Iterative Operator-splitting Methods Research Articles

变系数2D对流扩散方程的高阶迭代算子分裂方法

本文针对变系数2D对流扩散方程，呈现了一种新颖的高阶迭代算子分裂方法。该方法结合了经典迭代格式和Zassenhaus乘积公式。傅立叶谱方法和维数分裂格式用于空间算子。数值实验验证了所提出的方法通过加权方法可以达到高阶精度。此外，新方法不仅可以减少误差而且能够节省大量的CPU时间。 In this paper, a novel higher order iterative operator splitting method is presented for the 2D convection diffusion equation with the variable coefficient. The proposed scheme combines the classical iterative scheme and Zassenhaus product formula for the temporal discretization. And Fourier pseudo spectral method and dimensional splitting scheme are applied for the spatial operators. The numerical results verified that the proposed method can get second order accuracy by weighted iterative scheme. Besides, the new method not only can reduce numerical error but also save a lot of CPU time than the classical iterative method.

PDE/ODE modeling and simulation to determine the role of diffusion in long-term and -range cellular signaling.

BackgroundWe study the relevance of diffusion for the dynamics of signaling pathways. Mathematical modeling of cellular diffusion leads to a coupled system of differential equations with Robin boundary conditions which requires a substantial knowledge in mathematical theory. Using our new developed analytical and numerical techniques together with modern experiments, we analyze and quantify various types of diffusive effects in intra- and inter-cellular signaling. The complexity of these models necessitates suitable numerical methods to perform the simulations precisely and within an acceptable period of time.MethodsThe numerical methods comprise a Galerkin finite element space discretization, an adaptive time stepping scheme and either an iterative operator splitting method or fully coupled multilevel algorithm as solver.ResultsThe simulation outcome allows us to analyze different biological aspects. On the scale of a single cell, we showed the high cytoplasmic concentration gradients in irregular geometries. We found an 11 % maximum relative total STAT5-concentration variation in a fibroblast and a 70 % maximum relative pSTAT5-concentration variation in a fibroblast with an irregular cell shape. For pSMAD2 the maximum relative variation was 18 % in a hepatocyte with a box shape and 70 % in an irregular geometry. This result can be also obtained in a cell with a box shape if the molecules diffuse slowly (with D=1 μm2/s instead of D=15 μm2/s). On a scale of cell system in the lymph node, our simulations showed an inhomogeneous IL-2 pattern with an amount over three orders of magnitude (10−3−1 pM) and high gradients in face of its fast diffusivity. We observed that 20 out of 125 cells were activated after 9 h and 33 in the steady state. Our in-silico experiments showed that the insertion of 31 regulatory T cells in our cell system can completely downregulate the signal.ConclusionsWe quantify the concentration gradients evolving from the diffusion of the molecules in several signaling pathways. For intracellular signaling pathways with nuclear accumulation the size of cytoplasmic gradients does not indicate the change in gene expression which has to be analyzed separately in future. For intercellular signaling the high cytokine concentration gradients play an essential role in the regulation of the molecular mechanism of the immune response. Furthermore, our simulation results can give the information on which signaling pathway diffusion may play a role. We conclude that a PDE model has to be considered for cells with an irregular shape or for slow diffusing molecules. Also the high gradients inside a cell or in a cell system can play an essential role in the regulation of the molecular mechanisms.

Coupled Navier–Stokes molecular dynamics simulation: Theory and applications based on iterative operator-splitting methods

In this paper, we contribute a multi-scale method based on an iterative operator splitting method, which takes into account the disparity between the macro- and micro-scopic scales. We couple the Navier–Stokes equations with molecular dynamics equations, while taking into account their underlying scales. Combining relaxation methods and averaging techniques, we can optimize the computational effort. The motivation arose from modeling fluid transport under the influence of a multiscale problem that has to be solved with smaller time scales, e.g., a non-Newtonian flow problem. The application concerned colloidal damping or fluid–solid problems, where we study an area where the Navier–Stokes equations have too little information about the stream field and we need at least the Boltzmann equation to obtain information about the whole density field. A novel research field is for example that of carbon nanotubes, where we have to couple macro- and micro-models and obtain a fluid–solid area which uses the Lennard–Jones fluid model.The proposed method to solve such delicate problems enables simulations in which the continuum flow aspects of the flow are described by the Navier–Stokes equations at time-scales appropriate for this level of modeling, while the viscous stresses within the Navier–Stokes equations are the result of molecular dynamics simulations, with much smaller time-scales. The main benefit of the proposed method is that time-dependent flows can then be modeled with a computational effort which is significantly less than if the complete flow were to be modeled at the molecular level, as a result of the different time-scales at the continuum and molecular levels, enabled by the application of the iterative operator-splitting method.We carry out a convergence analysis for the splitting methods, see also [30].Finally, we present numerical results for the modified methods and applications to real-life flow problems.

