Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification

Two different explicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order, 5-point Forward Time Centred Space (FTCS) explicit formula, and the (1,9) FTCS explicit scheme which is generally second-order, but is fourth order when the diffusion number takes the value s = (1/6). These schemes are economical to use, are second-order and have bounded range of stability. The range of stability for the 5-point formula is less restrictive than the (1,9) FTCS explicit scheme. The results of numerical experiments are presented, and accuracy and Central Processor (CPU) times needed for each of the methods are discussed. These schemes use less central processor times than the second-order fully implicit method for two-dimensional diffusion with temperature overspecification. We also give error estimates in the maximum norm for each of these methods.     

Finite volume methods with local refinement for convection-diffusion problems

Computing - Based on approximation of the balance relation for convection-diffusion problems, finite difference schemes on rectangular locally refined grids are derived and studied. A priori...     

Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

In this article, we extend a Milstein finite difference scheme introduced in 8 Giles, M. B. and Reisinger, C. 2012. Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIAM Financ. Math., 3(1): 572–592. (doi:10.1137/110841916)[Crossref] , [Google Scholar] for a certain linear stochastic partial differential equation (SPDE) to semi-implicit and fully implicit time-stepping as introduced by Szpruch 32 Szpruch, L. 2010. Numerical approximations of nonlinear stochastic systems PhD Thesis, University of Strathclyde [Google Scholar] for stochastic differential equations (SDEs). We combine standard finite difference Fourier analysis for partial differential equations with the linear stability analysis in 3 Buckwar, E. and Sickenberger, T. 2011. A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. Math. Comput. Simulation, 81: 1110–1127. (doi:10.1016/j.matcom.2010.09.015)[Crossref], [Web of Science ®] , [Google Scholar] for SDEs to analyse the stability and accuracy. The results show that Crank–Nicolson time-stepping for the principal part of the drift with a partially implicit but negatively weighted double Itô integral gives unconditional stability over all parameter values and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries.     

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $\ell ^2$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax–Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel’s conjectures in Turkel (16(2):109–129, 1977).     

Conservative difference schemes for diffusion problems with boundary and interface conditions

For 0≤x≤1, 0≤t≤T we consider the diffusion equation $$\gamma (x)u_t (x, t) - (B u)_x (x, t) = f(x, t)$$ with (alternatively)B u:=(a(x)u) _x +b(x)u ora(x)u _x +β(x)u. There are given initial valuesu(x,0), influx rates?(B u) (0,t) and (B u) (1,t) across the lateral boundaries and an influx rate (B u) (ζ?0,t)?(B u) (ζ+0,t) at an interface ζ∈(0, 1) where the elsewhere smooth functions γ,a, b, β are allowed to have jump discontinuities.a and γ are assumed to be positive. Interpretingu(x, t) as temperature and γ(x) u (x, t) as energy density we can easily express the total energy \(E(t) = \int\limits_0^1 {\gamma (x) u (x, t)} \) in terms of integrals of the given data. We describe and analyse explicit and implicit one-step difference schemes which possess a discrete quadrature analogue exactly matchingE(t) at the time grid points. These schemes also imitate the isotonic dependence of the solution on the data. Hence stability can be proved by Gerschgorin's method and, under appropriate smoothness assumptions, convergence is 0 ((Δx)²+Δt).     

A difference scheme with high accuracy in time for fourth-order parabolic equations

A novel finite difference method is developed for the numerical solution of fourth-order parabolic partial differential equations in one and two space variables. The method is seen to evolve from a multiderivative method for second-order ordinary differential equations.The method is tested on three model problems, with constant coefficients and variable coefficients, which have appeared in the literature.     

Stability and convergence of difference schemes for multi-dimensional parabolic equations with variable coefficients and mixed derivatives

ABSTRACT

We present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results.     

Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method

A difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimates of high order (orderp), which hold uniformly in the singular perturbation parameterε. The method is based on the use of a defect-correction technique and special adaptively graded and patched meshes, with meshsizes varying betweenh andε ^3/2 h whenp=2, whereh is the meshsize, used in the part of the domain where the solution is smooth, andε ^3/2 h is the final meshsize in the boundary layer. Similar constructions hold for interior layers. The correction operator is a monotone operator, enabling the estimate of the error of optimal order in maximum norm. The total number of meshpoints used in ad-dimensional problem isO(ε ^?s)h ^?d+O(h ^?d), wheres is 1/p or 1/2p, respectively in the case of boundary or interior layer.     

