Differential Equations II

Exercises and solutions for a second year differential equations course.

Addressing endogeneity in the relationship between health and employment through a simultaneous equations model

Este documento analiza la relación de doble causalidad entre salud y empleo y su comportamiento dinámico usando datos de Estados Unidos tomados del PSID (Pane Study of Income Dynamics). Este estudio usa dos variables dependientes (Estado de salud auto-reportado y Empleo), las cuales son estimadas usando un modelo probit bivariado para abordar el problema de endegeneidad presente en dicha relación. Los resultados muestran evidencia significativa de la existencia de dicha endogeneidad y del impacto positivo que tiene sobre la probabilidad de ser empleado tener un buen estado de salud y vicesersa, sin embargo, el impacto de la situación de empleo sobre el estado de salud se encuentra que no es significativa.

A variational approach for calculating Franck-Condon factors including mode-mode anharmonic coupling

We have implemented our new procedure for computing Franck-Condon factors utilizing vibrational configuration interaction based on a vibrational self-consistent field reference. Both Duschinsky rotations and anharmonic three-mode coupling are taken into account. Simulations of the first ionization band of Cl O2 and C4 H4 O (furan) using up to quadruple excitations in treating anharmonicity are reported and analyzed. A developer version of the MIDASCPP code was employed to obtain the required anharmonic vibrational integrals and transition frequencies

Variational calculation of static and dynamic vibrational nonlinear optical properties

The vibrational configuration interaction method used to obtain static vibrational (hyper)polarizabilities is extended to dynamic nonlinear optical properties in the infinite optical frequency approximation. Illustrative calculations are carried out on H2 O and N H3. The former molecule is weakly anharmonic while the latter contains a strongly anharmonic umbrella mode. The effect on vibrational (hyper)polarizabilities due to various truncations of the potential energy and property surfaces involved in the calculation are examined

Variational calculation of vibrational linear and nonlinear optical properties

A variational approach for reliably calculating vibrational linear and nonlinear optical properties of molecules with large electrical and/or mechanical anharmonicity is introduced. This approach utilizes a self-consistent solution of the vibrational Schrödinger equation for the complete field-dependent potential-energy surface and, then, adds higher-level vibrational correlation corrections as desired. An initial application is made to static properties for three molecules of widely varying anharmonicity using the lowest-level vibrational correlation treatment (i.e., vibrational Møller-Plesset perturbation theory). Our results indicate when the conventional Bishop-Kirtman perturbation method can be expected to break down and when high-level vibrational correlation methods are likely to be required. Future improvements and extensions are discussed

A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations

In this paper a cell by cell anisotropic adaptive mesh technique is added to an existing staggered mesh Lagrange plus remap finite element ALE code for the solution of the Euler equations. The quadrilateral finite elements may be subdivided isotropically or anisotropically and a hierarchical data structure is employed. An efficient computational method is proposed, which only solves on the finest level of resolution that exists for each part of the domain with disjoint or hanging nodes being used at resolution transitions. The Lagrangian, equipotential mesh relaxation and advection (solution remapping) steps are generalised so that they may be applied on the dynamic mesh. It is shown that for a radial Sod problem and a two-dimensional Riemann problem the anisotropic adaptive mesh method runs over eight times faster.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Weak constraints in four-dimensional variational data assimilation

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state.We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.

Four-dimensional variational assimilation of ozone profiles from the Microwave Limb Sounder on the Aura satellite

Ozone profiles from the Microwave Limb Sounder (MLS) onboard the Aura satellite of the NASA's Earth Observing System (EOS) were experimentally added to the European Centre for Medium-range Weather Forecasts (ECMWF) four-dimensional variational (4D-var) data assimilation system of version CY30R1, in which total ozone columns from Scanning Imaging Absorption Spectrometer for Atmospheric CHartographY (SCIAMACHY) onboard the Envisat satellite and partial profiles from the Solar Backscatter Ultraviolet (SBUV/2) instrument onboard the NOAA-16 satellite have been operationally assimilated. As shown by results for the autumn of 2005, additional constraints from MLS data significantly improved the agreement of the analyzed ozone fields with independent observations throughout most of the stratosphere, owing to the daily near-global coverage and good vertical resolution of MLS observations. The largest impacts were seen in the middle and lower stratosphere, where model deficiencies could not be effectively corrected by the operational observations without the additional information on the ozone vertical distribution provided by MLS. Even in the upper stratosphere, where ozone concentrations are mainly determined by rapid chemical processes, dense and vertically resolved MLS data helped reduce the biases related to model deficiencies. These improvements resulted in a more realistic and consistent description of spatial and temporal variations in stratospheric ozone, as demonstrated by cases in the dynamically and chemically active regions. However, combined assimilation of the often discrepant ozone observations might lead to underestimation of tropospheric ozone. In addition, model deficiencies induced large biases in the upper stratosphere in the medium-range (5-day) ozone forecasts.

A high-order arbitrarily unstructured finite-volume model of the global atmosphere: tests solving the shallow-water equations

Simulations of the global atmosphere for weather and climate forecasting require fast and accurate solutions and so operational models use high-order finite differences on regular structured grids. This precludes the use of local refinement; techniques allowing local refinement are either expensive (eg. high-order finite element techniques) or have reduced accuracy at changes in resolution (eg. unstructured finite-volume with linear differencing). We present solutions of the shallow-water equations for westerly flow over a mid-latitude mountain from a finite-volume model written using OpenFOAM. A second/third-order accurate differencing scheme is applied on arbitrarily unstructured meshes made up of various shapes and refinement patterns. The results are as accurate as equivalent resolution spectral methods. Using lower order differencing reduces accuracy at a refinement pattern which allows errors from refinement of the mountain to accumulate and reduces the global accuracy over a 15 day simulation. We have therefore introduced a scheme which fits a 2D cubic polynomial approximately on a stencil around each cell. Using this scheme means that refinement of the mountain improves the accuracy after a 15 day simulation. This is a more severe test of local mesh refinement for global simulations than has been presented but a realistic test if these techniques are to be used operationally. These efficient, high-order schemes may make it possible for local mesh refinement to be used by weather and climate forecast models.

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.