An Asymptotic-Numerical Method for a Class of Weakly Coupled System of Singularly Perturbed Convection-Diffusion Equations

A weakly coupled convection dominated system of m-equations is analyzed. A higher order accurate asymptotic-numerical method is presented. The solutions of convection dominated problem are known to exhibit multi-scale character. There exist narrow region across the boundary of the domain where the solution exhibit steep gradient. This region is termed as boundary layer region and the solution of problem is said to have a boundary layer. Outside of this region, the solution of system behaves smoothly. To capture this multi-scale nature given system is factorized into two explicit systems. The degenerate system of initial value problems (IVPs), obtained by setting ??=?0, corresponds to the smooth solution, which lies outside of boundary layers. For solution inside boundary layers, a system of boundary value problems (BVPs) is obtained using stretching transformation. Regardless of this simple factorization, solutions of these systems preserve the key features of the given coupled system. Runge–Kutta method is used to solve the degenerate system of IVPs, whereas the system of BVPs is solved analytically. Stability and consistency of the proposed method is established. A uniform convergence of higher order is obtained. Possible extension to differential difference equations are also brought to attention. A comparative study of the present method with some state of art existing numerical schemes is carried out by means of several test problems. The results so obtained demonstrate the effectiveness and potential of present approach.     

Fast Iterative Solution of Saddle Point Problems in Optimal Control Based on Wavelets

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries.Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only ₂-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the ₂-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization.In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows N _J on that level.Finally, numerical results are provided.     

Performance comparison of the NWF and DC methods for implementing High-Resolution schemes in a fully coupled incompressible flow solver

This paper reports on the use of the Normalized Weighting Factor (NWF) method and the Deferred Correction (DC) approach for the implementation of High Resolution (HR) convective schemes in an implicit, fully coupled, pressure-based flow solver. Four HR schemes are realized within the framework of the NWF and DC methods and employed to solve the following three laminar flow problems: (i) lid-driven flow in a square cavity, (ii) sudden expansion in a square cavity, and (iii) flow in a planar T-junction, over three grid systems with sizes of 10⁴, 5 × 10⁴, and 3 × 10⁵ control volumes. The merit of both approaches is demonstrated by comparing the computational costs required to solve these problems using the various HR schemes on the different grid systems. Whereas previous attempts to use the NWF method in a segregated flow solver failed to produce converged solutions, current results clearly demonstrate that both methods are suitable for utilization in a coupled flow solver. In terms of CPU efficiency, there is no global and consistent superiority of any method over another even though the DC method outperformed the NWF method in two of the three test problems solved.     

Preconditioning trace coupled 3d‐1d systems using fractional Laplacian

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work, we consider a simplified model problem of a 3d‐1d coupling and the main objective is to construct algorithms that may utilize standard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three‐dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well‐defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the system. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.     

Optimization of a Class of Nonlinear Dynamic Systems: New Efficient Method without Lagrange Multipliers

This paper deals with optimization of a class of nonlinear dynamic systems with n states and m control inputs commanded to move between two fixed states in a prescribed time. Using conventional procedures with Lagrange multipliers, it is well known that the optimal trajectory is the solution of a two-point boundary-value problem. In this paper, a new procedure for dynamic optimization is presented which relies on tools of feedback linearization to transform nonlinear dynamic systems into linear systems. In this new form, the states and controls can be written as higher derivatives of a subset of the states. Using this new form, it is possible to change constrained dynamic optimization problems into unconstrained problems. The necessary conditions for optimality are then solved efficiently using weighted residual methods.     

Improved T − ψ nodal finite element schemes for eddy current problems

The aim of this paper is to propose improved T − ψ finite element schemes for eddy current problems in the three-dimensional bounded domain with a simply-connected conductor. In order to utilize nodal finite elements in space discretization, we decompose the magnetic field into summation of a vector potential and the gradient of a scalar potential in the conductor; while in the nonconducting domain, we only deal with the gradient of the scalar potential. As distinguished from the traditional coupled scheme with both vector and scalar potentials solved in a discretizing equation system, the proposed decoupled scheme is presented to solve them in two separate equation systems, which avoids solving a saddle-point equation system like the traditional coupled scheme and leads to an important saving in computational effort. The simulation results and the data comparison of TEAM Workshop Benchmark Problem 7 between the coupled and decoupled schemes show the validity and efficiency of the decoupled one.     

Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

One‐dimensional models of gravity‐driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first‐order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic‐wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first‐order quasi‐linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so‐called Lagrangian‐remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341–B1368) for a multiclass Lighthill–Whitham‐Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian‐antidiffusive remap (L‐AR) schemes for the polydisperse sedimentation model is constructed. These L‐AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1109–1136, 2016     

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

     

Operator-splitting galerkin methods for one kind of oin reaction model for the pollution in groundwater

Abstract. The operator-splitting methods for the mathematic model of one kind of oin reactionsfor the problem of groundwater are considered. Optimal error estimates in Lz and Hl norm areobtained for the approximation solution.     

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)     

Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Several choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two‐phase medium at the microscopic scale. This system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single‐phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial‐exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two‐scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings. Copyright © 2008 John Wiley & Sons, Ltd.     

