Multiscale Asymptotic Analysis Research Articles

Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media

In this paper we develop a multiple scattering model for elastic waves in random anisotropic media. It relies on a kinetic approach of wave propagation phenomena pertaining to the situation whereby the wavelength is comparable to the correlation length of the weak random inhomogeneities—the so-called weak coupling limit. The waves are described in terms of their associated energy densities in the phase space position × wave vector. They satisfy radiative transfer equations in this scaling, characterized by collision operators depending on the correlation structure of the heterogeneities. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators, along the same lines as Bal (2005). The model accounts for all possible polarizations of waves in anisotropic elastic media and their interactions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide. Thus it embodies isotropic elasticity which was considered in several previous publications. Some particular anisotropic cases of engineering interest are derived in detail.

Pricing vulnerable options under a stochastic volatility model

Abstract In this paper, we consider the pricing of vulnerable options when the underlying asset follows a stochastic volatility model. We use multiscale asymptotic analysis to derive an analytic approximation formula for the price of the vulnerable options and study the stochastic volatility effect on the option price. A numerical experiment result is presented to demonstrate our findings graphically.

Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

An efficient parallel multiscale numerical algorithm is proposed for a parabolic equation with rapidly oscillating coefficients representing heat conduction in composite material with periodic configuration. Instead of following the classical multiscale asymptotic expansion method, the Fourier transform in time is first applied to obtain a set of complex-valued elliptic problems in frequency domain. The multiscale asymptotic analysis is presented and multiscale asymptotic solutions are obtained in frequency domain which can be solved in parallel essentially without data communications. The inverse Fourier transform will then recover the approximate solution in time domain. Convergence result is established. Finally, a novel parallel multiscale FEM algorithm is proposed and some numerical examples are reported.

Multiscale analysis of a perpetual American option with the stochastic elasticity of variance

A perpetual American option is considered under a generalized model of the constant elasticity of variance model where the constant elasticity is perturbed by a small fast mean-reverting Ornstein–Uhlenbeck process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on option prices as well as optimal exercise prices. Our results improve the existing option price structure in view of flexibility and applicability through the market price of risk. The revealed results may provide useful information on real option problems.

Kinetic Modeling for Transport of Elastic Waves in Anisotropic Heterogeneous Media

This paper deals with the derivation of transport equations for high-frequency elastic bulk waves propagating in anisotropic het- erogeneous media. The former describe the evolution of the energy density associated to those waves in the phase space posi- tion × wave vector. The model accounts for all possible polarization of waves in an anisotropic elastic medium and their inter-actions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide on acoustic axes. Thus it embodies isotropic elasticity which was considered in several previous publications. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators.

Multiscale Asymptotic Analysis and Computation of Optimal Control for Elliptic Systems with Constraints

This paper reports the multiscale analysis and numerical algorithms for a class of distributed elliptic optimal control problems with constraints. Multiscale asymptotic expansions are presented, and an explicit rate of convergence is derived. A multiscale numerical method is developed. Numerical experiments are carried out to validate the predicted convergence results.

Localized elastic buckling: non-linearities versus inhomogeneities

Buckling of an elastic beam column resting on an inhomogeneous Winkler foundation is revisited from the standpoint of multiscale asymptotic analysis. Despite the complexity posed by the presence of nonlinearity and variable coefficients in the governing eigenproblems, our analytical strategy is shown to capture successfully both the linear and the incipient post-buckling regimes. In particular, the present investigation produces a non-linear amplitude equation with variable coefficients that accounts for the spatial inhomogeneity in the normal restoring force of the foundation. Localization is here due to the spectral properties of the corresponding linearized buckling operator, whereas the role of the non-linearities remains confined to that of amplifying such behaviour.

Optimal Boussinesq model for shallow-water waves interacting with a microstructure

In this paper, we consider the propagation of water waves in a long-wave asymptotic regime, when the bottom topography is periodic on a short length scale. We perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced Boussinesq systems parametrized by a free parameter that is the depth at which the velocity is evaluated. We obtain explicit expressions for the coefficients of the resulting effective Korteweg-de Vries (KdV) equations. We show that it is possible to choose the free parameter of the reduced model so as to match the KdV limits of the full and reduced models. Hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem, which has a microstructure. We also discuss the impact of the rough bottom on the effective wave propagation. In particular, nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case.

