A brief survey on numerical methods for solving singularly perturbed problems

The International Workshop on Multi-Rate Processes and Hysteresis conference series focuses on singular perturbation problems and hysteresis as common strongly nonlinear phenomena occurring in mathematical, physical, economical, engineering and information systems. The term ‘strongly nonlinear’ means, in particular, that linearization will not encapsulate the observed phenomena. Singular perturbation problems and hysteresis can be manifested at different stages of the same or similar processes. Furthermore, a number of fundamental hysteresis models can be considered as a limit of time relaxation processes, or admit an approximation by a differential equation, which is singular with respect to a particular parameter. However, interaction between researchers in the areas of systems with time relaxation and systems with hysteresis (and between the ‘multi-rate’ and ‘hysteresis’ research communities) has so far been limited, and there is little cross-fertilization of ideas. It is the aim of the conference series to fill this gap.The 5th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2010) was hosted by the Pollack Mihály Faculty of Engineering, University of Pécs, Hungary, from 31 May to 3 June 2010, on the occasion of Pécs being the Cultural Capital of Europe in 2010. The workshop was organized in cooperation with University College Cork, Ireland, which hosted all of the previous Workshops: International Workshop on Multi-rate Processes and Hysteresis (University College, Cork, Ireland, 31 March–5 April 2008). Proceedings are published in Journal of Physics: Conference Series volume 138. See http://euclid.ucc.ie/appliedmath/murphys2008/murphys2008.htm; International Workshop on Multi-rate Processes and Hysteresis (University College, Cork, Ireland, 3–8 April 2006). Proceedings are published in Journal of Physics: Conference Series volume 55. Further information is available at http://Euclid.ucc.ie/murphys2006.htm; International Workshop on Hysteresis and Multi-scale Asymptotic (University College, Cork, Ireland, 17–21 March 2004). Proceedings are published in Journal of Physics: Conference Series volume 22. Further details are available at http://Euclid.ucc.ie/hamsa2004.htm; International Workshop on Relaxation Oscillations and Hysteresis (University College, Cork, Ireland, 1–6 April 2002). The related collection of invited lectures was published as a volume Singular Perturbations and Hysteresis, SIAM, Philadelphia, 2005. International Workshop on Geometrical Methods of Nonlinear Analysis and Semiconductor Laser Dynamics (University College Cork, Ireland, 5–6 April 2001). A collection of invited papers has been published as a special issue of Proceedings of the Russian Academy of Natural Sciences: Nonlinear dynamics of laser and reacting system, available at http://euclid.ucc.ie/appliedmath/gmna2001/ProcGMNA2001p1.pdf. Among the aims of this and previous workshops were: to bring together the leading experts in singular perturbation and hysteresis phenomena in applied problems; to discuss important problems in the areas of reacting systems, semiconductor lasers, shock phenomena, economic modelling, fluid mechanics, electrical engineering and modelling biological systems with emphasises on hysteresis and singular perturbations; to learn and share modern techniques in areas of common interest. The International Workshop on Multi-rate Processes and Hysteresis (Pollack Mihály Faculty of Engineering, University of Pécs, Hungary, 31 May–3 June 2010) brought together about 50 scientists who are actively researching the areas of dynamical systems with hysteresis and singular perturbations with applications to physical, engineering and economic systems. The countries represented at the Workshop included the Czech Republic, Germany, Hungary, Ireland, Israel, Italy, Poland, Romania, Russia, the United Kingdom and USA. Workshop photo 31 May 2010 Sponsorship of the Workshop by the Pollack Mihály Faculty of Engineering, University of Pécs (Hungary), University College Cork (Ireland), University of Pécs (Hungary), The University of Texas at Dallas (USA), and the Cultural Capital of Europe 2010, Pécs (Hungary), is gratefully acknowledged.The Editors and Organizers of the Workshop are sincerely grateful to Dr Géza Várady, Ms Andrea Zseni and Mr Ádám Schiffer of the Pollack Mihály Faculty of Engineering, University of Pécs, and Dr Alexander Pimenov of University College Cork for managing the organization of the conference and for the assistance in formatting of all the manuscripts.More information about the workshop can be found at http://murphys5.pmmk.pte.hu/Amalia Ivanyi, Péter Iványi, Dmitrii Rachinskii and Vladimir A Sobolev EditorsMURPHYS 2010, PMMK PTE, 31 May – 3 June 2010Sponsored by POLLACK MIHÁLY FACULTY OF ENGINEERING, UNIVERSITY OF PÉCS PÉCSI TUDOMÁNYEGYETEM UNIVERSITY OF PÉCS Cultural Capital of Europe 2010, Pécs, Hungary International Steering Committee Z I BalanovIsrael M BrokateGermany R CrossUK K DahmenUSA M DimianRomania G FriedmanUSA A Ivanyi (Co-Chairman)Hungary P Iványi (Co-Chairman)Hungary L KalachevUSA P KrejčíCzech Republic R O'Malley (Co-Chairman)USA A Pokrovskii (Co-Chairman)Ireland N PopovicUK D Rachinskii (Co-Chairman)Ireland S S SazhinUK V Sobolev (Co-Chairman)Russia S SzabóHungary C VisoneItalyInternational Program Committee G AlmásiHungary Z BalanovIsrael M BrokateGermany R CrossUK K DahmenUSA M DimianRomania G FriedmanUSA A Ivanyi (Co-Chairman)Hungary P Iványi (Co-Chairman)Hungary S JeneiHungary G KádárHungary L KalachevUSA R KersnerHungary G KovácsHungary P KrejčíCzech Republic P M KuczmannHungary P P O'KaneIreland R O'Malley (Co-Chairman)USA A Pokrovskii (Co-Chairman)Ireland N PopovicUK D Rachinskii (Co-Chairman)Ireland B V H ToppingUK V C VisoneItaly

