On the Stability Analysis of Decoupled Solution Schemes

Many problems in engineering, physics or other disciplines require an integrated treatment of coupled fields. These problems are characterised by a dynamic interaction among two or more physically or computationally distinct components, where the undergoing mathematical model commonly consists of a system of coupled PDE. Considerable progress has been made in the development of appropriate computational schemes to solve such coupled PDE systems. These attempts have resulted in various monolithic and decoupled numerical solution approaches. Despite the unconditional stability offered by implicit monolithic solution strategies, their use is not always recommended. The reason mainly lies in the complexity of the resulting system of equations and the limited flexibility in choosing appropriate time integrators for individual components. This has motivated the elaboration of tailored decoupled solution schemes, which follow the idea of splitting the problem into several sub-problems. But selection of the way of splitting can have a direct influence on the stability of the resulting solution algorithm. This necessitates the stability analysis of such an algorithm. Here, we introduce a general framework for the stability analysis of decoupled solution schemes. The approach is then used to study the stability behaviour of established decoupling strategies applied to typical volume- and surface-coupled problems, namely, coupled problems of thermoelasticity, porous media dynamics and structure-structure interaction. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled Multi-field and Multi-rate Problems – Numerical Solution and Stability Analysis

The numerical solution of coupled differential equation systems is usually done following a monolithic or a decoupled algorithm. In contrast to the holistic monolithic solvers, the decoupled solution strategies are based on breaking down the system into several subsystems. This results in different characteristics of these families of solvers, e. g., while the monolithic algorithms provide a relatively straight-forward solution framework, unlike their decoupled counterparts, they hinder software re-usability and customisation. This is a drawback for multi-field and multi-rate problems. The reason is that a multi-field problem comprises several subproblems corresponding to interacting subsystems. This suggests exploiting an individual solver for each subproblem. Moreover, for the efficient solution of a multi-rate problem, it makes sense to perform the temporal integration of each subproblem using a time-step size relative to its evolution rate. Nevertheless, decoupled solvers introduce additional errors to the solution and, thus, they must always be accompanied by a thorough stability analysis. Here, tailored solution schemes for the decoupled solution of multi-field and multi-rate problems are proposed. Moreover, the stability behaviour of the solutions obtained from these methods are studied. Numerical examples are solved and the reliability of the outcome of the stability analysis is investigated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability Analysis of Decoupled Solution Strategies for Coupled Multi-field Problems – A General Framework

A coupled partial differential equation (PDE) system, stemming from the mathematical modelling of a coupled phenomenon, is usually solved numerically following a monolithic or a decoupled solution method. In spite of the potential unconditional stability offered by monolithic solvers, their usage for solving complex problems sometimes proves cumbersome. This has motivated the development of various partitioned and staggered solution strategies, generally known as decoupled solution schemes. To this end, the problem is broken down into several isolated yet communicating sub-problems that are independently advanced in time, possibly by different integrators. Nevertheless, using a decoupled solver introduces additional errors to the system and, therefore, may jeopardise the stability of the solution [1]. Consequently, to scrutinise the stability of the solution scheme becomes a pertinent step in proposing decoupled solution strategies. Here, we endeavour to present a practical stability analysis algorithm, which can readily be used to reveal the stability condition of numerical solvers. To illustrate its capabilities, the algorithm is then utilised for the stability analysis of solution schemes applied to multi variate coupled PDE systems resulting from the mathematical modelling of surface- and volume-coupled multi-field problems. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Acceleration of partitioned coupling schemes for problems of thermoelasticity

A partitioned coupling scheme for problems of thermo-elasticity at finite strains is presented. The coupling between the mechanical and thermal field is one of the most important multi-physics problem. Typically two different strategies are used to find an accurate solution for both fields: Partitioned or staggered coupling schemes, in which the mechanics and heat transfer is treated as a single field problem, or a monolithic solution of the full problem. Monolithic formulations have the drawback of a non-symmetric system which may lead to extremely large computational costs. Because partitioned schemes avoid this problem and allow for numerical formulations which are more flexible, we consider a staggered coupling algorithm which decouples the mechanical and the thermal field into partitioned symmetric sub-problems by means of an isothermal operator-split. In order to stabilize and to accelerate the convergence of the partitioned scheme, two different methods are employed: dynamic relaxation and a reduced order model quasi-Newton method. A numerical simulation of a quasi-static problem is presented investigating the performance of accelerated coupling schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peer methods