Elongational perturbations on nematic liquid crystal polymers under a weak shear

The two-dimensional Smoluchowski equation is employed to study the effect of elongational perturbations on nematic liquid crystal polymers under a weak shear. We use the multiscale asymptotic analysis to show that (1) when the elongational perturbation is small relative to the weak shear, the orientational probability density function (pdf) tumbles periodically only in an intermediate range of polymer concentration; outside this intermediate range (i.e., for very small and very large polymer concentration) the orientational pdf converges to a steady state and there is no tumbling. (2) When the elongational perturbation is about 20% of the shear rate or larger, the intermediate range of tumbling disappears and the orientational pdf always converges to a steady state regardless of the polymer concentration. Our theoretical predictions are consistent with various earlier results based on the Leslie–Ericksen theory [C. V. Chaubal and L. G. Leal, J. Non-Newtonian Fluid Mech. 82, 22 (1999)] or analogous 3D numerical simulations [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett. 93, 088301 (2004); M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech. 118, 17 (2004)].

Multiscale Models with Moisture and Systematic Strategies for Superparameterization

Abstract The accurate parameterization of moist convection presents a major challenge for the accurate prediction of weather and climate through numerical models. Superparameterization is a promising recent alternative strategy for including the effects of moist convection through explicit turbulent fluxes calculated from a cloud-resolving model. Basic scales for cloud-resolving modeling are the microscales on the order of 10 km in space on time scales on the order of 15 min, where vertical and horizontal motions are comparable and moist processes are strongly nonlinear (meso-gamma scale). In this paper, systematic multiscale asymptotic analysis is utilized to develop simplified microscale mesoscale dynamic (MMD) models for interaction between the microscales and spatiotemporal mesoscales on the order of 100 km and 2.5 h (meso-beta scale). The new MMD models lead to a systematic framework for superparameterization for numerical weather prediction (NWP) generalizing the traditional column modeling framework. The MMD formulation also provides a flexible systematic framework for devising new parameterization strategies for NWP intermediate between the two extremes of column modeling and detailed cloud-resolving modeling. It is also established here that these MMD models fit crudely into the recent systematic multiscale framework developed to explain the observed larger-scale statistical self-similarity of tropical convection, and therefore provide a systematic framework for superparameterization. Finally, it is shown that the new MMD models have the structure of a heterogeneous multiscale method so that many numerical techniques recently developed in the applied mathematics literature can be applied to this formulation.

Multiscale Modeling of Solid Rocket Motors: Computational Aerodynamic Methods for Stable Quasi-Steady Burning

We consider the problem of efficiently computing the stable burnback of a solid rocket motor when the motor is in the quasi-steady burning regime of operation. When the motor is modeled as a cavity, filled with a compressible fluid with injection from the solid-propellant surface, the problem is a standard one in steady computational aerodynamics. For large rockets, the problem of quasi-steady solid rocket motor core flow is analogous to the steady aerodynamic flow past a large aircraft flying at speeds such that compressible flow effects must be included. One can adopt advanced time-integration strategies that have been applied to commercial or military aircraft to solid rocket motor grain burning to compute a series of steady flows as the grain burns back to near completion. The slow regression of the burning solid-propellant surface is analogous to the motion of control surfaces on the aircraft. With straightforward two-timing, multiscale asymptotic analysis we develop a reduced quasi-steady description of the quasi-steady burning that is extensible to three dimensions. We show how to integrate reduced equations for stable quasi-steady motor burning and use the multigrid method as a representative method to obtain the steady flows in the frozen configurations.

Multiscale Asymptotic Analysis and Numerical Simulation for the Second Order Helmholtz Equations with Rapidly Oscillating Coefficients Over General Convex Domains

The multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper. A multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer. A multiscale numerical algorithm and a postprocessing technique are given. Finally, numerical results show that the method presented in this paper is effective and reliable.

Numerical study of pattern formation in weakly damped parametric surface waves

We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.

Interaction of a solitary wave with an external force

Using the forced Korteweg-de Vries equation as a model, we investigate the interaction of a solitary wave with an isolated moving external force of small amplitude. We use both analysis and numerical calculations. Our analysis is based on a multi-scale perturbation asymptotic analysis and is carried through to either the first- or second-order, depending on whether the first or more accurate second-order expression for the solitary wave speed is used. The perturbation theory leads to a pair of autonomous equations for the solitary wave amplitude and position and these are analyzed in detail for the two extreme cases when the lengthscale of the external force is either much greater, or much less, than that of the solitary wave. The theory predicts that the main régimes are passage, when the solitary wave passes through the external force with some amplitude modulation, repulsion, when the solitary wave is reflected by the external force with a significant amplitude change, and trapping, when the solitary wave remains either partially or totally trapped at the location of the external force. These theoretical predictions are confirmed in our numerical simulations of the forced Korteweg-de Vries equation.