Giuseppina Settanni

Numerical simulation of the whispering gallery modes in prolate spheroids

Abstract In this paper, we discuss the progress in the numerical simulation of the so-called ‘whispering gallery’ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of ‘whispering gallery’ phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm–Liouville problem. We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries. We illustrate the performance of the method by means of numerical experiments.

Asymptotical computations for a model of flow in saturated porous media

We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.

High Order Finite Difference Schemes for the Numerical Solution of Eigenvalue Problems for IVPs in ODEs

In this short note we describe how to apply high order finite difference methods to the solution of eigenvalue problems with initial conditions. Finite differences have been successfully applied to both second order initial and boundary value problems in ODEs. Here, based on the results previously obtained, we outline an algorithm that at first computes a good approximation of the eigenvalues of a linear second order differential equation with initial conditions. Then, for any given eigenvalue, it determines the associated eigenfunction.

A Deferred Correction Approach to the Solution of Singularly Perturbed BVPs by High Order Upwind Methods: Implementation Details

In this note we give implementation details on the computation of the monitor function which is used inside a code based on high order upwind methods. The considered strategy is based on deferred correction and allows to compute an approximation of the error for a method of order p with essential no additional computational cost.

Hybrid x-space: a new approach for MPI reconstruction.

Magnetic particle imaging (MPI) is a new medical imaging technique capable of recovering the distribution of superparamagnetic particles from their measured induced signals. In literature there are two main MPI reconstruction techniques: measurement-based (MB) and x-space (XS). The MB method is expensive because it requires a long calibration procedure as well as a reconstruction phase that can be numerically costly. On the other side, the XS method is simpler than MB but the exact knowledge of the field free point (FFP) motion is essential for its implementation. Our simulation work focuses on the implementation of a new approach for MPI reconstruction: it is called hybrid x-space (HXS), representing a combination of the previous methods. Specifically, our approach is based on XS reconstruction because it requires the knowledge of the FFP position and velocity at each time instant. The difference with respect to the original XS formulation is how the FFP velocity is computed: we estimate it from the experimental measurements of the calibration scans, typical of the MB approach. Moreover, a compressive sensing technique is applied in order to reduce the calibration time, setting a fewer number of sampling positions. Simulations highlight that HXS and XS methods give similar results. Furthermore, an appropriate use of compressive sensing is crucial for obtaining a good balance between time reduction and reconstructed image quality. Our proposal is suitable for open geometry configurations of human size devices, where incidental factors could make the currents, the fields and the FFP trajectory irregular.

Devising efficient numerical methods for oscillating patterns in reaction-diffusion systems

In this paper, we consider the numerical approximation of a reaction-diffusion system 2D in space whose solutions are patterns oscillating in time or both in time and space. We present a stability analysis for a linear test heat equation in terms of the diffusion d and of the reaction timescales given by the real and imaginary parts α and β of the eigenvalues of J ( P e ) , the Jacobian of the reaction part at the equilibrium point P e . Focusing on the case α = 0 , β ? 0 , we obtain stability regions in the plane ( ? , ? ) , where ? = λ ( h ; d ) h t , ? = β h t , h t time stepsize, λ lumped diffusion scale depending also from the space stepsize h and from the spectral properties of the discrete Laplace operator arising from the semi-discretization in space. In space we apply the Extended Central Difference Formulas (ECDFs) of order p = 2 , 4 , 6 . In time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators: the Explicit Euler and ADI methods, the symplectic Euler and a partitioned Runge-Kutta method that are symplectic in the absence of diffusion. Hence, by estimating λ , for each method we derive stepsize restrictions h t ? F m e t ( h ; d , β , p ) in terms of the stability curve F m e t depending on diffusion and reaction timescales and from the approximation order in space. For the same schemes, we provide also a dispersion error analysis. We present numerical simulations for the test heat equation and for the Lotka-Volterra PDE system with solutions oscillating only in time for the presence of a centre-type dynamics. In these cases, the implicit-symplectic schemes provide the best choice. We solve also the Schnakenberg model with spatial patterns oscillating in space and time in the presence of an attractive limit cycle due to the Turing-Hopf instability. In this case, all schemes attain closed orbits in the phase space, but the Explicit ADI method is the best choice from the computational point of view.

Variable-step finite difference schemes for the solution of Sturm–Liouville problems

We discuss the solution of regular and singular Sturm–Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm–Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.

A finite differences MATLAB code for the numerical solution of second order singular perturbation problems

We show the main features of the MATLAB code HOFiD_UP for solving second order singular perturbation problems. The code is based on high order finite differences, in particular on the generalized upwind method. Within its simplicity, it uses order variation and continuation for solving any difficult nonlinear scalar problem. Several numerical tests on linear and nonlinear problems are considered. The best performances are reported on problems with perturbation parameters near the machine precision, where most of the codes for two-point BVPs fail.

A Stepsize Variation Strategy for the Solution of Regular Sturm‐Liouville Problems

In this note we show how a simple stepsize variation strategy improves the solution algorithm of regular Sturm‐Liouville problems. We suppose the eigenvalue problem is approximated by variable stepsize finite difference schemes and the obtained algebraic eigenvalue problem is solved by a matrix method estimating the first eigenvalues and eigenvectors of sparse matrices. The variable stepsize strategy is based on an equidistribution of the error (approximated by two methods with different orders). The results show a marked reduction of the number of points and, consequently, a much lower computational cost, with respect to the algorithm obtained using constant stepsize.

Calculations of the morphology dependent resonances

∗Dipartimento di Matematica, Universita di Bari, Via E. Orabona 4, I-70125 Bari, Italy †Institut Computational Mathematics, TU Braunschweig, Pockelsstrasse 14, D-38106 Braunschweig, Germany ∗∗Dipartimento di Matematica e Fisica ‘E. De Giorgi’ , Universita del Salento, Via per Arnesano, I-73047 Lecce, Italy ‡Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria