A finite difference method with non-uniform timesteps for fractional diffusion equations

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behavior of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.     

Alternative forms with orders of the lax equivalence theorem in Banach spaces

The Lax equivalence theorem on the convergence of the solution of the discrete problem to that of the given properly posed initial-value problem states that if the difference scheme is consistent, then stability is necessary and sufficient for convergence. In a recent paper by Butzer-Weis this theorem was equipped with orders in the setting of arbitrary Banach spaces in the sense that consistency, stability and convergence were considered with orders. By modifying the concepts involved suitably, an alternative form of the Lax theorem reads that consistency and stability is equivalent to convergence. This result is also generalized to one containing orders, in fact, both forms of the Lax theorem are valid under the same definitions of consistency, stability, and convergence with orders. An example is given showing that the latter theorem is in a certain sense best possible.     

An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.     

On two upwind finite-difference schemes for hyperbolic equations in non-conservative form

Several finite difference-schemes for approximating solutions of initial value problems associated with systems of linear hyperbolic differential equations are considered. Common features of the schemes is the approximation of the space-like derivatives according to the behavior of the characteristics (upwind schemes for hyperbolic equations). The analysis of standard properties (consistency, stability, convergence, dissipativity, phase error) of finite difference schemes is performed. In addition, extensions of certain upwind schemes to nonlinear equations, extensions to several space-like dimensions by splitting methods and two implicit finite difference schemes are considered.     

The dynamic stresses induced by moving interfacial cracks

Elastodynamic problems involving moving mixed boundary conditions are considered. In particular, uniform and nonuniform propagation in Mode I, II and III types of motion of semi-infinite cracks along the interface of two dissimilar half-spaces are treated. The equations of motion are transformed to a new coordinate system in which the moving tip of the crack appears always at the origin of the coordinates. An implicit three-level numerical method of solution is given which is proved to be more efficient than a previous explicit one. Furthermore, an implicit method for the numerical formulation of the boundary conditions is presented and is shown to yield better results than a previous formulation. The stability analysis of the proposed finite difference approximation is given, and stability criteria are presented as well as a proof of the convergence of the iterative process involved in the numerical formulation of the boundary and interface conditions. The reliability of the present method of solution is examined in several situations where analytical results are known.     

Assessment of implicit operators for the upwind point Gauss-Seidel method on unstructured meshes

The effect of the numerical dissipation level of implicit operators on the stability and convergence characteristics of the upwind point Gauss-Seidel (GS) method for solving the Euler equations was studied through the von Neumann stability analysis and numerical experiments. The stability analysis for linear model equations showed that the point GS method is unstable even for very small CFL numbers when the numerical dissipation level of the implicit operator is equivalent to that of the explicit operator. The stability restriction is rapidly alleviated as the dissipation level of the implicit operator increases. The instability predicted by the linear stability analysis was further amplified as the flow problems became stiffer due to the presence of the shock wave or the refinement of the mesh. It was found that for the efficiency and the robustness of the upwind point GS method, the numerical flux of the implicit operator needs to be more dissipative than that of the explicit operator.     

Hopscotch procedures for a fourth-order parabolic partial differential equation differential equation

In a recent paper (McKee, 1975) the Hopscotch method was applied to solve the fourth-order parabolic (beam) equation. Several computational schemes were discussed which prove to be conditionally stable with the stability range no better than that of the usual explicit scheme.By using two different nets in this paper, the Hopscotch algorithm is applied to the decomposed form of the equation (Richtmyer, 1959) and shown to provide a stable computational scheme.     

离散自适应重复控制: 收敛性分析与实现

针对周期已知情形下的离散周期时变系统, 提出一种自适应重复控制方法, 参数估计采用带死区修正的重复学习投影算法. 关键技术引理在分析离散自适应控制系统时起到了关键作用, 通过推广这一引理, 文中给出重复域关键技术引理, 用于证明离散自适应重复控制系统的稳定性和收敛性. 理论分析表明, 系统的输入和输出信号均有界; 且当周期数趋于足够大时, 跟踪误差收敛于一邻域中, 其半径为干扰的界. 在直线电机实验装置上的应用结果验证了 所提出重复控制方法的有效性.     

An implicit Lyapunov control for finite‐dimensional closed quantum systems

In this paper, the state convergence problem for closed quantum systems is investigated. We consider two degenerate cases, where the internal Hamiltonian of the system is not strongly regular or the linearized system around the target state is not controllable. Both the cases are closely related to practical systems such as one‐dimensional oscillators and coupled two spin systems. An implicit Lyapunov‐based control strategy is adopted for the convergence analysis. In particular, two kinds of Lyapunov functions are defined by implicit functions and their existences are guaranteed by a fixed point theorem. The convergence analysis is investigated by the LaSalle invariance principle for both cases. Moreover, the two Lyapunov functions are unified in a general form, and the characterization of the largest invariant set is presented. Finally, simulation studies are included to show the effectiveness and advantage of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation for the two-dimensional and three-dimensional Riesz space fractional diffusion equations with delay and a nonlinear reaction term

ABSTRACT

In this research, we consider the alternating direction implicit method for solving the two-dimensional and three-dimensional Riesz space fractional diffusion equations with delay and a nonlinear reaction term. The corresponding theoretical results including stability and convergence are provided. Moreover, the convergence order of the proposed method is improved by using the Richardson extrapolation method. The numerical results are presented to show the robustness and effectiveness of the numerical method.     