Generalizations and error analysis of the iterative operator splitting method

AbstractThe properties of iterative splitting with two bounded linear operators have been analyzed by Faragó et al. For more than two operators, iterative splitting can be defined in many different ways. A large class of the possible extensions to this case is presented in this paper and the order of accuracy of these methods are examined. A separate section is devoted to the discussion of two of these methods to illustrate how this class of possible methods can be classified with respect to the order of accuracy.

Iterative Operator-splitting Methods with Embedded Discretization Schemes

In this paper we describe a computation of iterative operator-splitting method, which are known as competitive splitting methods, see [11] and [10]. We derived a closed form, based on commutators for the iterative method. The time discretization schemes apply extrapolation schemes and Pade approximations to the exp-functions. Spatial discretization schemes considered Lax-Wendroff methods and are combined with the iterative schemes. The error analysis describe the approximation errors. Numerical examples of ordinary and partial differential equations support the fast computation ideas.

Coupling methods for heat transfer and heat flow: Operator splitting and the parareal algorithm

We propose an operator splitting method for coupling heat transfer and heat flow equations. This work is motivated by the need to couple independent industrial heat transfer solvers (e.g., the Aura-Fluid software package) and heat flow solvers (e.g., Openfoam). Such packages are often used to simulate the influence of solar heat in car bodies and are coupled by A-B splitting techniques. The main goal of this work is the acceleration of the coupled software system by iterative operator splitting methods and additional time-parallelism using the parareal algorithm. We present these new splitting techniques along with some novel convergence results and test the splitting-parareal combination on various numerical problems.

Iterative operator splitting method for capillary formation model in tumor angiogenesis problem: Analysis and application

Iterative operator splitting method is used to solve numerically the mathematical model for capillary formation in tumor angiogenesis problem. The method is based on first splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes. The algorithms are obtained by applying the proposed method to the given model problem. The explicit local error bounds are derived to show consistency. We also explained the stability by constructing the stability functions. The obtained numerical results show that iterative splitting method provides high accuracy and efficiency with respect to other classical methods in the literature. Copyright © 2011 John Wiley & Sons, Ltd.

Stabilization theory of iterative operator-splitting methods

In this paper we present a stabilization theory of iterative operator-splitting methods for linear and nonlinear differential equations. Continuous formulation is described and also the stability for linear and nonlinear cases. We apply linearization techniques to adduce proof of the linear theory. Iterative methods are applied to linearize and couple operator equations. A general theory is derived for linear and nonlinear iterative-splitting methods. Test examples verify the underlying theoretical results.

Iterative operator-splitting methods for nonlinear differential equations and applications

In this article, we consider iterative operator-splitting methods for nonlinear differential equations with bounded and unbounded operators. The main feature of the proposed idea is the embedding of Newton's method for solving the split parts of the nonlinear equation at each step. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator-splitting methods by providing improved results and convergence rates. We apply our results to deposition processes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1026–1054, 2011

Operator-splitting methods in respect of eigenvalue problems for nonlinear equations and applications for Burgers equations

In this article we consider iterative operator-splitting methods for nonlinear differential equations with respect to their eigenvalues. The main focus of the proposed idea is the fixed-point iterative scheme that linearizes our underlying equations. On the basis of the approximated eigenvalues of such linearized systems we choose the order of the operators for our iterative splitting scheme. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator-splitting methods by providing improved results and convergence rates. We apply our results to deposition processes.

