## PROBLEMAS DE EVOLUCION: MODELOS, APLICACIONES Y NUEVAS TECNICAS ASINTOTICAS Y NUMERICAS DE RESOLUCION

MTM2014-52859-P

Programa Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia
Subprograma Subprograma Estatal de Generación del Conocimiento
Convocatoria Proyectos de I+D dentro del Subprograma Estatal de Generación del Conocimiento (2014)
Año convocatoria 2014
Unidad de gestión Dirección General de Investigación Científica y Técnica
Centro beneficiario UNIVERSIDAD PÚBLICA DE NAVARRA (UPNA)
Centro realización UNIVERSIDAD PUBLICA DE NAVARRA

## Publicaciones

Found(s) 27 result(s)
Found(s) 1 page(s)

#### Orthogonal basis with a conicoid first mode for shape specification of optical surfaces

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
• Navarro, Rafael
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed., This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union MTM2014-52859 and FIS2014-58303.

#### Multipoint flux mixed finite element methods for slightly compressible flow in porous media

• Arrarás Ventura, Andrés
• Portero Egea, Laura
In this paper, we consider multipoint flux mixed finite element discretizations for slightly compressible Darcy flow in porous media. The methods are formulated on general meshes composed of triangles, quadrilaterals, tetrahedra or hexahedra. An inexact Newton method that allows for local velocity elimination is proposed for the solution of the nonlinear fully discrete scheme. We derive optimal error estimates for both the scalar and vector unknowns in the semidiscrete formulation. Numerical examples illustrate the convergence behavior of the methods, and their performance on test problems including permeability coefficients with increasing heterogeneity., This work was partially supported by MINECO grant MTM2014-52859-P.

#### Avoiding the order reduction when solving second-order in time PDEs with Fractional Step Runge–Kutta–Nyström methods

• Moreta, M. Jesús
• Bujanda Cirauqui, Blanca
• Jorge Ulecia, Juan Carlos
We study some of the main features of Fractional Step Runge–Kutta–Nyström methods when they are used to integrate Initial–Boundary Value Problems of second order in time, in combination with a suitable spatial discretization. We focus our attention on the order reduction phenomenon, which appears if classical boundary conditions are taken at the internal stages. This drawback is specially hard when time dependent boundary conditions are considered. In this paper we present an efficient technique, very simple and computationally cheap, which allows us to avoid the order reduction; such technique consists in modifying the boundary conditions for the internal stages of the method., M.J. Moreta had financial support from MTM 2015-66837-P. B. Bujanda had financial support from TEC 2013-45585-C2-1-R. J.C. Jorge had financial support from MTM 2014-52859.

#### Convergent and asymptotic methods for second-order difference equations with a large parameter

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
This is a post-peer-review, pre-copyedit version of an article published in Mediterranean Journal of Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s00009-018-1267-9, We consider the second-order linear difference equation y(n+2)−2ay(n+1)−Λ2y(n)=g(n)y(n)+f(n)y(n+1) , where Λ is a large complex parameter, a≥0 and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large |Λ|of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind., This research was supported by the Spanish Ministry of Economía y Competitividad, project MTM2014-52859-P. The Universidad Pública de Navarra is acknowledged by its financial support.

#### A simplification of the stationary phase method: application to the Anger and Weber functions

• López García, José Luis
The main difficulty in the practical use of the stationary phase method in asymptotic expansions of
integrals is originated by a change of variables. The coefficients of the asymptotic expansion are the coefficients of
the Taylor expansion of a certain function implicitly defined by that change of variables. In general, this function is
not explicitly known, and then the computation of those coefficients is cumbersome. Using the factorization of the
exponential factor used in previous works of [Tricomi, 1950], [Erdélyi and Wyman, 1963], and [Dingle, 1973], we
obtain a variant of the method that avoids that change of variables and simplifies the computations. On the one hand,
the calculation of the coefficients of the asymptotic expansion is remarkably simpler and explicit. On the other hand,
the asymptotic sequence is as simple as in the standard stationary phase method: inverse powers of the asymptotic
variable. New asymptotic expansions of the Anger and Weber functions Jλx(x) and Eλx(x) for large positive x and
real parameter λ 6= 0 are given as an illustration., This research was supported by the Spanish Ministry of Economía y Competitividad, project MTM2014-52859-P. The Universidad Pública de Navarra is acknowledged by its financial support.

#### On a modifcation of Olver's method: a special case

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
This is a post-peer-review, pre-copyedit version of an article published in Constructive Approximation. The final authenticated version is available online at: https://doi.org/10.1007/s00365-015-9298-y, We consider the asymptotic method designed by Olver (Asymptotics and
special functions. Academic Press, New York, 1974) for linear differential equations of
the second order containing a large (asymptotic) parameter : xm y −2 y = g(x)y,
with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, especially the
cases m = 0, ±1, giving the Poincaré-type asymptotic expansions of two independent
solutions of the equation. The case m = 2 is different, as the behavior of the solutions
for large is not of exponential type, but of power type. In this case, Olver’s theory
does not give many details. We consider here the special case m = 2. We propose
two different techniques to handle the problem: (1) a modification of Olver’s method
that replaces the role of the exponential approximations by power approximations,
and (2) the transformation of the differential problem into a fixed point problem from
which we construct an asymptotic sequence of functions that converges to the unique
solution of the problem. Moreover, we show that this second technique may also be
applied to nonlinear differential equations with a large parameter., The Dirección General de Ciencia y Tecnología (REF.MTM2014-52859) is acknowledged for its financial support.

