
Daniele FUNARO
Professore Ordinario Dipartimento di Scienze Chimiche e Geologiche  Sede Dipartimento di Scienze Chimiche e Geologiche

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2023
 A Dynamic Representation of mRNA Nucleotides Clarifies the Conundrum of Codon Redundancy
[Articolo su rivista]
Funaro, D.
abstract
The deciphering of the genetic code takes place through the reading of the nitrogenous bases, which are four in number. In most cases, the bases are taken three by three, thus generating 64 possible combinations with repetition. Each combination (codon) allows for the synthesis of a specific amino acid. Since the latter are only 21 in number, the codonamino acid conversion table shows a strong redundancy. Countless efforts have been made to understand the true encryption mechanism. Here, we want to add our version, which consists of associating a periodic sound based on three notes to each codon. RNA now becomes a dynamic object and not just a list of static instructions. In addition to a different interpretation of the genetic code, there is also a considerable reduction in redundancy, given that the number of periodic sounds that can be produced with three notes drops to 20 (with the addition of four pure frequencies). Finally, we discuss the possibility of how these sounds can be generated and travel inside the double helix, and possibly emitted as biophotons.
2023
 Electromagnetic Displacements Rotating Inside an Annular Region
[Relazione in Atti di Convegno]
Fatone, L.; Funaro, D.
abstract
2023
 Electromagnetic fields simulating a rotating sphere and its exterior with implications to the modeling of the heliosphere
[Articolo su rivista]
Fatone, L.; Funaro, D.
abstract
Vector displacements expressed in spherical coordinates are proposed. They correspond to electromagnetic fields in vacuum that globally rotate about an axis and display many circular patterns on the surface of a ball. The fields satisfy the set of Maxwell's equations, and some connections with magnetohydrodynamics can also be established. The solutions are extended with continuity outside the ball. In order to avoid peripheral velocities of arbitrary magnitude, as it may happen for a rigid rotating body, they are organized to form successive encapsulated shells, with substructures recalling ballbearing assemblies. A recipe for the construction of these solutions is provided by playing with the eigenfunctions of the vector Laplace operator. Some applications relative to astronomy are finally discussed.
2023
 Newtonian Forces Exerted by Electromagnetic Waves Traveling into Matter
[Articolo su rivista]
Funaro, Daniele
abstract
Electromagnetic waves, developing in vacuum or into matter, produce dynamical alterations of the spacetime metric. This is a consequence of Einstein’s equation, that we are able to solve explicitly in some circumstances. Solutions are in fact obtained by plugging on the righthand side of the equation some appropriate energy tensors. Hence, the passage of a wave generates both electrodynamics and ‘gravitational’ (local and temporary) modifications of the molecular lattice of a dielectric. If the wave or the dielectric body are asymmetric, we could theoretically obtain a distribution of Newtonianlike forces with nonzero resultant. This hypothesis suggested a laboratory experiment where an electromagnetic signal applied to a ring with a particular geometry imparts a directional thrust in apparent violation of the actionreaction principle. This test was recently realized with success. Therefore, the present theoretical approach, once appropriately refined, may constitute a crucial referring point for further developments.
2023
 Spacetime deformations of electromagnetic nature are far from negligible
[Working paper]
Funaro, Daniele
abstract
We would like to collect a series of considerations concerning the influence, which we believe is relevant, that phenomena of an electromagnetic nature have on the geometry of spacetime. The approach, supported by the critical observation of already wellknown properties and sustained by theoretical elements, leads us to attribute a primary role to electric and magnetic interactions. In fact, a seemingly small interpretative effort can help link theories elegantly and mathematically which at the moment appear independent. Einstein’s equations have a validity that goes beyond the cosmological one and, with the appropriate corrections, they can clarify what really happens inside
matter, starting with the bonds that dominate at the molecular level all the way up the description of the macroscopic characteristics. This analysis allows us to leave the context hitherto reserved exclusively for quantum mechanics and to place the study of the structure of matter in a more classic framework.
2022
 A decisionmaking machine learning approach in Hermite spectral approximations of partial differential equations
[Articolo su rivista]
Fatone, Lorella; Funaro, Daniele; Marco Manzini, Gian
abstract
The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is underresolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is timedependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decisionmaking process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications. As a matter of fact we also show at the end an application in the framework of plasma physics.
2022
 An Efficient RingShaped Electromagnetic Thruster
[Articolo su rivista]
Funaro, Daniele; Chiolerio, Alessandro
abstract
An electromagnetic thruster is proposed and successfully tested. Its design is inspired by theoretical considerations whose qualitative predictions are well matched with the experimental results. The efficiency is higher than any other device so far reported in the literature, producing a directional thrust of about 2.7 × 10^{−6}m, where m is the mass of the thruster itself, with a nominal power injected of about 10 Watts. The prototype has the shape of a ring and is powered by both radio frequency signals and a stationary high voltage. Improvements and generalizations
can be easily devised by adjusting the geometry of the device.