Automatic global error control of numerical schemes is examined. A new approach to this problem is presented. Namely, the problem is reformulated so that the global error is controlled by the numerical method itself rather than by the user. This makes it possible to find numerical solutions satisfying various accuracy requirements in a single run, which so far was considered unrealistic. On the other hand, the asymptotic equality of local and global errors, which is the basic condition of the new method for efficiently controlling the global error, leads to the concept of double quasi-consistency. This requirement cannot be satisfied within the classical families of numerical methods. However, the recently proposed peer methods include schemes with this property. There exist computational procedures based on these methods and polynomial interpolation of fairly high degree that find the numerical solution in a single run. If the integration stepsize is sufficiently small, the error of this solution does not exceed the prescribed tolerance. The theoretical conclusions of this paper are supported by the numerical results obtained for test problems with known solutions.     

Characteristic-based schemes for dispersive waves I. The method of characteristics for smooth solutions

In order to embark on the development of numerical schemes for stiff problems, we have studied a model of relaxing heat flow. To isolate those errors unavoidably associated with discretization, a method of characteristics is developed, containing three free parameters depending on the stiffness ratio. It is shown that such “decoupled” schemes do not take into account the interaction between the wave families and hence result in incorrect wave speeds. We also demonstrate that schemes can differ by up to two orders of magnitude in their rms errors even while maintaining second-order accuracy. We show that no method of characteristics solution can be better than second-order accurate. Next, we develop “coupled” schemes which account for the interactions, and here we obtain two additional free parameters. We demonstrate how coupling of the two wave families can be introduced in simple ways and how the results are greatly enhanced by this coupling. Finally, numerical results for several decoupled and coupled schemes are presented, and we observe that dispersion relationships can be a very useful qualitative tool for analysis of numerical algorithms for dispersive waves. © 1993 John Wiley & Sons, Inc.     

Cluster synchronization for directed community networks via pinning partial schemes

In this paper, we focus on driving a class of directed networks to achieve cluster synchronization by pinning schemes. The desired cluster synchronization states are no longer decoupled orbits but a set of un-decoupled trajectories. Each community is considered as a whole and the synchronization criteria are derived based on the information of communities. Several pinning schemes including feedback control and adaptive strategy are proposed to select controlled communities by analyzing the information of each community such as indegrees and outdegrees. In all, this paper answers several challenging problems in pinning control of directed community networks: (1) What communities should be chosen as controlled candidates? (2) How many communities are needed to be controlled? (3) How large should the control gains be used in a given community network to achieve cluster synchronization? Finally, an example with numerical simulations is given to demonstrate the effectiveness of the theoretical results.     

Staggered solution of fluid-porous-media interaction using the method of local Lagrange multipliers

Surface interaction among non-overlapping bulk-fluid and porous-medium bodies occurs in different situations, e. g., the interaction of blood with a blood vessel wall, a body of water with an earth dam structure, or acoustic waves with acoustic panels used in soundproofing. These are multi-field phenomena, comprising various surface- and volume-coupling mechanisms that should be reflected in the corresponding mathematical models. These models, together with appropriate initial and boundary values, assemble a coupled problem, the solution of which reveals the behaviour of the system under external excitations. The solution is commonly done numerically, following a monolithic or a decoupled approach. Here, the focus is on the latter. To design an efficient decoupled scheme, different types of coupling within the problem are addressed. These are the volume coupling between the degrees of freedom (DOF) within each subdomain, and the surface coupling between the DOF on the common boundaries. In particular, the latter constrains the feasible space of the solution of the problem. In this regard, local Lagrange multipliers (LLM) are employed to reformulate the problem in an unconstrained form. Unlike other domain decomposition methods which are based on using global Lagrange multipliers, the LLM method yields a complete separation of the subdomains and, consequently, facilitates parallel solution of the sub-problems. Moreover, within the subdomains, the penalty method is used to decouple pressure from other DOF. This procedure, on the one hand, reduces the size of the problem that should be solved at the interface and, on the other hand, removes the burden of using mixed finite elements within the subsystems. In the next step, the stability behaviour of the resulting staggered approach is analysed, and the unconditional stability of the method is established. Finally, the method is employed to solve a benchmark example, and using the numerical results, the reliability of the outcomes of the stability analysis is investigated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modified sensitivity equation method for the Euler equations in presence of shocks

The continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti-diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: one which takes into account the contact wave and another that neglects it. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application to uncertainty propagation is investigated and the different numerical schemes are compared.     

Boundary control problems for quasilinear systems of hyperbolic equations

For quasilinear systems of hyperbolic equations, the nonclassical boundary value problem of controlling solutions with the help of boundary conditions is considered. Previously, this problem was extensively studied in the case of the simplest hyperbolic equations, namely, the scalar wave equation and certain linear systems. The corresponding problem formulations and numerical solution algorithms are extended to nonlinear (quasilinear and conservative) systems of hyperbolic equations. Some numerical (grid-characteristic) methods are considered that were previously used to solve the above problems. They include explicit and implicit conservative difference schemes on compact stencils that are linearizations of Godunov’s method. The numerical algorithms and methods are tested as applied to well-known linear examples.     

Second-order,loosely coupled methods for fluid-poroelastic material interaction

This work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time-dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin–Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for the coupled problem. The first method uses the second-order backward differentiation formula (BDF2) for implicit integration, while treating the interface terms explicitly using a second-order extrapolation formula. The second method is the Crank–Nicolson and Leap-Frog (CNLF) method, where the Crank–Nicolson method is used to implicitly advance the solution in time, while the coupling terms are explicitly approximated by the Leap-Frog integration. We show that the BDF2 method is unconditionally stable and uniformly stable in time, while the CNLF method is stable under a CFL condition. Both schemes are validated using numerical simulations. Second-order convergence in time is observed for both methods. Simulations over a longer period of time show that the errors in the solution remain bounded. Cases when the structure is poroviscoelastic and poroelastic are included in numerical examples.     

电力系统中地网腐蚀诊断的一种新的数学模型及其仿真计算

对电力系统中具有重大应用价值的地网腐蚀诊断问题抽象出仿真求解的一种新的数学模型:即求解带约束的非线性隐式方程组模型.但由于问题本身的物理特性决定了所建立的数学模型具有以下特点:一是非线性方程组为欠定方程组,而且非线性程度非常高;二是方程组的所有函数均为隐函数;三是方程组附加若干箱约束条件.这种特性给模型分析与算法设计带来巨大困难.对于欠定方程组的求解,文中根据工程实际背景,尽可能地扩充方程的个数,使之成为超定方程组,然后对欠定方程组和超定方程组分别求解并进行比较.将带约束的非线性隐函数方程组求解问题,转化为无约束非线性最小二乘问题,并采用矩阵求导等技术和各种算法设计技巧克服隐函数的计算困难,最后使用拟牛顿信赖域方法进行计算.大量的计算实例表明,文中所提出的数学模型及求解方法是可行的.与目前广泛采用的工程简化模型相比较,在模型和算法上具有很大优势.     

High-order time integration methods in molecular dynamics

The mechanical behaviour of molecular structures can be described with stiff differential equations, which can not be solved analytically. Several numerical time integration schemes can be found in the literature. The aim of this paper is to present the class of partitioned Runge-Kutta methods applied in molecular dynamics. This class of methods includes a wide range of explicit and implicit, single- and multi-stage, symplectic and non-symplectic, low- and high-order time integration schemes. Also most of the classical methods like explicit and implicit Euler, explicit and implicit midpoint rule, Störmer-Verlet and Newmark are also partitioned Runge-Kutta methods. The schemes are implemented in a finite element code which can serve as a numerical platform for molecular dynamics. This code is used to show the sensitivity of the simulations to the accuracy of the initial values. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New schemes for the finite-element dynamic analysis of piezoelectric devices

