# Workshop Programme

## for period 26 - 30 March 2007

### The Theory of Highly Oscillatory Problems

26 - 30 March 2007

Timetable

 Monday 26 March 08:30-10:00 Registration 10:00-11:00 Berry, M (Bristol) Three recent results on asymptotics of oscillations Sem 1 The results are separate, and apparently paradoxical, and have implications for physics. First, when two exponentials compete, their interference can be dominated by the contribution with smaller exponent. Second, repeated differentiation of almost all functions in a wide class generates trigonometric oscillations (‘almost all functions tend to cosx’). Third, it is possible to find band-limited functions that oscillate arbitrarily faster than their fastest Fourier component (‘superoscillations’). 11:00-11:30 Coffee 11:30-12:15 Engquist, B (Texas at Austin) Heterogeneous multi-scale methods Sem 1 Continuum simulations of solids or fluids for which some atomistic information is needed are typical example of multi-scale problems with very large ranges of scales. For such problems it is necessary to restrict the simulations on the micro-scale to a smaller subset of the full computational domain. The heterogeneous multi-scale method is a framework for developing and analyzing numerical methods that couple computations from very different scales. Local micro-scale simulations on small domains supply missing data to a macro-scale simulation on the full domain. Examples are local molecular dynamics computations that produce data to a continuum macro-scale model, or a highly oscillatory dynamical system for which a local estimate of resonances is enough to supply data for a smoother evolution of averages. We will focus on dynamical systems. 12:30-13:30 Lunch at Wolfson Court 14:15-15:00 MacKay, RS (Warwick) Hamiltonian slow manifolds with internal oscillation Sem 1 Many Hamiltonian systems possess special families of solutions which can be described approximately as slowly drifting periodic orbits. Examples include the gravitational three-body problem, the interaction of two identical charged particles in a magnetic field, and the propagation and interaction of discrete breathers (time-periodic spatially localised motions in Hamiltonian networks of units). Theory will be presented for how to improve a zeroth order manifold of approximate solutions to r-th order for any r > 0, meaning one that contains all nearby periodic orbits of nearby period and has error of order the r-th power of the drift field (even with a small constant for r=1). In the normally hyperbolic case an exactly invariant nearby submanifold can be constructed. If there are normally elliptic directions, however, this is impossible in general but the above r-th order approximations can be achieved provided that the normal frequencies avoid all multiples of that of the approximately periodic motion and are fast compared with the drift. An effective Hamitlonian is derived to describe the drift of the orbits. Applications to the above fields will be given. An introduction has been published as R.S.MacKay, Slow Manifolds, in Energy localisation and transfer, eds T.Dauxois, A.Litvak-Hinenzon, R.S.MacKay, A.Spanoudaki (World Sci Publ Co, 2004), 149-192. 15:00-15:30 Tea 15:30-16:15 Ainsworth, M (Strathclyde) Dispersive and dissipative behaviour of Galerkin approximation using high order polynomials Sem 1 We consider the dispersive properties of Galerkin finite element methods for wave propagation. The dispersive properties of conforming finite element scheme are analysed in the setting of the Helmholtz equation and an explicit form the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behaviour depending on the order of the polynomials used relative to the mesh-size and the wave number. Quantitative estimates are obtained for the behaviour and rates of decay of the dispersion error in the differing regimes. 16:15-17:00 Dobrokhotov, S (Russian Academy of Sciences) Generalization of the Maslov Theory for Localized Asymptotics and Tsunami waves Sem 1 We suggest a new asymptotic representation for the solutions to the multidimensional wave equations with variable velocity with localized initial data. This representation is the generalization of the Maslov canonical operator based also on a simple relationship between fast decaying and fast oscillating solutions, and on boundary layer ideas. It establishes the connection between initial localized perturbations and wave profiles near the wave fronts including the neighborhood of backtracking (focal or turning) and self intersection points. We show that wave profiles are related with a form of initial sources and also with the Lagrangian manifolds organized by the rays and wavefronts. In particular we discuss the influence of such topological characteristics like the Maslov and Morse indices to metamorphosis of the profiles after crossing the focal points. We apply these formulas to the problem of a propagation of tsunami waves in the frame of so-called piston model''. Finally we suggest a fast asymptotically-numerical algorithm for simulation of tsunami wave over nonuniform bottom. Different scenarios of the distribution of the waves are considered, the wave profiles of the front are obtained in connection with the different shapes of the source and with the diverse rays generating the fronts. It is possible to use suggested algorithm to predict in real time the zones of the beaches where the amplitude of the tsunami wave has dangerous high values. In