Consistency of iterative operator‐splitting methods: Theory and applications

AbstractIn this article, we describe a different operator‐splitting method for decoupling complex equations with multidimensional and multiphysical processes for applications for porous media and phase‐transitions. We introduce different operator‐splitting methods with respect to their usability and applicability in computer codes. The error‐analysis for the iterative operator‐splitting methods is discussed. Numerical examples are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Modified Jacobian Newton Iterative Method: Theory and Applications

This article proposes a new approach to the construction of a linearization method based on the iterative operator-splitting method for nonlinear differential equations. The convergence properties of such a method are studied. The main features of the proposed idea are the linearization of nonlinear equations and the application of iterative splitting methods. We present an iterative operator-splitting method with embedded Newton methods to solve nonlinearity. We confirm with numerical applications the effectiveness of the proposed iterative operator-splitting method in comparison with the classical Newton methods. We provide improved results and convergence rates.

Operator-splitting methods respecting eigenvalue problems for convection-diffusion and wave equations

We discuss iterative operator-splitting methods for convection-diffusion and wave equations motivated from the eigenvalue problem to decide the splitting process. The operator-splitting methods are well-know to solve such complicated multi-dimensional and multi-physical problems. Often the problem, how to decouple the underlying operators, is not understood well enough. We propose a method based on computing the eigenvalues for the simpler problem to decide the splitting operators and the time steps. We present the analysis and the numerical results.

A domain decomposition method based on the iterative operator splitting method

In this article a new approach is proposed for constructing a domain decomposition method based on the iterative operator splitting method. The convergence properties of such a method are studied. The main feature of the proposed idea is the decoupling of space and time. We present a multi-iterative operator splitting method that combines iteratively the space and time splitting. We confirm with numerical applications the effectiveness of the proposed iterative operator splitting method in comparison with the classical Schwarz waveform relaxation method as a standard method for domain decomposition. We provide improved results and convergence rates.

Additive and iterative operator splitting methods and their numerical investigation

As an alternative to the classical splitting methods, two new splitting schemes have been developed recently: the additive and the iterative splitting. In this paper we discuss the most important properties, the advantages and disadvantages of these schemes, and investigate their performance on simple examples as well as on more complex physical models.

Seismic sources and waves using iterative operator splitting methods

AbstractIn this paper, we discuss iterative operator‐splitting methods for wave equations motivated from a realistic problem in seismic sources and waves. The operator splitting methods are well‐known to solve such complicated multi‐dimensional and multi‐physical problems, while decomposing into simpler and computable equations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear iterative operator splitting methods and applications for nonlinear differential equations

AbstractIn this paper we discuss decomposition methods that are used for solving nonlinear differential equations. The motivation arose from the need to decouple nonlinear operator equations into simpler, computable operator equations [1]; [2]. We consider iterative operator‐splitting methods for the decoupling of the differential equations and we apply iterative steps to achieve linearisation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations

In this paper we design higher-order time integrators for systems of stiff ordinary differential equations. We combine implicit Runge–Kutta and BDF methods with iterative operator-splitting methods to obtain higher-order methods. The idea of decoupling each complicated operator in simpler operators with an adapted time scale allows to solve the problems more efficiently. We compare our new methods with the higher-order fractional-stepping Runge–Kutta methods, developed for stiff ordinary differential equations. The benefit is the individual handling of each operator with adapted standard higher-order time integrators. The methods are applied to equations for convection–diffusion reactions and we obtain higher-order results. Finally we discuss the applications of the iterative operator-splitting methods to multi-dimensional and multi-physical problems.

Operator splitting methods for wave equations

The motivation for our studies is coming from simulation of earthquakes, that are modelled by elastic wave equations. In our paper we focus on stiff phanomenons for the wave equations. In the course of this article we discuss iterative operator splitting methods for wave equations motivated by realistic problems dealing with seismic sources and waves. The operator splitting methods are well-known to solve this kind of multidimensional and multiphysical problems. We present the consistency analysis for iterative methods as theoretical background with respect to the underlying boundary conditions. From an algorithmic point of view we discuss the the decoupling and non-decoupling method with respect to the eigenvalues. We verify our methods with test examples for which analytical solutions can be derived. Multidimensional examples are presented for realistic applications for the wave equation. Finally we discuss the results.

Iterative operator-splitting methods for linear problems

The operator-splitting methods are based on splitting of the complex problem into a sequence of simpler tasks. A useful method is the iterative splitting method which ensures a consistent approximation in each step. In our paper, we suggest a new method which is based on the combination of the splitting time interval and the traditional iterative operator splitting. We analyse the local splitting error of the method. Numerical examples are given in order to demonstrate the method.