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Abstract We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H ¹-norm and sub-optimal L ²-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L ²(0,T;H ¹)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20     

Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to illustrate and validate the optimal efficiency and robustness of the performance of the present multigrid strategy.     

Causal inference for time series analysis: problems,methods and evaluation

Knowledge and Information Systems - Time series data are a collection of chronological observations which are generated by several domains such as medical and financial fields. Over the years,...     

Linear stability analysis in fluid-structure interaction with transpiration. Part I: Formulation and mathematical analysis

The aim of this work is to provide a new Linearization Principle approach particularly suited for problems in fluid-structure stability. The complexity here, and the main difference with respect to the classical approach, comes from the fact that the full non-linear fluid equations are written in a moving (i.e. time dependent) domain. The underlying idea of our approach uses transpiration techniques [J. Fluid Mech. 4 (1958) 383; G. Mortchéléwicz, Application of linearized Euler equations to flutter, in: 85th AGARD SMP Meeting, Aalborg, Denmark, 1997; P. Raj, B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis, in: AIAA 24th Applied Aerodynamics Conference, number 93-3506, Monterey, Canada, 1993; AIAA 27(6) (1989) 777], with the formalization and linearization recently developed in [Rév. Européenne Élém. Finis, 9(6-7) (2000) 681, A. Dervieux (Ed.), Fluid-Structure Interaction, Kogan Page Science, London, 2003 (Chapter 3)]. This allows us to obtain a new grid independent coupled spectral problem involving the linearized Navier-Stokes equations and those of a reduced linear structure. The coupling is realized through specific transpiration conditions acting on a fixed interface, while keeping a fixed fluid domain. We provide a rigorous mathematical treatment of this eigenproblem. We prove that the corresponding eigenmodes, characterizing the free evolution of the system, can be obtained from the characteristic values of a compact operator acting on a Hilbert space. Moreover, we localize the eigenfrequencies of the system in a parabolic region of the complex plan centered along the positive real axis.     

数字水印协议分析及网络世界相关问题辨析

深入分析了Chin-LaungLei等提出的高效和匿名的数字水印协议以及S.C.Cheung和Hanif Curreem提出的互联网环境中同时适用于一手和二手市场的数字水印协议,指出其缺点和安全漏洞.对网络空间数字作品交易过程中的匿名性问题,二手市场问题,进行了深入细致的分析和探讨.提出在网络空间进行数字作品电子交易时,基于PIG-CA的数字证书本身就为普通消费者提供了匿名保护以及在网络空间数字作品的二手市场没有存在意义的观点.     

Error estimates for two-phase stefan problems in several space variables,I: Linear boundary conditions

The enthalpy formulation of two-phase Stefan problems, with linear boundary conditions, is approximated by C⁰-piecewise linear finite elements in space and backward-differences in time combined with a regularization procedure. Error estimates of L²-type are obtained. For general regularized problems an order ε^1/2 is proved, while the order is shown to be ε for non-degenerate cases. For discrete problems an order h²ε⁻¹+h+τε^−1/2+τ^2/3 is obtained. These orders impose the relations ε∼τ∼h^4/3 for the general case and ε∼h∼τ^2/3 for non-degenerate problems, in order to obtain rates of convergence h^2/3 or h respectively. Besides, an order h|log h|+τ^1/2 is shown for finite element meshes with certain approximation property. Also continuous dependence of discrete solutions upon the data is proved. This work was supported by the “Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)” of Argentina.     

Numerical stability of finite difference algorithms for electrochemical kinetic simulations: Matrix stability analysis of the classic explicit,fully implicit and Crank-Nicolson methods and typical problems involving mixed boundary conditions

The stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference discretizations of example diffusional initial boundary value problems from electrochemical kinetics has been investigated using the matrix method of stability analysis. Special attention has been paid to the effect of the discretization of the mixed, linear boundary condition with time-dependent coefficients on stability, assuming the two-point forward-difference approximations for the gradient at the left boundary (electrode). Under accepted assumptions one obtains the usual stability criteria for the classic explicit and fully implicit methods. The Crank-Nicolson method turns out to be only conditionally stable in contrast to the current thought regarding this method.     