this connection we also discuss the following questions: the problems of the regularization of the wave field near focal points; ill-possed problems appearing in the geometry of the wavefronts; the inverse problem connected with the possibility of reconstruction of the source via the measurement of the tsunami wave profile on the shelf etc. This work was done together with S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov and was partially supported by RFBR grant N 05-01-00968 and Agreement Between University "La Sapienza", Rome and Institute for Problems in Mechanics RAS, Moscow. Bibliography [1] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, The description of tsunami waves propagation based on the Maslov canonical operator, Doklady Mathematics, 2006, v.74, N 1, pp. 592-596 [2] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, Asymptotic theory of tsunami waves: geometrical aspects and the generalized Maslov representation, Publications of Kyoto Research Mathematical Institute, Vol.4, page 118-153, 2006, ISSN 1880-2818. [3] S.Dobrokhotov, S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov Explicit asymptotics for tsunami waves in framework of the piston model, Russ. Journ. Earth Sciences, 2006, v.8, ES403, pp.1-12 [4] S.Dobrokhotov, S.Sinitsyn, B.Tirozzi, Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. I. The Cauchy Problem, Russ. Jour.Math.Phys., v.14, N1, 2007, pp.28-56 17:00-17:30 Klein, C (MPI) Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation Sem 1 The aim of this talk is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate. Related Links http://www.mis.mpg.de/preprints/2005/prepr2005_124.html - preprint 17:30-18:30 Welcome Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Tuesday 27 March Chair: A Iserles 09:00-09:45 Lubich, C (Tuebingen) Numerical oscillations over long times Sem 1 09:45-10:30 Hairer, E (Geneve) Conservation of energy and actions in numerical discretizations of nonlinear wave equations Sem 1 For numerical discretizations of nonlinearly perturbed wave equations the long-time near-conservation of energy, momentum, and harmonic actions is studied. The proofs are based on the technique of modulated Fourier expansions in time. Rigorous statements on the long-time conservation properties are shown under suitable numerical non-resonance conditions and under a CFL condition. The time step need not be small compared to the inverse of the largest frequency in the space-discretized system. This is joint work with Christian Lubich and David Cohen. 10:30-11:00 Cohen, D (NTNU) Highly oscillatory Hamiltonian systems with non-constant mass matrix Sem 1 We will present a class of numerical methods (based on the trigonometric methods) for such Hamiltonian problems. We will then present a frequency expansion of the numerical solution: the modulated Fourier expansion. The system that determines the coefficients of this expansion has two formal invariants which are related to the total energy and the oscillatory energy of the original system. This allows us to prove the near-conservation of the total and the oscillatory energy for the numerical schemes over very long time intervals. Related Links http://www.math.ntnu.no/~cohen - homepage 11:00-11:30 Coffee 11:30-12:15 Reich, S (Potsdam) Highly oscillatory PDEs, slow manifolds and regularized PDE formations Sem 1 The main motivation of my talk is provided by geophysical fluid dynamics. The underlying Euler or Navier-Stokes equations display oscillatory wave dynamics on a wide range of temporal and spatial scales. Simplified models are often based on the idea of balance and the concept of a slow manifold. Examples are provided by hydrostatic and geostrophic balance. One would also like to exploit these concepts on a computational level. However, slow manifolds are idealized objects that do not fully characterize the complex fluid behavior. I will describe a novel regularization technique that makes use of balance and slow manifolds in an adaptive manner. The regularization approach is based on a reinterpretation of the (linearly) implicit midpoint rule as an explicit time-stepping method applied to a regularized set of Euler equations. Adaptivity can be achieved by means of a predictor-corrector interpretation of the regularization. 12:30-13:30 Lunch at Wolfson Court 14:15-15:00 Sjostrand, J (Ecole Polytechnique) Tunnel effect for Kramers-Fokker-Planck type operators Sem 1 For a large class of Kramers-Fokker-Planck type operators, we determine in the semiclassical (here the low temperature) limit the full asymptotic expansion of the splitting between the lowest eigenvalue (0) and the next one. In a previous work with Herau and C. Stolk we did so in the case when a certain potential (or exponent of a Maxwellian) has preciely one local minimum and then the splitting is "large". In this new work we treat the case when the potential has two local minima. Then the splitting is exponentially small and related to a tunnel effect between the minima. Our most direct source of inspiration has been works by Herau-F.Nier, B.Helffer-Nier, but our methods are quite different. Joint work with F Herau and M Hitrik. 