2022
 Ball Lightning as Plasma Vortexes: A Reinforcement of the Conjecture
[Articolo su rivista]
Funaro, Daniele
abstract
The idea that ball lightning is a plasma manifestation has been put forth by many authors. One of the major drawbacks of this approach concerns with the stability of these structures. After showing the theoretical existence of pure rotating electromagnetic waves in equilibrium, we argue that these can reasonably provide a robust stabilizing scaffolding for the development of ball lightning phenomena.
2022
 HighOrder Discretization of Backward Anisotropic Diffusion and Application to Image Processing
[Articolo su rivista]
Funaro, Daniele; Fatone, Lorella
abstract
Anisotropic diffusion is a wellrecognized tool in digital image processing, including edge detection and focusing. We present here a particular nonlinear timedependent operator together with an appropriate highorder discretization for the space variable. In just a single step, the procedure emphasizes the contour lines encircling the objects, paving the way to accurate reconstructions at a very low cost. One of the main features of such an approach is the possibility of relying on a rather large set of invariant discontinuous images, whose edges can be determined without introducing any approximation.
2022
 How and why non smooth solutions of the 3D Navier–Stokes equations could possibly develop
[Articolo su rivista]
Funaro, Daniele
abstract
Fluid configurations in threedimensions, displaying a plausible decay of regularity in a finite time, are suitably built and examined. Vortex rings are the primary ingredients in this study. The full Navier–Stokes system is converted into a 3D scalar problem, where appropriate numerical methods are implemented in order to figure out the behavior of the solutions. Further simplifications in 2D and 1D provide interesting toy problems, that may be used as a starting platform for a better understanding of blowup phenomena.
2022
 The Impact of a Pervasive Electrodynamical Background on Biological Interactions
[Working paper]
Funaro, Daniele
abstract
It is assumed that the background electromagnetic radiation is not just a noise, but a structured dynamical environment which is modified by the presence of matter, and, at the same time, promotes aggregations of elements at various scales of complexity. The process starts at atomic and molecular level, and extends its domain to evolved assemblies, up to the level of living beings. The first consequence of this interpretation is the fundamental observation that the whole is not strictly the sum of its parts, but a new arrangement, where electromagnetic waves resonate at frequencies different from those of the components, emitting photons in new energy ranges. The second crucial argument, is the possibility of establishing alternative means of biological communications either by direct way (biophotons), or through a suitable excitation of the electromagnetic habitat. A corollary of this analysis is that DNA can be interpreted as an active transmitting entity, rather that a mere list of crude instructions.
2022
 The SpaceTime Outside a Pulsating Charged Sphere
[Articolo su rivista]
Funaro, D.
abstract
We consider the problem of determining the dynamics of the electromagnetic field generated outside a ball whose charge changes depending on time. We are in conditions of perfect symmetry and the electric field is considered to be radial. This is not a simplification since, under such a hypothesis, the magnetic field does not develop. Thus, it is first necessary to find out the appropriate modeling equations. These are obtained by writing a suitable energy tensor that combines the classical electromagnetic stressenergy tensor with a special kind of mass tensor. The next step is to show that it is possible to solve Einstein’s equations by plugging the new tensor on the righthand side. Interesting connections with some classical solutions related to black holes are finally established.
2021
 Stability and Conservation Properties of HermiteBased Approximations of the VlasovPoisson System
[Articolo su rivista]
Funaro, D.; Manzini, G.
abstract
Spectral approximation based on HermiteFourier expansion of the VlasovPoisson model for a collisionless plasma in the electrostatic limit is provided by adding highorder artificial collision operators of LenardBernstein type. These differential operators are suitably designed in order to preserve the physicallymeaningful invariants (number of particles, momentum, energy). In view of timediscretization, stability results in appropriate norms are presented. In this study, necessary conditions link the magnitude of the artificial collision term, the number of spectral modes of the discretization, as well as the timestep. The analysis, carried out in full for the Hermite discretization of a simple linear problem in onedimension, is then partly extended to cover the complete nonlinear VlasovPoisson model.
2020
 Electromagnetic Waves in Annular Regions
[Articolo su rivista]
Funaro, Daniele
abstract
In suitable bounded regions immersed in vacuum, time periodic wave solutions solving a full set of electrodynamics equations can be explicitly computed. Analytical expressions are available in special cases, whereas numerical simulations are necessary in more complex situations. The attention here is given to selected threedimensional geometries, which are topologically equivalent to a toroid, where the behavior of the waves is similar to that of fluiddynamics vortex rings. The results show that the shape of the sections of these rings depends on the behavior of the eigenvalues of a certain elliptic differential operator. Timeperiodic solutions are obtained when at least two of such eigenvalues attain the same value. The solutions obtained are discussed in view of possible applications in electromagnetic whispering galleries or plasma physics.