Computing stability domains in the space of time delay and controller coefficients for FOPDT and SOPDT systems

In this paper, a method is proposed to compute the maximum allowable time delay for first-order plus dead time and second-order plus dead time systems, in order to maintain stability. Designing first order controllers for such systems to preserve stability for a longer time delay is the main aim of this paper. The procedure uses the properties of the phase diagram of the open-loop transfer functions of such systems. The effects of the variation of the controller coefficients on the maximum allowable time delay of the systems are investigated and the stability domains in the space of the uncertain delay and the controller coefficients are computed. The results can be used to design first-order robust controllers for first and second order processes containing uncertain delays.     

Exhaustive stability analysis in a consensus system with time delay and irregular topologies

A consensus problem and its stability are studied for a group of agents with second-order dynamics and communication delays. The communication topologies are taken as irregular but always connected and undirected. The delays are assumed to be quasi-static and the same for all the interagent channels. A decentralised, PD-like control structure is proposed to create a consensus in the position and velocity of the agents. We present an interesting factorisation feature for the characteristic equation of the system which simplifies the stability analysis considerably from a prohibitively large dimensional problem to a manageable small scale. It facilitates a rare stability picture in the space of the control parameters and the delay, utilising a paradigm named cluster treatment of characteristic roots (CTCR). The influence of the individual factors on the absolute and relative stability of the system is studied. This leads to the introduction of two novel concepts: the most exigent eigenvalue, which refers to the one that defines the delay stability margin of the system, and the most critical eigenvalue, which is the one that dictates the consensus speed of the system. It is observed that the most exigent eigenvalue is not always the most critical, and this feature may be used as a design tool for the control logic. Case studies and simulations results are presented to verify these concepts.     

Recent experiences with error estimation and adaptivity, Part II: Error estimation for h-adaptive approximations on grids of triangles and quadrilaterals

In Part I, residual and flux projection error estimators for finite element approximations of scalar elliptic problems were reviewed; numerical studies on the performance of these estimators were presented for finite element approximations of the solution of Poisson's equation on uniform grids of hierarchic triangles of order p (1 p 7). Here further numerical experiments are given which also include error estimators for the vector-valued problem of plane elastostatics and implementations for h-adaptive grids of triangles and quadrilaterals which are constructed using an algorithm of equidistribution of error coupled with h-refinement or h-remeshing schemes. A detailed numerical study of several flux-projectors for h-adaptive grids of bilinear and biquadratic quadrilaterals is conducted; a flux equilibration iteration, which may be employed in some cases to improve flux projection estimates, is also included. FAor the case of grids of quadrilaterals, several versions of the element residual estimators, which differ by the approximate flux employed for the calculation of the boundary integral term in the definition of the local problems, are compared. The numerical experiments confirm the good overall performance of residual estimates and indicate that flux projection estimates, which are now operational in several commercial codes, may be divergent when they are employed to estimate the error in even order h-adaptive approximations.     

A fast algorithm for color space conversion and rounding error analysis based on fixed-point digital signal processors

The YCbCr (luminance, chrominance-blue, and chrominance-red) color space is adopted in video codecs or transmission, while the HSV (hue, saturation and value) color space is used in some video analysis algorithms. In this paper, a fast algorithm based on fixed-point digital signal processors (DSPs) is proposed for YCbCr to HSV conversion. Floating-point multiplications are replaced with fixed-point shifts, 16-bit fixed-point multiplications, and additions. To compensate for rounding error and convert floating-point multiplication into 16-bit fixed-point multiplication in the calculation, tables occupying 225 bytes and 256 bytes are established, respectively. According to the experimental results, the error caused by the proposal is within the acceptable range. At the same time, the proposed algorithm is about 10 times faster than traditional YCbCr to HSV conversion algorithm on a fixed-point digital signal processor (DSP) platform and about 1.41 times faster on a personal computer (PC) platform.