15:00-15:30 Tea 15:30-16:15 Mahalov, A (Arizona State) Fast singular oscillating limits of the 3D Navier-Stokes and Euler equations: Global regularity and three-dimensional Euler dynamics Sem 1 In the first part of the talk, we study highly oscillatory problems for the incompressible 3D Navier-Stokes and prove existence on infinite time intervals of regular strong solutions; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold. The global existence is proven using techniques of fast singular oscillating limits, Lemmas on restricted convolutions and the Littlewood-Paley dyadic decomposition. In the second part of the talk, we analyze regularity and dynamics of highly oscillatory problems for the 3D Euler equations. Detailed proofs can be found in the following references: Asymptotic Analysis, vol. 15, No. 2, p. 103-150, 1997 (with A. Babin and B. Nicolaenko). Indiana University Mathematics Journal, vol. 48, No. 3, p. 1133-1176, 1999 (with A. Babin and B. Nicolaenko). Indiana University Mathematics Journal, vol. 50, p. 1-35, 2001 (with A. Babin and B. Nicolaenko). Russ. Mathematical Surveys, vol. 58, No. 2 (350), p. 287-318, 2003 (with B. Nicolaenko). Methods and Applications of Analysis, vol. 11, No. 4, p. 605-633, 2004 (with B. Nicolaenko, C. Bardos and F. Golse). Hokkaido Mathematical Journal, vol. 35, No. 2, p. 321-364, 2006 (with Y. Giga, K. Inui and S. Matsui). Annals of Math. Studies, Princeton University Press (Editors: J. Bourgain and C. Kenig), In Press (with Y. Giga and B. Nicolaenko). 16:15-17:00 Quispel, R (La Trobe) Some remarks concerning exponential integrators Sem 1 TBA 17:00-17:30 Babych, N (Bath) Spectral results for membrane with perturbed stiffness and mass density Sem 1 We study the spectrum of a nonhomogeneous membrane that consists of two parts with strongly different stiffness and mass density. The small parameter describes the quotient of stiffness coefficients. The M-th power of the parameter is comparable to the ratio of densities. We show that the asymptotic behaviour of eigenvalues and eigenfunctions depends on rate M. We distinguish three cases M<1, M=1 and M>1. The strong resolvent convergence of perturbed operators leads to loss of the completeness of limit eigenfunction system in both cases when M is different from 1. Therefore the limit operators describe only a part of the prelimit membrane vibrations. With an eye to close this gap we use the WKB-asymptotic expansions with a quantized small parameter to prove the existence of other kind of eigenvibrations, namely high frequency vibrations. In the critical case M=1 the limit operator is a nonself-adjoint one, nevertheless the perturbed operators are self-adjoint in a certain topology. The multiplicity of spectrum and structure of root spaces are investigated. Complete asymptotic expansions for eigenelements are constructed and justified in each case of the perturbations. 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Thursday 29 March Chair: E Hairer 09:00-09:45 Zuazua, E (Autonoma) Finite difference approximation of homogenization problems for elliptic equations Sem 1 In this talk, the problem of the approximation by finite differences of solutions to elliptic problems with rapidly oscillating coefficients and periodic boundary conditions will be discussed. The mesh-size is denoted by $h$ while $\ee$ denotes the period of the rapidly oscillating coefficient. Using Bloch wave decompositions, we analyze the case where the ratio $h/\ee$ is rational. We show that if $h/\ee$ is kept fixed, being a rational number, even when $h,\ee\to 0$, the limit of the numerical solution does not coincide with the homogenized one obtained when passing to the limit as $\ee\to 0$ in the continuous problem. Explicit error estimates are given showing that, as the ratio $h/\ee$ approximates an irrational number, solutions of the finite difference approximation converge to the solutions of the homogenized elliptic equation. We consider both the 1-d and the multi-dimensional case. Our analysis yields a quantitative version of previous results on numerical homogenization by Avellaneda, Hou and Papanicolaou, among others. This is a joint work with Rafael Orive 09:45-10:30 Venakides, S (Duke) Rigorous leading order asymptotic solutions of the semiclassical focusing NLS equation Sem 1 Solving an integrable nonlinear differential equation reduces to solving the matrix Riemann-Hilbert problem (RHP) that arises from the invese scattering procedure. Typically, this means finding a matrix m(z) that is analytic in the closed complex plane except on a given oriented contour on which m(z) has a jump discontinuity: the value of m(z) left of the contour equals its value on the right multiplied by a given "jump matrix'' V(z). The space-time variables appear as parameters in the jump matrix; the complex variable z is the spectral variable of the linear differential operator that effects the linearization of the nonlinear system through the Lax pair associated with the system. We derive an explicit formula for the leading asymptotic (oscillatory) solution of the semiclassical focusing nonlinear Schroedinger equation for a class of initial data and prove an error estimate. We extend the analysis when time is large, the asymptotic formula is then expressed in terms of elementary functions. We first outline the notion of a Lax pair and how the RHP arises naturally.We then describe the steepest descent procedure that leads to an explicitly solvable asymptotic RHP through jump matrix factorizations and contour deformations. We also outline the g-funciton mechanism and describe how it determines the successful deformations. 