. In order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step, we present a first and a second order semi-analytical Fourier spectral (SAFS) methods for solving the Allen-Cahn equation. The core idea of the methods is to decompose the original equation into linear and nonlinear subequations, which have closed-form solutions in the Fourier and physical spaces, respectively. Both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi-implicit Fourier spectral method.

Journal ArticleDOI
TL;DR: In this article, a new two-dimensional variable-order fractional percolation equation is considered, and a new implicit numerical method and an alternating direct method for the 2D model is proposed.
Abstract: Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcy's law with variable order Riemann-Liouville fractional derivatives; this leads to a new variable-order fractional percolation equation.In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

Journal ArticleDOI
TL;DR: This work developed a deterministic model for a woven gas diffusion layer and a stochastic model for the catalyst layer based on clusterization of carbon particles and verified that both of the models developed accurately recover the experimental values of the permeability.
Abstract: This work represents a step towards reliable algorithms for reconstructing the micro-morphology of electrode materials of high temperature proton exchange membrane fuel cells and for performing pore-scale simulations of fluid flow (including rarefaction effects). In particular, we developed a deterministic model for a woven gas diffusion layer (GDL) and a stochastic model for the catalyst layer (CL) based on clusterization of carbon particles. We verified that both of the models developed accurately recover the experimental values of the permeability, without any special ad hoc tuning. Moreover, we investigated the effect of catalyst particle distributions inside the CL on the degree of clusterization and on the microscopic fluid flow, which is relevant for the modeling of degradation (e.g. loss of phosphoric acid). The three-dimensional pore-scale simulations of the fluid flow for the direct numerical calculation of the permeability were performed by the lattice Boltzmann method (LBM).

Journal ArticleDOI
TL;DR: A Petrov-Galerkin discretisation is studied of an ultra-weak variational formulation of the convection-diffusion equation in a mixed form that is well-posed in the limit case of a pure transport problem.
Abstract: A Petrov-Galerkin discretisation is studied of an ultra-weak variational formulation of the convection-diffusion equation in a mixed form. To arrive at an implementable method, the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space. To prevent that this latter space has to be taken increasingly large for vanishing diffusion, a formulation is constructed that is well-posed in the limit case of a pure transport problem. Numerical experiments show approximations that are very close to the best approximations to the solution from the trial space, uniformly in the size of the diffusion term.

Journal ArticleDOI
TL;DR: The vaccination period and the latent period are assumed to have arbitrary distributions that are represented by age-specific rates leaving the vaccinated and the exposed classes, resulting in an SVEIR model with population dynamics.
Abstract: The vaccination period (ie, the time period retaining acquired immunity following vaccination) and the latent period are two important factors affecting disease dynamics Classical compartmental disease models unrealistically assume that these periods are exponentially distributed In this paper, these two ages are assumed to have arbitrary distributions that are represented by age-specific rates leaving the vaccinated and the exposed classes, resulting in an SVEIR model with population dynamics We investigate the global behavior of this model, and derive its basic reproduction number The basic reproduction number completely determines the global dynamics of the model, ie, the disease-free equilibrium is globally asymptotically stable and the disease always dies out when the basic reproduction number is less than unity; whereas when the basic reproduction number is larger than unity, there exists a unique endemic equilibrium which is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists The contributions of the vaccine-wane rate to the basic reproduction number and the level of the endemic equilibrium are displayed

Journal ArticleDOI
TL;DR: A study of four fractional generalized Cattaneo equations from Compte and Metzler (1997) called GCE, G CE I, GCE II, and GCE III and also a fractional version of the parabolic heat equation has observed that when the fractional order is large enough, these equations give temperatures less than absolute zero.
Abstract: The classical parabolic heat equation based on Fourier's law implies infinite heat propagation speed. To remedy this physical flaw, the hyperbolic heat equation is used, but it may instead predict temperatures less than absolute zero. In recent years, fractional heat equations have been proposed as generalizations of heat equations of integer order. By simulating a 1D model problem of size on the order of a thermal energy carrier's mean free path length, we have done a study of four fractional generalized Cattaneo equations from Compte and Metzler (1997) called GCE, GCE I, GCE II, and GCE III and also a fractional version of the parabolic heat equation. We have observed that when the fractional order is large enough, these equations give temperatures less than absolute zero. But if the fractional order is small enough, GCE I does not have this problem when the domain length is comparable to the mean free path length. With larger size, GCE I and GCE III also give non-oscillating solutions for both small and large values of the fractional order.

Journal ArticleDOI
TL;DR: In this article, the authors systematically studied one square jet (AR=1) and four rectangular jets with an aspect ratio of width over height AR=1.5, 2,2.5 and 3 respectively using the lattice Boltzmann method for direct numerical simulation.
Abstract: In this work we systematically study one square jet (AR=1) and four rectangular jets with an aspect ratio of width over height AR=1.5,2,2.5, and 3 respectively using the lattice Boltzmann method for direct numerical simulation. Focuses are on various flow properties on transverse planes downstream to investigate the correlation between the downstream velocity and secondary flow. Three distinct regions of jet development are identified in all the five jets. As the length of the PC (potential core) region maintains about the same, that of the CD (characteristic decay) region strongly depends on the jet aspect-ratio (AR) and Reynolds number (Re). The 45^o and 90^o axis-switching occur in the CD region, with the former followed by the latter at the early and late stages of the CD region respectively. The half-width streamwise velocity contour reveals that 45^o axis-switching is mainly contributed by the corner effect, whereas the aspect-ratio (elliptic) feature affects the shape of the jet when 45^o axis-switching occurs. The close examinations of flow pattern and vorticity contour, as well as the correlation between streamwise velocity and vorticity, indicate that 90^o axis-switching results from the boundary effect. Specific flow patterns for 45^o and 90^o axis-switching are identified to reveal the mechanism of the axis-switchings respectively.

Journal ArticleDOI
TL;DR: A tensor decomposition approach is combined with Chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases.
Abstract: This paper focuses on the curse of dimensionality in the numerical solution of the stationary Fokker–Planck equation for systems with state-independent excitation. A tensor decomposition approach is combined with Chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. Following the enforcement of the normality condition via a penalty method, the discretized system is solved using alternating least squares algorithm. Numerical results for a variety of systems, including separable/non-separable systems, linear/nonlinear systems and systems with/without closed-form stationary solutions up to ten dimensional state-space are presented to illustrate the effectiveness of the proposed method.