Initial Value Method Research Articles

Numerical transformation methods: Blasius problem and its variants

Blasius problem is the simplest nonlinear boundary-layer problem. We hope that any approach developed for this epitome can be extended to more difficult hydrodynamics problems. With this motivation we review the so called Töpfer transformation, which allows us to find a non-iterative numerical solution of the Blasius problem by solving a related initial value problem and applying a scaling transformation. The applicability of a non-iterative transformation method to the Blasius problem is a consequence of its partial invariance with respect to a scaling group. Several problems in boundary-layer theory lack this kind of invariance and cannot be solved by non-iterative transformation methods. To overcome this drawback, we can modify the problem under study by introducing a numerical parameter, and require the invariance of the modified problem with respect to an extended scaling group involving this parameter. Then we apply initial value methods to the most recent developments involving variants and extensions of the Blasius problem.

Parameter and state estimation of experimental chaotic systems using synchronization

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time-dependent and time-independent coupling and discuss the use of periodic impulse coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time-dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. For time-independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time-dependent coupling appears to have benefits in synchronizing long data sets with minimal impact, while the initial value method with time-independent coupling tends to be substantially faster, more flexible, and easier to use. We also describe a method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time-dependent injected current. They are expected to be of value in nonchaotic systems as well. Software is available on request.

Tunneling by a semiclassical initial value method with higher order corrections

Tunneling through the one-dimensional Eckart barrier is treated by applying a higher order semiclassical approximation which adds corrections proportional to powers of ℏ to the Herman–Kluk (HK) initial value approximation. Although the usual, zero-order HK treatment is very poor in this case, the first- and second-order corrections substantially improve the accuracy of the computed tunneling probabilities. To investigate how this works, the HK expression is shown to be equivalent, for the present purposes, to a formula involving a single integral over the initial momentum, with an integrand that has a simple analytical form. Similarly, the most important part of the first-order correction term is shown to be expressible in a very simple form. For a particular range of energies, the integral can be analyzed in terms of a steepest descent treatment along a path through a caustic. In this way, it is verified that the zero-order approximation does not approach the correct classical limit as ℏ → 0, but the first-order term, which does not vanish in this limit, improves the accuracy of the result. More generally, the corrections of each order contain terms that are of all orders in ℏ1/3, including those that are O(ℏ0) and which survive in the classical limit. The infinite sum over such terms is performed analytically and shown to yield the correct classical limit for the tunneling amplitude.

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.

A Novel Approach to the Solar Interior‐Atmosphere Eigenvalue Problem

In this paper we introduce a new approach to study the interaction of solar eigenoscillations, with particular emphasis on the f-mode, with random inhomogeneities caused by flows and magnetic field near the solar surface. We present an initial value method to derive a general dispersion relation for a class of models where the magnetic atmosphere is overlying an arbitrary static solar interior. In these models the interior part is treated parametrically and does not need to be specified before we obtain the dispersion relation. In order to demonstrate the applicability of the proposed method, an analytical solution of the dispersion relation is given for an incompressible interior with constant density.

The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes

B-spline methods are Linear Multistep Methods based on B-splines which have good stability properties [F. Mazzia, A. Sestini, D. Trigiante, B-spline multistep methods and their continuous extensions, SIAM J. Numer Anal. 44 (5) (2006) 1954–1973] when used as Boundary Value Methods [L. Brugnano, D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, J. Comput. Appl. Math. 66 (1–2) (1996) 97–109; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 1998]. In addition, they have an important feature: if k is the number of steps, it is always possible to associate to the numerical solution a C k spline of degree k + 1 collocating the differential equation at the mesh points. In this paper we introduce an efficient algorithm to compute this continuous extension in the general case of a non-uniform mesh and we prove that the spline shares the convergence order with the numerical solution. Some numerical results for boundary value problems are presented in order to show that the use of the information given by the continuous extension in the mesh selection strategy and in the Newton iteration makes more robust and efficient a Matlab code for the solution of BVPs.

NONLINEAR APPROACH TO APPROXIMATE ACOUSTIC BOUNDARY ADMITTANCE IN CAVITIES

In this paper, an algorithm is derived to solve a problem of inverse acoustics. It considers the damped acoustic boundary value problem, i.e. the Helmholtz equation and admittance boundary condition, in order to approximate the boundary admittance of interior domains. The algorithm is implemented by using a finite element method and tested for two-dimensional cavities with arbitrary shapes. The admittance condition is reconstructed based on sound pressure measurements. The solution of the arising nonlinear system of equations is obtained by applying the Newton method following a presetting method for finding reasonable initial boundary admittance values. A residual norm accounts for the objective function. Its first- and second-order sensitivities are determined analytically by using a modal decomposition in order to avoid direct inversion of the system matrix. The experiment is simulated by taking sound pressure data of the forward solution as inputs for the inverse problem. Test examples show that very few measurement points are necessary to reproduce piecewise constant boundary admittance values very accurately. Then, the admittance boundary condition is applied to reproduce the sound pressure distribution in the cavity. Again, it becomes obvious that only a few measurement points are required to reconstruct the sound pressure field.