Chapter 1 Introduction to singularly perturbed problems: introduction examples of singularly perturbed problems convection-diffusion problems momentum conservation laws Prandtl equations problem of a thin beam problems of the shock wave structure Burger's equation one dimensional steady reaction-diffusion-convection model Orr-Sommerfeld problem diffusion-drift motion problem idealized problems semilinear problem weakly-coupled systems of ordinary differential equations autonomous equation equation with a power function multiplying the second derivative general idealized problem invariants of equations singular functions definition of the singular functions examples of singular functions layer-type functions notion of layers definition of layers examples of layers partition of layers scale of a layer classification of layers basic approaches to analyze problems with a small parameter method of multivariable asymptotic expansions method of matched asymptotic expansions expansion via differential inequalities numerical methods method of layer-damping transformations comments. Chapter 2 Background for qualitative analysis: introduction differential inequalities scalar problems systems of the second order theorems of inverse monotonicity first order equations second order equations requirements imposed on estimates of the derivatives formulation of an optimal univariate transformation necessary bounds for the first derivative bounds on the higher derivatives uniform bounds on the total variation inequality relations comments. Chapter 3 Estimates of the solution derivatives to semilinear problems: introduction initial problem smooth problem nonsmooth terms second order equations strong ellipticity problem with the condition f(x,u) = xg(x,u) problem of population dynamics theory generalization to mixed boundary conditions and dependence on e equation with a power function affecting the second derivative power singularities exponential singularity generalization to elliptic and parabolic equations estimates of the solution derivatives comments. Chapter 4 Problems for ordinary quasilinear equations: introduction autonomous boundary value problem preliminary results boundary layers interior layers nonautonomous equation estimates of the first derivative graphical chart for localizing the layers example of the problem analysis of the limit solution properties of the limit solution comments. (Part contents).

On Some Basic Concepts in the Analysis of Singular Perturbations

We focus primarily on the convergence analysis of singular perturbation for a class of double phase obstacle inclusions with mixed boundary conditions and two set-valued terms defined respectively in the domain $ \Omega $ and on the boundary $ \partial\Omega_{2} $ . More precisely, we introduce a family of singular perturbation problems corresponding to the double phase obstacle inclusion under consideration. Then we prove the existence of solutions to the singular perturbation problems. Finally, a convergence theorem is established which shows that the solution set of the double phase inclusion can be approximated by the solution set of the singular perturbation problem in the Kuratowski sense.

On the Nipp polyhedron algorithm for solving singular perturbation problems

Stability and Consistency Analysis of Difference Methods for Singular Perturbation Problems

Shock Calculations and the Numerical Solution of Singular Perturbation Problems

The perturbation problem that describes the effect of a weak magnetic field on stellar adiabatic oscillation is considered. This perturbation problem is singular when the magnetic field does not vanish at the stellar surface, and a regular perturbation scheme fails where the magnetic pressure is comparable to the thermodynamic pressure. The application of the Method of Matched Asymptotic Expansion is used to obtain expressions for the eigenfunctions and the eigenfrequencies.

