An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Increasing Accuracy and Efficiency for Regularized Navier-Stokes Equations

In this article, a finite element scheme for the family of time relaxation models, that represent a regularization of Navier-Stokes equations, is developed, analyzed and numerically tested. The proposed finite element scheme combines three ideas: (i) the use of an incompressible filter, for better consistency outside the periodic domains, (ii) a second order accurate linearization for the nonlinear term, that allows to solve only one linear system per time step, and (iii) a stabilization in time term that compliments well the linearization. A complete numerical analysis of the scheme, that includes the computability of its numerical solutions, its stability, and velocity error estimates, is given. This is followed by numerical experiments that confirm the theoretical convergence rates and show the advantage of the proposed scheme.     

A numerical method for the Landau–Lifshitz equation with magnetostriction

We present a numerical scheme for Landau–Lifshitz–Gilbert equation coupled with the equation of elastodynamics. The considered physical model describes the behaviour of ferromagnetic materials when magnetomechanical coupling is taken into account. The time‐discretization is based on the backward Euler method with projection. In the numerical approximation, the two equations are decoupled by a suitable linearization in order to solve the magnetic and mechanic part separately. The resulting semi‐implicit scheme is linear and allows larger time‐steps than explicit methods. We prove stability and error estimates for the presented time discretization in 2D. Finally, we test the accuracy of the scheme on an academic numerical example with known exact solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Discretization effects on approximate deconvolution modeling for LES

We study the applicability of low‐order schemes with the approximate deconvolution model (ADM) for large‐eddy simulation. As a test case compressible decaying isotropic turbulence is considered. Results obtained with low‐order finite difference schemes and a pseudospectral scheme are compared with filtered well‐resolved direct numerical simulation (DNS) data. It is found that even for low‐order schemes very good results can be obtained if the cutoff wavenumber of the filter is adjusted to the modified wavenumber of the differentiation scheme.     

Total bounded variation‐based Poissonian images recovery by split Bregman iteration

This paper presents a new total bounded variation regularization‐based Poissonian images deconvolution scheme. Computationally, an extended split Bregman iteration is described to obtain the optimal solution recursively. Moreover, the rigorous convergence analysis of the proposed algorithm is also expatiated here. Compared with the computational speed and the recovered results of the total variation‐based method, numerical simulations definitely demonstrate the competitive performance of the proposed strategy in Poissonian images restoration. Copyright © 2012 John Wiley & Sons, Ltd.     

A compact finite difference method for the solution of the generalized Burgers–Fisher equation

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A compact rational Krylov method for large‐scale rational eigenvalue problems

In this work, we propose a new method, termed as R‐CORK, for the numerical solution of large‐scale rational eigenvalue problems, which is based on a linearization and on a compact decomposition of the rational Krylov subspaces corresponding to this linearization. R‐CORK is an extension of the compact rational Krylov method (CORK) introduced very recently in the literature to solve a family of nonlinear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R‐CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size as the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R‐CORK is more efficient from the point of view of memory and orthogonalization than the classical rational Krylov method applied directly to the linearization. Taking into account that the R‐CORK method is based on a classical rational Krylov method, to implement implicit restarting is also possible, and we show how to do it in a memory‐efficient way. Finally, some numerical examples are included in order to show that the R‐CORK method performs satisfactorily in practice.     

Energy analysis and improved regularity estimates for multiscale deconvolution models of incompressible flows

This paper presents new analytical results and the first numerical results for a recently proposed multiscale deconvolution model (MDM) recently proposed. The model involves a large‐eddy simulation closure that uses a novel deconvolution approach based on the introduction of two distinct filtering length scales. We establish connections between the MDM and two other models, and, on the basis of one of these connections, we establish an improved regularity estimate for MDM solutions. We also prove that the MDM preserves Taylor‐eddy solutions of the Navier–Stokes equations and therefore does not distort this particular vortex structure. Simulations of the MDM are performed to examine the accuracy of the MDM and the effect of the filtering length scales on energy spectra for three‐dimensional homogeneous and isotropic flows. Numerical evidence for all tests clearly indicates that the MDM gives very accurate coarse‐mesh solutions and that this multiscale approach to deconvolution is effective. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient numerical approach to solve Schrödinger equations with space fractional derivative

We design and analyze an efficient numerical approach to solve the coupled Schrödinger equations with space‐fractional derivative. The numerical scheme is based on leap‐frog in time direction and Fourier method in spatial direction. The advantage of the numerical scheme is that only a linear equation needs to be solved for each time step size, and we proved that the energy and mass of space‐fractional coupled Schrödinger equations (SFCSEs) are conserved in the case of full‐discrete scheme. Moreover, we also analyze the error estimate of the numerical scheme, and numerical solutions converge with the order in L² norm. Numerical examples are illustrated to verify the theoretical results.     