2020
 Highly Directive Biconic Antennas Embedded in a Dielectric
[Articolo su rivista]
Chiolerio, Alessandro; Diazzi, Lorenzo; Funaro, Daniele
abstract
Designing antennas suitable for generating highly directive electromagnetic signals has become a fundamental task. This is particularly relevant for the development of efficient and sustainable pointtopoint communication channels, and for energy transfer. Indeed, these are nowadays expanding areas of research. In order to deal with said particular wave phenomena, an extension of the electrodynamics equations is taken into account, where exact solitonic type solutions are admitted. These waves may have compact support and travel along a straight line, without dissipation, at the speed of light. The result suggests the design of biconic type antennas having specific properties that are numerically examined in this paper. The cones, supplied with an oscillating source, are embedded in a dielectric material of suitable shape, with the purpose of driving the signal in the proper direction. The computations based on the extended model are aimed toward simulating the possibility of generating peculiar wave behaviors, in view of practical implementations in the framework of pointtopoint communications or wireless power transmission.
2020
 On the Use of Hermite Functions for the Vlasov–Poisson System
[Relazione in Atti di Convegno]
Fatone, Lorella; Funaro, Daniele; Manzini, Gianmarco
abstract
We apply a secondorder semiLagrangian spectral method for the Vlasov–Poisson system, by implementing Hermite functions in the approximation of the distribution function with respect to the velocity variable. Numerical tests are performed on a standard benchmark problem, namely the twostream instability test case. The approach uses two independent sets of Hermite functions, based on Gaussian weights symmetrically placed with respect to the zero velocity level. An experimental analysis is conducted to detect a reasonable location of the two weights in order to improve the approximation properties.
2019
 A Semi‑Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials
[Articolo su rivista]
Fatone, Lorella; Funaro, Daniele; Manzini, Gianmarco
abstract
In this work, we apply a semiLagrangian spectral method for the Vlasov–Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss secondorder accurateintime schemes, obtained by coupling spectral techniques in the space–velocity domain with a BDF timestepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the twostream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the timeadvancing procedure.
2019
 ArbitraryOrder TimeAccurate SemiLagrangian Spectral Approximations of the VlasovPoisson System
[Articolo su rivista]
Funaro, Daniele; Fatone, Lorella; Manzini, Gianmarco
abstract
The VlasovPoisson system, modeling the evolution of noncollisional plasmas in the electrostatic limit, is approximated by a semiLagrangian technique. Spectral methods of periodic type are implemented through a collocation approach. Groups of particles are represented by the Fourier Lagrangian basis and evolve, for a single timestep, along an highorder accurate representation of the local characteristic lines. The timeadvancing technique is based on truncated Taylor series that can be, in principle, of any order of accuracy, or by coupling the phase space discretization with highorder accurate Backward Dierentiation Formulas (BDF) as in the methodoflines framework.
At each timestep, particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs. Thus, these techniques combine excellent features of spectral approximations with highorder time integration, and the resulting method has excellent conservation properties. Indeed, we can prove that the total number of particles, which is proportional to the total mass and charge, is conserved up to the machine precision. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.
2019
 From Photons to Atoms  The Electromagnetic Nature of Matter
[Monografia/Trattato scientifico]
Funaro, Daniele
abstract
Motivated by a revision of the classical equations of electromagnetism that allow for the inclusion of solitary waves in the solution space, the material collected in this book examines the consequences of adopting the modified model in the description of atomic structures. The possibility of handling "photons" in a deterministic way indeed gives a chance to review the foundations of quantum physics. Atoms and molecules are described as aggregations of nuclei and electrons joined through organized photon layers resonating at various frequencies, explaining how matter can absorb or emit light quanta. Some established viewpoints are subverted, offering an alternative scenario. The analysis seeks to provide an answer to many technical problems in physical chemistry and, at the same time, to raise epistemological questions.
2018
 A Model for Ball Lightning Derived from an Extension of the Electrodynamics Equations
[Relazione in Atti di Convegno]
Funaro, Daniele
abstract
Ball lightning is an impressive natural electromagnetic phenomenon occurring in atmosphere under suitable circumstances. Its origin, composition and stability issues are a matter of debate, due to presence of many evidences still unexplained. An attempt to provide a model, in alternative to the ones already available, is here presented. The aim is to interpret the phenomenon through a fluid plasma selftrapped in very stable toroidshaped regions. To this end, a suitable adaptation of the equations ruling electrodynamics is taken into account.