10:30-11:00 Malham, SJA (Heriot-Watt) Numerically evaluating the Evans function by Magnus integration Sem 1 We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane. In this situation, a substantial portion of the computational effort---the numerical evaluation of the iterated integrals which appear in the Magnus series---can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated a priori. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers. 11:00-11:30 Coffee 11:30-12:15 Efendiev, Y (Texas A and M) Multiscale numerical methods using limited global information Sem 1 In this talk, I will describe multiscale numerical methods for flows in heterogeneous porous media. The main idea of these methods is to construct local basis functions that can capture the small scale information when they are coupled via some global formulation. I will discuss the use of local boundary conditions, oversampling methods and the use of global information in constructing basis functions. Applications of these methods to stochastic equations will be also discussed. This is joint work with J. Aarnes, T. Hou, L. Jiang, V. Ginting. 12:30-13:30 Lunch at Wolfson Court 14:00-14:30 Matthews, P (Nottingham) Exponential integrators and functions of the matrix exponential Sem 1 Exponential integrators are the most efficient class of methods for the time-stepping of stiff, semilinear, oscillatory PDEs such as the KdV equation. They solve the stiff, linear part of the PDE exactly. In the case of periodic boundary conditions, a Fourier spectral method can be used, so the linear part is diagonal and the methods can be applied straightforwardly. For other spatial discretizations, functions of the matrix exponential are required, which are susceptible to rounding errors. Several methods for evaluating these functions will be discussed. 14:30-15:00 Hoang, VH (Cambridge) High dimensional finite element methods for elliptic problems with highly oscillating coefficients Sem 1 Elliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work and memory requirement comparable to standard FEM for single scale problems in $\Omega$ while it gives numerical approximations of the correct homogenized limit as well as of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results. 15:00-15:30 Tea 15:30-16:15 Liu, H (Texas at Austin) A multiscale method for stiff ordinary differential equations with resonance Sem 1 We introduce a multiscale method to compute the effective behavior of a class of stiff and highly oscillatory ODEs. The oscillations may be in resonance with one another and thereby generate some hidden slow dynamics. Our method relies on correctly tracking a set of slow variables whose dynamics is effectively closed, and is sufficient to approximate the effective behavior of the ODEs. This set of variables is found by our numerical methods. We demonstrate our algorithms by a few examples that include a commonly studied problem of Fermi, Pasta, and Ulam (FPU). 16:15-17:00 Smyshlyaev, VP (Bath) Localization and propagation in high contrast highly oscillatory media via 'non-classical' homogenization Sem 1 We discuss fundamental mathematical tools for analysis of localization and propagation effects in highly oscillatory media with high contrasts. With the underlying two small parameters of the oscillations and the contrast, there is a "critical" scaling when the phenomena at the micro and macro scales are coupled in a non-trivial way, with "unusual" effects observed in an asymptotically explicit way. The related mathematical tool is that of a "non-classical" (high-contrast) homogenization, accounting for "high-frequency" oscillations, as opposes to the "classical" homogenization whose scope is limited by dealing in effect with low frequencies only. Those tools include "non-classical" two-scale asymptotic expansions, two-scale operator and spectral convergence, and two-scale compactness (with the latter building on, among others, recent deep ideas of V.V. Zhikov [1]). We illustrate this on the problem of wave localization in high contrast periodic media with a defect [2] (a problem relevant to photonic crystal fibres). We also discuss the use of these techniques to the problem of "slowing down" of wave packets in high contrast highly-oscillatory media (the so-called "slow light" effect, with relevance to coupled resonances and metastability), and other prospects. Joint work with Ilia V. Kamotski. [1] V.V. Zhikov, On an extension of the method of two-scale convergence and its applications, (Russian) Mat. Sbornik 191 (2000), 31-72; English translation in Sbornik Math. 191 (2000), 973-1014; V.V. Zhikov, Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients. (Russian) Algebra i Analiz 16 (2004), 34-58; [2] I.V. Kamotski and V.P. Smyshlyaev, Localised eigenstates due to defects in high contrast periodic media via homogenisation. BICS preprint 3/06. http://www.bath.ac.uk/math-sci/bics/preprints/BICS06_3.pdf (2006) Related Links http://www.bath.ac.uk/math-sci/bics/preprints/BICS06_3.pdf - link to paper [2] 17:00-17:30 Matthies, K (Bath) Exponential estimates in averaging and homogenisation Sem 1 Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this talk we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging. In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions. Methods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schr\"odinger equations describing dispersion management, and second-order linear elliptic equations. Related Links http://www.maths.bath.ac.uk/~km230/ - Homepage with links to publications 20:00-18:00 Conference Dinner in the Dining Hall at Magdalene College