ε-Uniformly convergent fitted mesh finite difference methods for general singular perturbation problems

We consider a singular perturbation problem of Modica-Mortola functional as the thickness of diffused interface approaches to zero. We assume that sequence of functions have uniform energy and square-integral curvature bounds in two dimension. We show that the limit measure concentrates on one rectifiable set and has square integrable curvature.

The governing equations describing methane oxidation in a reverse flow reactor are given by a set of convective-diffusion equations with a nonlinear reaction term, where temperature and methane conversion are dependent variables. In this study, the process is assumed to be a one-dimensional pseudohomogeneous model and takes place with a certain reaction rate in which thewhole process ofthereactor is still workable. Thus, the reaction rate can proceed at a fixed temperature. Under these conditions, we can restrict ourselves to solving the equations for the conversion only. From the available data, it turns out that the ratio of the diffusion term to the reaction term is small. Hence, this ratio is considered as a small parameter in our model and this leads to a singular perturbation problem. Numerical difficulties will be found in the vicinity of a small parameter in front of a higher order term. Here, we present an analytical solutionby means of matched asymptotic expansions. The result shows that, up to and including the first order of approximation, the solution is in agreement with the exact and numerical solutions of the boundary value problem.

Adiabatic elimination is a standard tool in quantum optics, that produces an effective Hamiltonian for a relevant subspace of states, incorporating effects of its coupling to states with much higher unperturbed energy. It shares with techniques from other fields the emphasis on the existence of widely separated scales. Given this fact, the question arises whether it is feasible to improve on the adiabatic approximation, similarly to some of those other approaches. A number of authors have addressed the issue from the quantum optics/atomic physics perspective, and have run into the issue of non-hermiticity of the effective Hamiltonian improved beyond the adiabatic approximation. Even though non-hermitian Hamiltonians are interesting in their own right, this poses conceptual and practical problems. Here, we first briefly survey methods present in the physics literature. Next we rewrite the problems addressed by the adiabatic elimination technique to make apparent the fact that they are singular perturbation problems from the point of view of dynamical systems. We apply the invariant manifold method for singular perturbation problems to this case, and show that this method produces the equation named after Bloch in nuclear physics. Given the wide separation of scales, it becomes intuitive that the Bloch’s equation admits iterative/perturbative solutions. We show, using a fixed point theorem, that indeed the iteration converges to a perturbative solution that produces in turn an exact Hamiltonian for the relevant subspace. We propose thus several sequences of effective Hamiltonians, starting with the adiabatic elimination and improving on it. We show the origin of the non-hermiticity, and that it is inessential given the isospectrality of the effective non-hermitian operator and a corresponding effective hermitian operator, which we build. We propose an application of the introduced techniques to periodic Hamiltonians.

In this paper, linear and nonlinear singularly perturbed problems are studied by a numerical approach based on polynomial differential quadrature. The weighting coefficient matrix is acquired using Chebyshev polynomials. Different classes of perturbation problems are considered as test problems to show the accuracy of method. Then, the quadrature results are compared with analytical solutions of well-known existing solutions.

In §6.2 in the last chapter we briefly discussed, by way of example, some singular perturbation problems in linear differential equations which could be solved using the exponential method developed there. In this chapter we discuss some more generally applicable singular perturbation techniques. These are extremely powerful and let us consider nonlinear problems, involving a small or large parameter, with a view to extracting the principle features of the solutions which cannot be found by classical methods. A fundamental property of nonlinear problems that we shall be interested in is that the main features of the solutions are not contained in the linearized problem as we see in §7.2 below.

The singular perturbation problem $\varepsilon y'' = f(t,y,y')$, $0 < t < 1$, $y(0,\varepsilon )$, $y(1,\varepsilon )$ given, is studied under the principal assumption that $f_{y'y'} $ is never zero in the domain of interest. Solutions are shown to exhibit essentially two types of asymptotic behavior: (i) boundary layer behavior and/or (ii) smooth transition from one stable reduced root to another. In addition, an algorithm for the exact determination of conditions guaranteeing such behavior as well as several illustrative examples are discussed. The results are established by applying an extension of the classical Nagumo theory of differential inequalities.