2018
 High Frequency Electrical Oscillations in Cavities
[Articolo su rivista]
Funaro, Daniele
abstract
If the interior of a conducting cavity (such as a capacitor or a coaxial cable) is supplied with a very highfrequency electric signal, the information between the walls propagates with an appreciable delay, due to the ﬁniteness of the speed of light. The conﬁguration is typical of cavities having size larger than the wavelength of the injected signal. Such a non rare situation, in practice, may cause a break down of the performances of the device. We show that the classical Coulomb’s law and Maxwell’s equations do not correctly predict this behavior. Therefore, we provide an extension of the modeling equations that allows for a more reliable determination of the electromagnetic ﬁeld during the evolution process. The main issue is that, even in vacuum (no dielectric inside the device), the fast variation of the signal produces sinks and sources in the electric ﬁeld, giving rise to zones where the divergence is not zero. These regions are well balanced, so that their average in the domain is zero. However, this behavior escapes the usual treatment with classical electromagnetism.
2018
 Isospectral Domains for Discrete Elliptic Operators
[Articolo su rivista]
Fatone, Lorella; Funaro, Daniele
abstract
Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finitedimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.
2017
 Convergence of spectral discretizations of the VlasovPoisson system
[Articolo su rivista]
Manzini, Gianmarco; Funaro, Daniele; Delzanno, Gian Luca
abstract
We prove the convergence of a spectral discretization of the VlasovPoisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in detail for the 1D1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence under suitable regularity assumptions on the exact solution.
2015
 Optimal collocation nodes for fractional derivative operators
[Articolo su rivista]
Fatone, Lorella; Funaro, Daniele
abstract
Spectral discretizations of fractional derivative operators are examined, where the approximation basis is related to the set of Jacobi polynomials. The pseudospectral method is implemented by assuming that the grid, used to represent the function to be differentiated, may not be coincident with the collocation grid. The new option opens the way to the analysis of alternative techniques and the search for optimal distributions of collocation nodes, based on the operator to be approximated. Once the initial representation grid has been chosen, indications for how to recover the collocation grid are provided, with the aim of enlarging the dimension of the approximation space. As a result of this process, performances are improved. Applications to fractional type advectiondiffusion equation and comparisons in terms of accuracy and efficiency are made. As shown in the analysis, special choices of the nodes can also suggest tricks to speed up computations.
2015
 Simulation of Electromagnetic Scattering with Stationary or Accelerating Targets
[Articolo su rivista]
Funaro, Daniele; E., Kashdan
abstract
The scattering of electromagnetic waves by an obstacle is analyzed
through a set of partial differential equations combining the Maxwell's
model with the mechanics of fluids. Solitary type EM waves, having compact support, may easily be modeled in this context since they turn out
to be explicit solutions. From the numerical viewpoint, the interaction of
these waves with a material body is examined. Computations are carried
out via a parallel highorder finitedifferences code. Due to the presence of
a gradient of pressure in the model equations, waves hitting the obstacle
may impart acceleration to it. Some explicative 2D dynamical configurations are then studied, enabling the study of photonparticle iterations
through classical arguments.
2014
 Charging Capacitors According to Maxwell’s Equations: Impossible
[Articolo su rivista]
Funaro, Daniele
abstract
The charge of an ideal parallel capacitor leads to the resolution of the wave equation for the electric field with prescribed initial
conditions and boundary constraints. Independently of the capacitor’s
shape and the applied voltage, none of the corresponding solutions is
compatible with the full set of Maxwell’s equations. The paradoxical situation persists even by weakening boundary conditions, resulting in the
impossibility to describe a trivial phenomenon such as the capacitor’s
charging process, by means of the standard Maxwellian theory.
2014
 Trapping Electromagnetic Solitons in Cylinders
[Articolo su rivista]
Funaro, Daniele
abstract
Electromagnetic waves in vacuum or dielectrics can be confined in unbounded cylinders in such a way that they turn around the main axis. For particular choices of the cylinder's section, interesting stationary configurations may be assumed. By refining some results obtained in previous papers, additional more complex situations are examined here. For such peculiar guided waves an explicit expression is given in terms of Bessel's functions. Possible applications are in the development of whispering gallery resonators.
2010
 Numerical Simulation of Electromagnetic Solitons and their Interaction with Matter
[Articolo su rivista]
Funaro, Daniele
abstract
A suitable correction of the Maxwell model brings to anenlargement of the space of solutions, allowing for the existenceof solitons in vacuum. We review the basic achievements of thetheory and discuss some approximation results based on an explicitfinitedifference technique. The experiments in two dimensionssimulate travelling solitary electromagnetic waves, and show theirinteraction with conductive walls. In particular, the classicaldispersion, exhibited by the passage of a photon through a smallaperture, is examined.
2010
 Rotating Electromagnetic Waves in ToroidShaped Regions
[Articolo su rivista]
C., Chinosi; L., Della Croce; Funaro, Daniele
abstract
Electromagnetic waves, solving the full set of Maxwell equationsin vacuum, are numerically computed. These waves occupy a fixedbounded region of the three dimensional space, topologicallyequivalent to a toroid. Thus, their fluid dynamics analogs arevortex rings. An analysis of the shape of the sections of therings, depending on the angular speed of rotation and the majordiameter, is carried out. Successively, spherical electromagneticvortex rings of Hill's type are taken into consideration. For someinteresting peculiar configurations, explicit numerical solutionsare exhibited.