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.

A full numerical calculation of the Franz–Keldysh effect on magnetoexcitons in a bulk semiconductor

We have performed a full numerical calculation of the Franz–Keldysh (FK) effect on magnetoexcitons in a bulk GaAs semiconductor. By employing an initial value method in combination with the application of a perfect matched layer, the numerical effort and storage size are dramatically reduced due to a significant reduction in both computed domain and number of base functions. In the absence of an electric field, the higher magnetoexcitonic peaks show distinct Fano lineshape due to the degeneracy with continuum states of the lower Landau levels. The magnetoexcitons that belong to the zeroth Landau level remain in bound states and lead to Lorentzian lineshape, because they are not degenerated with continuum states. In the presence of an electric field, the FK effect on each magnetoexcitonic resonance can be identified for high magnetic fields. However, for low magnetic fields, the FK oscillations dominate the spectrum structure in the vicinity of the bandgap edge and the magnetoexcitonic resonances dominate the spectrum structure of higher energies. In the moderate electric fields, the interplay of FK effect and magnetoexcitonic resonance leads to a complex and rich structure in the absorption spectrum.

Sobolev’s Method for Hammerstein Integral Equations

Sobolev’s initial value method is used to solve the Hammerstain equation of the second kind.

A semiclassical hybrid approach to many particle quantum dynamics

We analytically derive a correlated approach for a mixed semiclassical many particle dynamics, treating a fraction of the degrees of freedom by the multitrajectory semiclassical initial value method of Herman and Kluk [Chem. Phys. 91, 27 (1984)] while approximately treating the dynamics of the remaining degrees of freedom with fixed initial phase space variables, analogously to the thawed Gaussian wave packet dynamics of Heller [J. Chem. Phys. 62, 1544 (1975)]. A first application of this hybrid approach to the well studied Secrest-Johnson [J. Chem. Phys. 45, 4556 (1966)] model of atom-diatomic collisions is promising. Results close to the quantum ones for correlation functions as well as scattering probabilities could be gained with considerably reduced numerical effort as compared to the full semiclassical Herman-Kluk approach. Furthermore, the harmonic nature of the different degrees of freedom can be determined a posteriori by comparing results with and without the additional approximation.

Exact Solution of Out-of-Plane Problems of an Arch with Varying Curvature and Cross Section

This paper deals with the exact solution of the differential equations for the out-of-plane behavior of an arch with varying curvature and cross section. The differential equations include the shear deformation effect. The cross section of the arch is doubly symmetric. Due to the double symmetry, in-plane and out-of-plane behavior will be uncoupled. However, a coupling of the out-of-plane bending and the torsional response will exist and will be discussed in this study. The governing differential equations of planar arches loaded perpendicular to their plane are solved exactly by using the initial value method. The analytical expressions of the fundamental matrix can be obtained for some cases. It is also possible to use these analytical expressions in order to obtain the displacements and the stress resultants for an arch with any loading and boundary conditions. The examples given in the literature are solved and the results are compared. The analytical expressions of the results are given for some examples.

Exact solution of free in-plane vibration of a stepped circular arch

This paper investigates the free in-plane vibration of stepped circular arches. The effects of axial extension, transverse shear deformation and rotatory inertia are included in the governing equations. The solution is obtained exactly by using the initial value method. The solution procedure is also applied to cases in which only one effect is considered in order to determine its contribution to the results. The results show that the aforementioned effects must be taken into account to get a better approximation to the actual behaviour of an arch, even if it is slender. The effects of step ratio, location of the step, boundary condition, opening angle and slenderness ratio on the frequency coefficients are studied. The mode transition phenomenon is also investigated and the mode shapes are given in figures. The examples in the literature are solved and comparisons are presented in the tables. Numerical solutions are obtained by using ANSYS which is a commercial finite element program. Experiments are also performed to verify the theoretical and numerical results.

Calculation of Point-to-Point Short-Time and Rare Trajectories with Boundary Value Formulation.

Sampling rare, short-time, and reactive trajectories is of considerable interest in molecular simulations. These trajectories, which are also called "activated", hop between stable states separated by energy or entropy barriers. Simulations of activated trajectories with random sampling of initial conditions are inefficient, since most initial conditions lead to trajectories that do not pass the barrier in short times. A boundary value formulation is proposed that selects these rare trajectories, making the sampling of point-to-point reactive trajectories more effective. Earlier boundary value formulations by one of us focused on computations of approximate trajectories. In the proposed method, trajectories are accurate even when we employ a relatively large integration step (by a factor of about 100 compared to initial value methods). The boundary value solutions to short-time reactive trajectories tend to be unique and have significant statistical weights compared to other reactive trajectories of the microcanonical ensemble. Three numerical examples are considered: a transition in the Mueller potential, a conformational change in alanine dipeptide, and an isomerization in a Lennard-Jones cluster.