New finite-element schemes are proposed for investigating harmonic and non-stationary problems for composite elastic and piezoelectric media. These schemes develop the techniques for the finite-element analysis of piezoelectric structures based on symmetric and partitioned matrix algorithms. In order to take account of attenuation in piezoelectric media, a new model is used which extends the Kelvin model for viscoelastic media. It is shown that this model enables the system of finite-element equations to be split into separate scalar equations. The Newmark scheme in a convenient formulation, which does not explicitly use the velocities and accelerations of the nodal degrees of freedom, is employed for the direct integration with respect to time of the finite-element equations of non-stationary problems. The results of numerical experiments are presented which illustrate the effectiveness of the proposed techniques and their implementation in the ACELAN finite-element software package.     

Two algorithms for solving systems of inclusion problems

The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all components of the systems being a sum of two maximal monotone operators. The algorithms are variants of the forward-backward splitting method and one being a hybrid with the alternating projection method. They consist of approximating the solution sets involved in the problem by separating half-spaces which is a well-studied strategy. The schemes contain two parts, the first one is an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part is the projection step, this being the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating half-spaces, the second chooses one component of the system and projects onto the separating half-space of this case. In the iterative process, the forward-backward operator is computed once per inclusion problem, representing a relevant computational saving if compared with similar algorithms in the literature. The convergence analysis of the proposed methods is given assuming monotonicity of all operators, without Lipschitz continuity assumption. We also present some numerical experiments.     

TWO-GRID CHARACTERISTIC FINITE VOLUME METHODS FOR NONLINEAR PARABOLIC PROBLEMS＊

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.     

Methods for parallel integration of stiff systems of odes

This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagnonal sub-system. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations.The main emphasis is on the formula of order 1, the decoupled implicit Euler formula. It is proved that this formula even for a wide range of multirate formulations has an asymptotic global error expansion permitting extrapolation. Besides, sufficient conditions for absolute stability are presented.     

Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics

We consider the fluid-structure interaction problem arising in haemodynamic applications. The finite elasticity equations for the vessel are written in Lagrangian form, while the Navier-Stokes equations for the blood in Arbitrary Lagrangian Eulerian form. The resulting three fields problem (fluid/ structure/ fluid domain) is formalized via the introduction of three Lagrange multipliers and consistently discretized by p-th order backward differentiation formulae (BDFp). We focus on partitioned algorithms for its numerical solution, which consist in the successive solution of the three subproblems. We review several strategies that all rely on the exchange of Robin interface conditions and review their performances reported recently in the literature. We also analyze the stability of explicit partitioned procedures and convergence of iterative implicit partitioned procedures on a simple linear FSI problem for a general BDFp temporal discretizations.     

Partitioned time stepping schemes for the non-stationary dual-fracture-matrix fluid flow model

This paper presents the coupled, and decoupled schemes for the naturally fractured reservoir consists of the triple-porosity medium. More specifically, the triple-porosity medium contains three contagious porous medium with more permeable macrofractures, less permeable microfractures, and matrix region which is often known as dual-fracture-matrix fluid flow model. Since the matrix has fluid communication with less permeable microfractures, and macrofratures are fed by the microfractures only, the global domain is divided into two subdomains by considering the traditional dual-porosity region and more permeable macrofractures region respectively. The flow and mass exchange between less permeable microfractures and more permeable macrofractures are modeled by two-fluid communication interface conditions while no-communication interface condition is imposed on between matrix and macrofractures region. The weak formulation and the well-posedness of the dual-fracture-matrix model are derived. Furthermore, coupled, implicit-explicit and data-passing partitioned schemes are proposed. The stability and the optimal convergence analysis are derived for both decoupled schemes. Five numerical examples are presented to illustrate the accuracy of the numerical methods and the applicability of the dual-fracture-matrix fluid flow model. Moreover, the parameter sensitivity analysis is performed in the fourth numerical example.