An approach to the Gummel map by vector extrapolation methods

The numerical approximation of nonlinear partial differential equations requires the computation of large nonlinear systems, that are typically solved by iterative schemes. At each step of the iterative process, a large and sparse linear system has to be solved, and the amount of time elapsed per step grows with the dimensions of the problem. As a consequence, the convergence rate may become very slow, requiring massive cpu-time to compute the solution. In all such cases, it is important to improve the rate of convergence of the iterative scheme. This can be achieved, for instance, by vector extrapolation methods. In this work, we apply some vector extrapolation methods to the electronic device simulation to improve the rate of convergence of the family of Gummel decoupling algorithms. Furthermore, a different approach to the topological ε-algorithm is proposed and preliminary results are presented.     

On a class of cyclic methods for the numerical integration of stiff systems of O.D.E.s

A class of cyclic linear multistep methods suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is developed. Particular attention is paid to the problem of deriving schemes which are almostA-stable, self starting, have relatively high orders of accuracy and contain a built in error estimate. These requirements demand that the linear multistep methods which are used are solved iteratively rather than directly in the usual way and an efficient method for doing this is suggested. Finally the algorithms are illustrated by application to a particular test problem.     

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L ² least-squares scheme or a weighted L ² least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L ² least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L ² least-squares scheme seems to be better than the iterative L ² least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.     

A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods (ABC-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in ABC-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. ABC-schemes are A- and L-stable methods of the second order, but there are ABC-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of ABC-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.     

Fractional-step <Emphasis Type="Italic">θ</Emphasis>-method for solving singularly perturbed problem in ecology

We consider a predator-prey model arising in ecology that describes a slow-fast dynamical system. The dynamics of the model is expressed by a system of nonlinear differential equations having different time scales. Designing numerical methods for solving problems exhibiting multiple time scales within a system, such as those considered in this paper, has always been a challenging task. To solve such complicated systems, we therefore use an efficient time-stepping algorithm based on fractional-step methods. To develop our algorithm, we first decouple the original system into fast and slow sub-systems, and then apply suitable sub-algorithms based on a class of θ-methods, to discretize each sub-system independently using different time-steps. Then the algorithm for the full problem is obtained by utilizing a higher-order product method by merging the sub-algorithms at each time-step. The nonlinear system resulting from the use of implicit schemes is solved by two different nonlinear solvers, namely, the Jacobian-free Newton-Krylov method and the well-known Anderson’s acceleration technique. The fractional-step θ-methods give us flexibility to use a variety of methods for each sub-system and they are able to preserve qualitative properties of the solution. We analyze these methods for stability and convergence. Several numerical results indicating the efficiency of the proposed method are presented. We also provide numerical results that confirm our theoretical investigations.     

A note on some computationally difficult set covering problems

Fulkerson et al. have given two examples of set covering problems that are empirically difficult to solve. They arise from Steiner triple systems and the larger problem, which has a constraint matrix of size 330 × 45 has only recently been solved. In this note, we show that the Steiner triple systems do indeed give rise to a series of problems that are probably hard to solve by implicit enumeration. The main result is that for ann variable problem, branch and bound algorithms using a linear programming relaxation, and/or elimination by dominance require the examination of a super-polynomial number of partial solutionsThis paper was written while the author was a CORE Fellow at the Université de Louvain, Louvain-la-Neuve, Belgium.     

Gauss-Newton and Inverse Gauss-Newton Methods for Coefficient Identification in Linear Elastic Systems

The “inverse problem” of determining parameter distributions in linear elastic structures has been explored widely in the literature. In the present article we discuss this problem in the context of a particular formulation of linear elastic systems, dividing the associated inverse problems into two classes which we call Case 1 and Case 2. In the first case the elastic parameters can be obtained by solving a certain set of linear algebraic equations, typically poorly conditioned. In the second case the corresponding problem involves nonlinear equations which usually must be solved by approximation methods, including the Gauss-Newton method for overdetermined systems. Here we discuss the application of this method and a related, empirically more stable, method which we call the inverse Gauss-Newton method. Convergence theorems are established and computational results for sample problems are presented.     

Splitting scheme for poroelasticity and thermoelasticity problems

Boundary value problems in thermoelasticity and poroelasticity (filtration consolidation) are solved numerically. The underlying system of equations consists of the Lamé stationary equations for displacements and nonstationary equations for temperature or pressure in the porous medium. The numerical algorithm is based on a finite-element approximation in space. Standard stability conditions are formulated for two-level schemes with weights. Such schemes are numerically implemented by solving a system of coupled equations for displacements and temperature (pressure). Splitting schemes with respect to physical processes are constructed, in which the transition to a new time level is associated with solving separate elliptic problems for the desired displacements and temperature (pressure). Unconditionally stable additive schemes are constructed by choosing a weight of a three-level scheme.     

Numerical analysis of the hyperbolic two-temperature model

The system of hyperbolic heat conduction problems is solved numerically. The explicit and fully implicit Euler type schemes for the time integration of the nonstationary problem are proposed and investigated. Space derivatives are approximated by using the finite volume method, resulting in conservative and monotonous discrete approximations of the second order of accuracy. The stability analysis is done in the L ₂ and energy norms for a simplified one-temperature equation and the system of two equations, describing the temperature and the flux. Results of numerical experiments are presented. This work was supported by the Lithuanian State Science and Studies Foundation within the projects B-03/2007, B-09/2007 and by the Agency for International Science and Technology Development Programmes in Lithuania within the EUREKA projects E!3691 OPTCABLES and E!3483 EULASNET LASCAN.