Analytical solutions of in-plane static problems for non-uniform curved beams including axial and shear deformations

Exact analytical solutions for in-plane static problems of planar curved beams with variable curvatures and variable cross-sections are derived by using the initial value method. The governing equations include the axial extension and shear deformation effects. The fundamental matrix required by the initial value method is obtained analytically. Then, the displacements, slopes and stress resultants are found analytically along the beam axis by using the fundamental matrix. The results are given in analytical forms. In order to show the advantages of the method, some examples are solved and the results are compared with the existing results in the literature. One of the advantages of the proposed method is that the high degree of statically indeterminacy adds no extra difficulty to the solution. For some examples, the deformed shape along the beam axis is determined and plotted and also the slope and stress resultants are given in tables.

Out-of-plane free vibration of a circular arch with uniform cross-section: Exact solution

In this study, free out-of-plane vibrations of a circular arch with uniform cross-section are investigated by taking into account the effects of transverse shear and rotatory inertia due to both flexural and torsional vibrations. The governing differential equations for out-of-plane vibration of uniform circular beams are solved exactly by using the initial value method. The solution does not depend on the boundary conditions. The same solution procedure is also used to obtain the results of other cases in which each effect is considered individually in order to assess its importance. The frequency coefficients are obtained for the first five modes of arches with various slenderness ratios and opening angles. The results show that the flexural and torsional rotatory inertia and shear deformation have very important effects on resonance frequencies, even if slender shallow arches are considered. It is concluded that the torsional rotatory inertia effect is the most significant effect to be included in the analysis. A phenomenon known as transition of modes from torsional into flexural is characterized by the sharp increment in resonance frequencies of modes that occurs at certain combinations of curvature and length of the arch. The mode transition phenomenon is shown in figures. Vibration problems for circular beams that have been analysed in the literature are solved and the results are compared in tables. The comparison shows good agreement between the results.

Economical error estimates for block implicit methods for ODEs via deferred correction

Deferred correction is a widely used tool for improving the numerical approximation to the solution of ODE problems [J.R. Cash, WSSIA 2 (1993) 113; J.R. Cash, M.H. Wright, SIAM J. Sci. Statist. Comput. 12 (1991) 971; M. Lentini, V. Pereyra, Math. Comp. 28 (1974) 981; B. Lindberg, BIT 20 (1980) 486; V. Pereyra, Numer. Math. 8 (1966) 376; V. Pereyra, Numer. Math. 10 (1967) 316; H.J. Stetter, Numer. Math. 29 (1978) 425; H.J. Stetter, in: Lecture Notes in Math. vol. 630, Springer, 1978, pp. 245–258; R.D. Skeel, SIAM J. Numer. Anal. 19 (1981) 171; R.D. Skeel, Numer. Math. 48 (1986) 1; P. Zadunaisky, Numer. Math. 27 (1976) 21]. Indeed, it allows to estimate the error due to the use of discrete methods. Such an estimate may be a global one, in the case of continuous BVPs, or a local one, when IVPs are to be approximated [L. Brugnano, in: Lecture Notes in Math., vol. 1196, Springer, 1997, pp. 78–89; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach, 1998]. Recently, it has been implemented in the computational code BiM [L. Brugnano, C. Magherini, J. Comput. Appl. Math. 164–165 (2004) 145, web page: http://math.unifi.it/~brugnano/BiM/index.html] for the numerical solution of stiff ODE-IVPs. In this paper we analyze deferred correction in connection with the methods used in that code, resulting in an overall simplification of the procedure, due to the properties of the underlying methods. The analysis is then extended to more general methods.

An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations

In this paper a numerical method is suggested to solve a class of boundary value problems for a weakly coupled system of singularly perturbed second order ordinary differential equations of reaction-diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of the well known perturbation method. Then initial value problems and terminal value problems (TVPs) are formulated such that their solutions are the terms of this asymptotic expansion. These problems are happened to be singularly perturbed problems and therefore exponentially fitted finite difference schemes are used to solve these problems. Since the boundary value problem is converted into a set of initial and TVPs and an asymptotic expansion approximation is used, the present method is termed as an asymptotic initial value method. Necessary error estimates are derived and examples provided to illustrate the method. The present method is easy to implement and well suited for parallel computing.

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.

Asymptotic initial value methods for two-parameter singularly perturbed boundary value problems for second order ordinary differential equations

In this paper some numerical methods are presented for singularly perturbed two-point boundary value problems for second order ordinary differential equations with two small parameters multiplying the derivatives. These methods are distinguished by the fact that, initial value problems and/or terminal value problems are constructed/deduced from the given boundary value problem and then solved by a fitted operator method. Further error estimates are derived. The methods suggested are simple to use and easy to implement. The important aspect is that some of the methods suggested are well suited for parallel computing. Examples are provided to illustrate the methods.