2008
 A Convergence Analysis for the Superconsistent Chebyshev Method
[Articolo su rivista]
L., Fatone; Funaro, Daniele; G. J., Yoon
abstract
The superconsistent collocation method is based on collocation nodes which are different from those used to represent the solution. The two grids are chosen in such a way that the continuous and the discrete operators coincide on a space as larger as possible (superconsistency). There are many documented situations in which this technique provides excellent numerical results. Unfortunately very little theory has been developed. Here, a theoretical convergence analysis for the superconsistent discretization of the second derivative operator, when the representation grid is the set of Chebyshev Gauss–Lobatto nodes is carried out. To this end, a suitable quadrature formula is introduced and studied.
2008
 Electromagnetism and the Structure of Matter
[Monografia/Trattato scientifico]
Funaro, Daniele
abstract
The classical theory of electromagnetism is entirely revised in this book by proposing a variant of Maxwell equations that allows solitonic solutions (photons). The Lagrangian is the standard one, but it is minimized on a constrained space that enforces the wave packets to follow the rules of geometrical optics. Exact solutions are explicitly shown; this opens a completely new perspective for the study of light wave phenomena. In the framework of general relativity, the equations are written in covariant form. A coupling with the metric is obtained through the Einstein equation, whose solutions are computed exactly in a lot of original situations. Finally, the explicit construction of elementary particles, consisting of rotating photons, is indicated. The results agree qualitatively and quantitatively with what it is actually observed. This opens the path to an understanding of the structure of matter and its properties, also aimed to provide a causal explanation to quantum phenomena.
2007
 FiniteDifferences Preconditioners for Superconsistent Pseudospectral Approximations
[Articolo su rivista]
L., Fatone; Funaro, Daniele; V., Scannavini
abstract
The superconsistent collocation method, which is based on a collocation grid different from the one used to represent the solution, has proven to be very accurate in the resolution of various functional equations. Excellent results can be also obtained for what concerns preconditioning. Some analysis and numerous experiments, regarding the use of finitedifferences preconditioners, for matrices arising from pseudospectral approximations of advectiondiffusion boundary value problems, are presented and discussed, both in the case of Legendre and Chebyshev representation nodes.
2006
 Analisi Numerica
[Voce in Dizionario o Enciclopedia]
Funaro, Daniele
abstract
2005
 A general class of finitedifference methods for the linear transport equation
[Articolo su rivista]
Funaro, Daniele; G., Pontrelli
abstract
A wide family of finitedifferences methods for the linear advection equation, based on a sixpoint stencil, is presented. The family depends on three parameters and includes most of the classical linear schemes. A stability and consistency analysis is carried out of the parameters. The problem of the determination of the parameters providing the best approximation is also addressed.
2005
 Finitedifference schemes for transportdominated equations using special collocation nodes
[Articolo su rivista]
Fatone, Lorella; Funaro, Daniele; R., Giova
abstract
We introduce finitedifference schemes based on a special upwindtype collocation grid, in order to obtain approximations of the solution of linear transportdominated advectiondiffusion problems. The method is well suited when the diffusion parameter is very small compared to the discretization parameter. A theory is developed and many numerical experiments are shown.
2004
 Superconsistent discretizations of integral type equations
[Articolo su rivista]
Funaro, Daniele
abstract
We recall some of the results, obtained by using the collocation method at special nodes, in the approximation of boundary value problems for partial differential equations. Similar techniques are applied for discretizing, by the Gaussian collocation method, a singular integral equation of the Carleman type. The idea, documented by some comparative tests, can be generalized to other approximation methods and/or integral equations of a different kind. It can also provide a link between different numerical approaches. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.
2003
 Superconsistent Discretizations with Application to Hyperbolic Equation
[Articolo su rivista]
Funaro, Daniele
abstract
A family of finite difference methods for the linear hyperbolic equations, constructed on a sixpoint stencil, is presented. The family depends on 3 parameters and includes many of the classical linear schemes. The approximation method is based on the use of two different grids. One grid is used to represent the approximated solution, the other (the collocation grid) is where the equation is to be satisfied. The two grids are related in such a way that the exact and the discrete operators have a common space which is as large as possible.
2002
 Superconsistent Discretizations
[Articolo su rivista]
Funaro, Daniele
abstract
We say that the approximation of a linear operator issuperconsistent when the exact and the discrete operatorscoincide on a functional space whose dimension is bigger thanthe number of degrees of freedom needed in the constructionof the discretization. By providing several examples we show how to build superconsitent schemes.