A roe-type Riemann solver for hyperbolic systems with relaxation based on time-dependent wave decomposition

Summary. The paper is devoted to the construction of a higher order Roe-type numerical scheme for the solution of hyperbolic systems with relaxation source terms. It is important for applications that the numerical scheme handles both stiff and non stiff source terms with the same accuracy and computational cost and that the relaxation variables are computed accurately in the stiff case. The method is based on the solution of a Riemann problem for a linear system with constant coefficients: a study of the behavior of the solutions of both the nonlinear and linearized problems as the relaxation time tends to zero enables to choose a convenient linearization such that the numerical scheme is consistent with both the hyperbolic system when the source terms are absent and the correct relaxation system when the relaxation time tends to zero. The method is applied to the study of the propagation of sound waves in a two-phase medium. The comparison between our numerical scheme, usual fractional step methods, and numerical simulation of the relaxation system shows the necessity of using the solutions of a fully coupled hyperbolic system with relaxation terms as the basis of a numerical scheme to obtain accurate solutions regardless of the stiffness. Received October 7, 1994 / Revised version received September 27, 1995     

Studies of a mathematical model for temperature-modulated bioluminescence tomography

We introduce and study a mathematical model for temperature-modulated bioluminescence tomography (TBT). The model is capable of self-adjusting values of experimental parameters that are used in the formulation. Major theoretical results of this article include: Solution existence of the model, convergence of numerical solutions, an iterative scheme based on linearization, studies of the solution limiting behaviours when normalized total energy function and/or some or all the energy percentages in individual spectral bands are known exactly. Several numerical examples are included to illustrate the improvement of the accuracy of the reconstructed bioluminescent source distribution due to the employment of measurements from multiple temperature distributions.     

Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation

A competitive nonstandard semi‐explicit finite‐difference method is constructed and used to obtain numerical solutions of the diffusion‐free generalized Nagumo equation. Qualitative stability analysis and numerical simulations show that this scheme is more robust in comparison to some standard explicit methods such as forward Euler and the fourth‐order Runge‐Kutta method (RK4). The nonstandard scheme is extended to construct a semi‐explicit and an implicit scheme to solve the full Nagumo reaction‐diffusion equation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 363–379, 2003.     

Error correction iterative method for the stationary incompressible MHD flow

A new iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations is derived in this paper. The method consists of two steps at each iteration step, we need first to solve the MHD equations by the Oseen-type iterative scheme, and then an error correction strategy is applied to control the error arising from the linearization of the nonlinear MHD equations. The new method not only maintains the advantage of the standard Oseen-type scheme but also possesses a rapid rate of convergence. It is proved that the convergence rate of the proposed method is increased greatly under the uniqueness condition. The uniform stability and convergence of the new scheme are analyzed. Ample numerical experiments are performed to validate the accuracy and the efficiency of the new numerical scheme.     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.     

三阶非线性KdV方程的交替分段显-隐差分格式

对三阶非线性KdV方程给出了一组非对称的差分公式,用这些差分公式与显、隐差分公式组合,构造了一类具有本性并行的交替分段显-隐格式A·D2证明了格式的线性绝对稳定性．对1个孤立波解、2个孤立波解的情况分别进行了数值试验．数值结果显示,交替分段显-隐格式稳定,有较高的精确度．     

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L ² and H ¹ fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.     

“Miniaturized” Linearizations for Quadratic 0/1 Problems

In order to solve a quadratic 0/1 problem, some techniques, consisting in deriving a linear integer formulation, are used. Those techniques, called “linearization”, usually involve a huge number of additional variables. As a consequence, the exact resolution of the linear model is, in general, very difficult. Our aim, in this paper, is to propose “economical” linear models. Starting from an existing linearization (typically the so-called “classical linearization”), we find a new linearization with fewer variables. The resulting model is called “Miniaturized” linearization. Based on this approach, we propose a new linearization scheme for which numerical tests have been performed.     

二维半线性反应扩散方程的交替方向隐格式

本文研究一类二维半线性反应扩散方程的差分方法.构造了一个二层线性化交替方向隐格式.利用离散能量估计方法证明了差分格式解的存在唯一性、差分格式在离散H~1模下的二阶收敛性和稳定性.最后给出两个数值例子验证了理论分析结果.     

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.