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We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter \(\sigma \in [0,1]\), that give back the original problem only when \(\sigma =1\). We compare the strategies by using well known test problems with two and three parameters.

In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton–Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton–Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ = 0 to τ = 1 . While following the involved imaginary path two numerical approaches are realized, the first is based on the Prufer angle technique, the second on high order finite difference schemes.

Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems.

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.

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 $\alpha" and $\beta$ of the eigenvalues of J(Pe), the Jacobian of the reaction part at the equilibrium point Pe. Focusing on the case $\alpha = 0,\beta \neq 0$, we obtain stability regions in the plane $(\xi ,\nu )$, where $\xi =\lambda (h;d)h_t$ , $\nU =\beta h_t$ , $h_t$ time stepsize, $\lambda$ 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 absence of diffusion. Hence, by estimating l , for each method we derive stepsize restrictions $h_t

In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg–de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p = 5 , solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p > 5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p = 5 . Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as p → 5 + . By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.

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.

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.

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.

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.

Magnetic particle imaging (MPI) is a new medical imaging technique capable of recovering the distribution of superparamagnetic particles from their measured induced signals [1]. In literature there are two main MPI reconstruction techniques: measurement-based (MB) and x-space (XS). In the first approach the unknown magnetic particles concentration is reconstructed in the harmonic-space using a System Function (SF), describing the relation between particle positions and the signal response [2, 3]. The second approach requires the knowledge of the field free point (FFP) exact position and velocity at all time steps during the scanning process [4, 5]. The x-space method is based on the assumption of ideal magnetic field shapes used for spatial encoding (selection field), and for signal excitation (focus-drive field). The realization of human size devices with an open geometry requires specific calibration procedures related to the methods used in the reconstruction phase. One of the advantages of open bore scanners would be an easier open access to the patient, especially in interventional scenarios with simultaneous and real-time scanning processes. In this case of geometry configurations with larger FOV, the exact velocity gridding for x-space MPI could be difficult to achieve during the whole scanning process. Hence, our proposal is an innovative technique named hybrid x-space (HXS) resulting from the combination of the measurement-based and the classical x-space approach, reducing and optimizing the calibration time by a compressive sensing technique using circulant matrices.

We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.

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.

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.

∗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

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.

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.

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.