2001
 A superconsistent Chebyshev collocation method for secondorder differential operators
[Articolo su rivista]
Funaro, Daniele
abstract
A standard way to approximate the model problem u = f, with u(+/1) = 0, is to collocate the differential equation at the zeros of Tn': x(i), i = 1,..., n  1, having denoted by T,, the nth Chebyshev polynomial. We introduce an alternative set of collocation nodes z(i), i = 1,..., n  1, which will provide better numerical performances. The approximated solution is still computed at the nodes {x(i)}, but the equation is required to be satisfied at the new nodes {z(i)}, which are determined by asking an extra degree of consistency in the discretization of the differential operator.
2000
 About 3D Spectral Element Computations
[Relazione in Atti di Convegno]
Funaro, Daniele
abstract
1999
 A note on secondorder finitedifference schemes on uniform meshes for advectiondiffusion equations
[Articolo su rivista]
Funaro, Daniele
abstract
An artificialviscosity finitedifference scheme is introduced for stabilizing the solutions of advectiondiffusion equations. Although only the linear onedimensional case is discussed, the method is easily susceptible to generalization. Some theory and comparisons with other wellknown schemes are carried out. The aim is, however, to explain the construction of the method, rather than considering sophisticated applications.
1999
 Spline approximation of advectiondiffusion problems using upwind type collocation nodes
[Articolo su rivista]
Funaro, Daniele; G., Pontrelli
abstract
A spline collocation method for linear advectiondiffusion equations is proposed. The method is based on an operatordependent collocation grid, and provides stabilized approximated solutions, with respect to the coefficient of the diffusive term, when problems are advection dominated. (C) 1999 Elsevier Science B.V. All rights reserved.
1998
 A Splitting Method for Unsteady Incompressible Viscous Fluids Imposing No Boundary Conditions on Pressure
[Articolo su rivista]
Funaro, D.; Giangi, M.; Mansutti, D.
abstract
We propose a timeadvancing scheme for the discretization of the unsteady incompressible NavierStokes equations. At any time step, we are able to decouple velocity and pressure by solving some suitable elliptic problems. In particular, the problem related with the determination of the pressure does not require boundary conditions. The divergence free condition is imposed as a penalty term, according to an appropriate restatement of the original equations. Some experiments are carried out by approximating the space variables with the spectral Legendre collocation method. Due to the special treatment of the pressure, no spurious modes are generated.
1997
 Improving the Performances of Implicit Schemes for Hyperbolic Equations
[Articolo su rivista]
Funaro, D.
abstract
Considering that, in the discretization of linear differential operators, one can choose suitable nodes of superconvergence for the evaluation of the residual, we apply this idea to firstorder operators associated with the approximation of hyperbolic equations, in order to improve some known implicit schemes.
1997
 Some remarks about the collocation method on a modified Legendre grid
[Articolo su rivista]
Funaro, Daniele
abstract
We compare the results obtained by applying the standard collocation method at the Legendre GaussLobatto nodes, for a model problem simulating a steady advectiondiffusion equation, with those obtained by collocating at a new set of nodes. These nodes are derived from a suitable modification of the Legendre grid, according to the magnitude of the ratio between the advective and the diffusive parts of the differential operator.
1997
 Spectral Elements for TransportDominated Equations
[Monografia/Trattato scientifico]
Funaro, Daniele
abstract
The book deals with the numerical approximation of various PDEs using the spectral element method, with particular emphasis for elliptic equations dominated by firstorder terms. It provides a simple introduction to spectral elements with additional new tools (upwind grids and preconditioners). Applications to fluid dynamics and semiconductor devices are considered, as well as in other models were transportdiffusion equations arise. The aim is to provide the reader with both introductive and more advanced material on spectral Legendre collocation methods. The book however does not cover all the aspects of spectral methods. Engineers, physicists and applied mathematicians may study how to implement the collocation method and use the results to improve their computational codes.
1994
 A fast solver for elliptic boundaryvalue problems in the square
[Articolo su rivista]
Funaro, D.
abstract
We approximate the solution of advectiondiffusion equations by collocation at a special grid related to the differential operator and the classical Legendre grid.
1994
 A novel numerical technique to investigate nonlinear guided waves: Approximation of nonlinear Schrödinger equation by nonperiodic pseudospectral methods
[Articolo su rivista]
De Veronico, C.; Funaro, D.; Reali, G. C.
abstract
A new numerical technique for investigating light waves guided by planar nonlinear dielectric films is presented. The method implements a multidomain spectral type approximation based on orthogonal algebraic polynomials, and makes possible to deal with nonperiodic boundary conditions and discontinuities of the data, overcoming the known deficiencies of the trigonometric polynomials. Results of preliminary numerical experiments for the solution of the nonlinear Schrödinger equation are presented.
1994
 An alternative approach to the analysis and the approximation of the NavierStokes equations
[Articolo su rivista]
Boffi, D.; Funaro, D.
abstract
A suitable restatement of the NavierStokes equations provides a new tool for their numerical approximation. Spectral methods are used in the experiments to show the effectiveness of the technique. The main achievement is the total elimination of spurious modes.
1994
 Spectral elements in the approximation of boundaryvalue problems in complex geometries
[Articolo su rivista]
Funaro, D.
abstract
1993
 A NEW SCHEME FOR THE APPROXIMATION OF ADVECTIONDIFFUSION EQUATIONS BY COLLOCATION
[Articolo su rivista]
Funaro, Daniele
abstract
A competitive algorithm, which allows the computation of approximated polynomial solutions of advectiondiffusion equations in the square, is presented. The equation is collocated at a special grid and the corresponding system is solved by a lowcost preconditioned iterative procedure. The method provides accurate results even when the solution presents sharp boundary layers.
1993
 A mathematical model of potential spreading along neuron dendrites of cerebellar granule cells
[Articolo su rivista]
Comincioli, V.; Funaro, D.; Torelli, A.; D'Angelo, E.; Rossi, P.
abstract
A differential model is proposed to describe the evolution of the potential in an electrically stimulated cerebellar granule cell. After suitable discretizations in the time and space variables, the results of some numerical experiments are shown.
1992
 Approximation by the Legendre collocation method of a model problem in electrophysiology
[Articolo su rivista]
Funaro, D.
abstract
We examine the polynomial approximation of the solution of a nonlinear differential problem modelling the evolution of the potential inside an electrically stimulated neuron. The collocation method at the Legendre GaussLobatto nodes is used for the discretization with respect to the space variable. © 1992.
1992
 Polynomial Approximation of Differential Equations
[Monografia/Trattato scientifico]
Funaro, Daniele
abstract
This book is a basic and comprehensive introduction to the use of spectral methods for the approximation of the solution to ordinary differential equations and timedependent boundaryvalue problems. The algorithms are presented and studied both from the point of view of the theoreticalanalysis of convergence and the numerical implementation. Unlike other texts devoted to the subject this is a concise introduction that is ideally suited to the novice and practitioner alike, enabling them to assimilate themethods quickly and efficiently.
1991
 APPROXIMATION OF SOME DIFFUSION EVOLUTIONEQUATIONS IN UNBOUNDEDDOMAINS BY HERMITE FUNCTIONS
[Articolo su rivista]
Funaro, Daniele; O., Kavian
abstract
Spectral and pseudospectral approximations of the heat equation are analyzed. The solution is represented in a suitable basis constructed with Hermite polynomials. Stability and convergence estimates are given and numerical tests are discussed.
1991
 CONVERGENCE RESULTS FOR PSEUDOSPECTRAL APPROXIMATIONS OF HYPERBOLIC SYSTEMS BY A PENALTYTYPE BOUNDARY TREATMENT
[Articolo su rivista]
Funaro, Daniele; D., Gottlieb
abstract
In a previous paper we have presented a new method of imposing boundary conditions in the pseudospectral Chebyshev approximation of a scalar hyperbolic equation. The novel idea of the new method is to collocate the equation at the boundary points as well as in the inner grid points, using the boundary conditions as penalty terms. In this paper we extend the above boundary treatment to the case of pseudospectral approximations to general constantcoefficient hyperbolic systems of equations, and we provide error estimates for the pseudospectral Legendre method. The same scheme can be implemented also in the general (even nonlinear) case.
1991
 Pseudospectral approximation of a PDE defined on a triangle
[Articolo su rivista]
Funaro, D.
abstract
A scheme for the approximation by collocation method of an elliptic equation defined on a triangle is proposed. Different solution techniques are examined.
1990
 COMPUTATIONAL ASPECTS OF PSEUDOSPECTRAL LAGUERRE APPROXIMATIONS
[Articolo su rivista]
Funaro, Daniele
abstract
Pseudospectral approximations in unbounded domains by Laguerre polynomials lead to illconditioned algorithms. We introduce a scaling function and appropriate numerical procedures to limit these phenomena.
1990
 Convergence analysis for pseudospectral multidomain approximations of linear advection equations
[Articolo su rivista]
Funaro, D.
abstract
Legendre and Chebyshev collocation schemes are proposed for the numerical approximation of first order linear hyperbolic equations, by a domain decomposition procedure. Spectral convergence estimates are provided both for Legendre and Chebyshev GaussLobatto nodes. © 1989 Oxford University Press.
1990
 Inverse inequalities for chebyshev approximations in L∞ norms
[Articolo su rivista]
Funaro, D.
abstract
Inverse inequalities in the space of polynomials, relating the maximum norm in [1,1] and weighted Sobolev norms, are shown.