三维Euler方程的隐式间断有限元算法

为了求解三维欧拉方程,对隐式时间离散格式间断有限元方法进行了研究。根据间断Galerkin有限元方法思想,构造内迭代SOR-LU-SGS隐式时间离散格式,结合当地时间步长技术、多重网格方法,实现了三维流场的计算。数值计算了ONERAM6机翼、大攻角尖前缘三角翼以及DLR-F4翼身组合体的亚声速绕流问题。结果表明,加入SOR内迭代步的LU-SGS隐式算法具有较大的优势,相较于GMRES算法所占用的内存少且收敛速度相当,是LU-SGS算法的三倍以上。针对三维算例,具有较好的稳定性和较高的收敛速度,能够给出准确的流场信息。与原方法相比,SOR-LU-SGS方法无论是在迭代步数上还是在CPU时间上,效率均有明显提高,适合于三维复杂流场计算。     

A hybrid finite element method for stokes flow: Part II—Stability and convergence studies

In Part I we have presented a hybrid finite element method based on an assumed stress field which has the features: (i) the unknowns in the final system of finite element equations are (a) the nodal velocities, and (b) the ‘constant term’ in the arbitrary pressure field over each element; (ii) ‘exact’ integrations were performed for each element.In the following we present studies of stability and convergence of the above hybrid finite element method.     

Globally stabilizing adaptive control design for nonlinearly-parameterized systems

In this note, a new adaptive control design is proposed for nonlinear systems that are possibly nonaffine and contain nonlinearly parameterized unknowns. The proposed control is not based on certainty equivalence principle which forms the foundation of existing and standard adaptive control designs. Instead, a biasing vector function is introduced into parameter estimate; it links the system dynamics to estimation error dynamics, and its choice leads to a new Lyapunov-based design so that affine or nonaffine systems with nonlinearly parameterized unknowns can be controlled by adaptive estimation. Explicit conditions are found for achieving global asymptotic stability of the state, and the convergence condition for parameter estimation is also found. The conditions are illustrated by several examples and classes of systems. Besides global stability and estimation convergence, the proposed adaptive control has the unique feature that it does not contains any robust control part which typically overpowers unknown dynamics, may be conservative, and also interferes with parameter estimation.     

基于前向后向算子分裂的稀疏性正则化图像超分辨率算法

提出了一种新的基于稀疏表示正则化的多帧图像超分辨凸变分模型, 模型中的正则项刻画了理想图 像在框架系统下的稀疏性先验, 保真项度量其在退化模型下与观测信号的一致性, 同时分析了最优解条件. 进一步, 基于前向后向算子分裂法提出了求解该模型的不动点迭代数值算法, 每一次迭代分解为仅对保真项的前向(显式)步与仅对正则项的后向(隐式)步, 从而大幅度降低了计算复杂性; 分析了算法的收敛性, 并采取序贯策略提高收敛速度. 针对可见光与红外图像序列进行了数值仿真, 实验结果验证了本文模型与数值算法的有效性.     

A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver

In this paper, we improve the Navier–Stokes flow solver developed by Sun et al. based on the spectral volume method (SV) in the following two aspects: the development of a more efficient implicit/p-multigrid solution approach, and the use of a new viscous flux formula. An implicit preconditioned LU-SGS p-multigrid method developed for the spectral difference (SD) Euler solver by Liang is adopted here. In the original SV solver, the viscous flux was computed with a local discontinuous Galerkin (LDG) type approach. In this study, an interior penalty approach is developed and tested for both the Laplace and Navier–Stokes equations. In addition, the second method of Bassi and Rebay (also known as BR2 approach) is also implemented in the SV context, and also tested. Their convergence properties are studied with the implicit BLU-SGS approach. Fourier analysis revealed some interesting advantages for the penalty method over the LDG method. A convergence speedup of up to 2-3 orders is obtained with the implicit method. The convergence was further enhanced by employing a p-multigrid algorithm. Numerical simulations were performed using all the three viscous flux formulations and were compared with existing high order simulations (or in some cases, analytical solutions). The penalty and the BR2 approaches displayed higher accuracy than the LDG approach. In general, the numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.     

An IMEX predictor–corrector method for pricing options under regime-switching jump-diffusion models

An efficient second-order method for pricing European and American options under regime-switching jump-diffusion models is presented and analysed for stability and convergence. The implicit–explicit (IMEX) nature of the proposed method avoids the need to invert a full matrix and leads to tridiagonal systems that can be efficiently solved by direct methods. The IMEX predictor–corrector method is coupled with the operator splitting method to solve the linear complementarity problem of the American options. Numerical experiments are performed to demonstrate the stability and second-order convergence of the method.     

A compact ADI Crank–Nicolson difference scheme for the two-dimensional time fractional subdiffusion equation

In this paper, a compact alternating direction implicit (ADI) Crank–Nicolson difference scheme is proposed and analysed for the solution of two-dimensional time fractional subdiffusion equation. The Riemann–Liouville time fractional derivative is approximated by the weighted and shifted Grünwald difference operator and the spatial derivative is discretized by a fourth-order compact finite difference method. The stability and convergence of the difference scheme are discussed and theoretically proven by using the energy method. Finally, numerical experiments are carried out to show that the numerical results are in good agreement with the theoretical analysis.     

Provably Convergent Methods for the Linear and Nonlinear Shape from Shading Problem

In this paper we present provably convergent algorithms for the linear and nonlinear Shape from Shading problem in the case of a Lambertian reflectance map. For the linear problem we discuss two explicit methods and one implicit method, for which we prove convergence for certain light directions. The method for the nonlinear Shape from Shading problem is based on a linear approximation of the image irradiance equation. For the resulting linear PDE the implicit method for the linear problem can be applied. We prove convergence of this method for all light directions.