1990
 LAGUERRE SPECTRAL APPROXIMATION OF ELLIPTIC PROBLEMS IN EXTERIOR DOMAINS
[Articolo su rivista]
O., Coulaud; Funaro, Daniele; O., Kavian
abstract
1990
 SOME RESULTS ABOUT THE PSEUDOSPECTRAL APPROXIMATION OF ONEDIMENSIONAL 4THORDER PROBLEMS
[Articolo su rivista]
Funaro, Daniele; W., Heinrichs
abstract
We analyze the pseudospectral approximation of fourth order problems. We give convergence results in the one dimensional case. Numerical experiments are shown in two dimensions for the approximation of the rhombic plate bending problem. Eigenvalues and preconditioning are also investigated.
1990
 Variational formulation for the Chebyshev pseudospectral approximation of Neumann problems
[Articolo su rivista]
Funaro, Daniele
abstract
A collocation scheme is proposed, on the basis of the Chebyshev nodes, for the approximation of elliptic problems with DirichletNeumann boundary conditions, deriving from a suitable variational formulation in weighted Sobolev spaces.
1988
 A NEW METHOD OF IMPOSING BOUNDARYCONDITIONS IN PSEUDOSPECTRAL APPROXIMATIONS OF HYPERBOLICEQUATIONS
[Articolo su rivista]
Funaro, Daniele; D., Gottlieb
abstract
A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.
1988
 An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods
[Articolo su rivista]
Funaro, Daniele; Alfio, Quarteroni; Paola, Zanolli
abstract
A domain decomposition method for secondorder elliptic problems is considered. An iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed. At each iteration, a relaxation is accomplished at the subdomain interfaces. In several circumstances, a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found. Moreover, when such a value is not available, an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed.This iterative method is then applied to the spectral collocation approximation of the differential problem. The same kind of convergence results are proven. Many numerical experiments show the effectiveness of the method proposed here.
1988
 DOMAIN DECOMPOSITION METHODS FOR PSEUDO SPECTRAL APPROXIMATIONS .1. 2ND ORDER EQUATIONS IN ONE DIMENSION
[Articolo su rivista]
Funaro, Daniele
abstract
Multidomain pseudo spectral approximations of second order boundary value problems in one dimension are considered. The equation is collocated at the Chebyshev nodes inside each subinterval. Different patching conditions at the interfaces are analyzed. Results of stability and convergence are given.
1988
 THE SCHWARZ ALGORITHM FOR SPECTRAL METHODS
[Articolo su rivista]
C., Canuto; Funaro, Daniele
abstract
Recently, the Schwarz alternating method has been successfully coupled to spatial discretizations of spectral type, in order to solve boundary value, problems in complex, geometries with infinite order accuracy. In this paper, a simple version of the method is considered. A proof of its convergence is given in the energy norm, exploiting the properties of discreteharmonic polynomials and a discrete maximum principle for spectral methods. More general situations can be handled theoretically in one space dimension.
1987
 A PRECONDITIONING MATRIX FOR THE CHEBYSHEV DIFFERENCING OPERATOR
[Articolo su rivista]
Funaro, Daniele
abstract
An efficient preconditioner for the Chebyshev differencing operator is considered. The corresponding preconditioned eigenvalues are real and positive and lie between 1 and ${\pi / 2}$. An eixpelicit formula for these eigenvalues and the corresponding eigenfunctions is given. The results are generalized to the case of operators related to Chebyshev discretizations of systems of linear differential equations.
1987
 Some results about the spectrum of the chebyshev differencing operator
[Relazione in Atti di Convegno]
Funaro, D.
abstract
A spectral method in space and a finitedifference scheme in time are employed to approximate the solution of the model equation: yt=yx. The operator ∂/∂x is discretized by the collocation method based on the Chebyshev nodes. The second order RungeKutta method is used for the operator ∂/∂t. It is known that the location, in the complex plane, of the eigenvalues of the collocation matrix is crucial for the stability. A simple way of computing the coefficients of the characteristic polynomial of that matrix is shown. An explicit computation of the roots gives indications on the choice of the time step. © 1987, Elsevier B.V.
1986
 A multidomain spectral approximation of elliptic equations
[Articolo su rivista]
Funaro, D.
abstract
A spectral approximation for the Poisson equation defined on Ω = ]−1, 1[×] −1,1[ is studied. The domain Ω is decomposed into two rectangular regions and the equation is collocated at the Legendre nodes in each domain. On the common boundary of the two subdomains, suitable conditions are imposed in order to obtain a unique solution from the resulting linear system. Different values of the discretization parameters are allowed in each rectangle. We prove the stability of the scheme and give convergence estimates. The rate of convergence in a single subdomain, depends only on the regularity of the exact solution therein. An efficient preconditioning matrix is proposed.
1983
 Error estimates for spectral approximation of linear advection equations over an ipercube
[Articolo su rivista]
Funaro, D.
abstract
Spectral and pseudospectral (collocation) approximations of the advection equations in an ipercube are presented. Collocation is imposed on the Chebyshev nodes. Stability and convergence results are given in Sobolev norms relative to some Jacobi weights. © 1984 Instituto di Elaborazione della Informazione del CNR.