#### Analytic formulas for the evaluation of the Pearcey integral

• López García, José Luis
• Pagola Martínez, Pedro Jesús
We can find in the literature several convergent and/or asymptotic
expansions of the Pearcey integral P(x, y) in different regions of the complex
variables x and y, but they do not cover the whole complex x and y planes.
The purpose of this paper is to complete this analysis giving new convergent
and/or asymptotic expansions that, together with the known ones, cover the
evaluation of the Pearcey integral in a large region of the complex x and y
planes. The accuracy of the approximations derived in this paper is illustrated
with some numerical experiments. Moreover, the expansions derived here are
simpler compared with other known expansions, as they are derived from a
simple manipulation of the integral definition of P(x, y)., This research was supported by the Spanish Ministry of “Economía y Competitividad”,
project MTM2014-52859-P. The Universidad Pública de Navarra is
acknowledged for its financial support.

#### Orthogonal basis for the optical transfer function

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
• Navarro, Rafael
We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 OTF perfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed., This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union MTM2014-52859 and FIS2014-58303.

#### The use of two-point Taylor expansions in singular one-dimensional boundary value problems I

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
We consider the second-order linear differential equation (x + 1)y′′ + f(x)y′ + g(x)y = h(x) in
the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed
Dirichlet-Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci
at x = ±1 containing the interval [−1, 1]. Then, the end point of the interval x = −1 may be a
regular singular point of the differential equation. The two-point Taylor expansion of the solution
y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential
equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary
value problem. This method is constructive and provides the two-point Taylor approximation
of the analytic solutions when they exist., The Ministerio de Economía y Competitividad (REF. MTM2014-52859-P) is acknowledged by its financial support.

#### Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
In previous papers [6–8,10], we derived convergent and asymptotic
expansions of solutions of second order linear differential equations with a
large parameter. In those papers we generalized and developed special cases
not considered in Olver’s theory [Olver, 1974]. In this paper we go one step
forward and consider linear differential equations of the third order: y
′′′ +aΛ2y′ +bΛ3y = f(x)y′ +g(x)y, with a, b ∈ C fixed, f′ and g continuous, and Λ
a large positive parameter. We propose two different techniques to handle the
problem: (i) a generalization of Olver’s method and (ii) the transformation of
the differential problem into a fixed point problem from which we construct an
asymptotic sequence of functions that converges to the unique solution of the
problem. Moreover, we show that this second technique may also be applied
to nonlinear differential equations with a large parameter. As an application
of the theory, we obtain new convergent and asymptotic expansions of the
Pearcey integral P(x, y) for large |x|., The Ministerio de Econom´ıa y Competitividad (REF. MTM2014-52859-P) is acknowledged by its financial support.

#### New asymptotic expansion and error bound for Stirling formula of reciprocal Gamma function

• Pagola Martínez, Pedro Jesús
Studying the problem about if certain probability measures are determinate by its moments
[4, 8, 10] is useful to know the asymptotic behavior of the probability densities for large
values of argument. This requires, previously, the knowledge of the asymptotic expansion of
reciprocal Gamma function 1/Γ(z) when ℜz is large and ℑz is fixed [8]. Then, the well known
Stirling formula for large |z| of the Gamma function Γ(z) or its reciprocal 1/Γ(z) is not appropriate
for this problem. So, the main aim of this paper is to obtain a new asymptotic expansion
for reciprocal Gamma function valid for large ℜz and establish a new explicit error bound for
the first term of this expansion, that is, the Stirling formula., This research was supported by the Spanish Ministry of Economía y Competitividad, project
MTM2014-52859. The Universidad Pública de Navarra is acknowledged by its financial support.

#### The asymptotic expansion of the swallowtail integral in the highly oscillatory region

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
The mathematical models of many short wavelength phenomena, specially wave propagation and optical diffraction,
contain, as a basic ingredient, oscillatory integrals with several nearly coincident stationary phase or saddle points. The
uniform approximation of those integrals can be expressed in terms of certain canonical integrals and their derivatives
[2,16]. The importance of these canonical diffraction integrals is stressed in [14] by means of the following sentence: The
role played by these canonical diffraction integrals in the analysis of caustic wave fields is analogous to that played by complex
exponentials in plane wave theory. Apart from their mathematical importance in the uniform asymptotic approximation of oscillatory integrals [12], the
canonical diffraction integrals have physical applications in the description of surface gravity waves [11], [17], bifurcation sets, optics, quantum mechanics, chemical physics [4] and acoustics (see [1], Section 36.14 and references there in). To our
knowledge, the first application of this family of integrals traces back to the description of the disturbances on a water surface produced, for example, by a traveling ship. These disturbances form a familiar pattern of bow and stern waves
which was first explained mathematically by Lord Kelvin [10] using these integrals., This research was supported by the Ministerio de Economía y Competitividad (MTM2014-52859) and the Universidad Pública de Navarra.

#### Geometric multigrid methods for Darcy–Forchheimer flow in fractured porous media

• Arrarás Ventura, Andrés
• Portero Egea, Laura
• Gaspar, F. J.
• Rodrigo, C.
In this paper, we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimensional smoothing and inter-grid transfer operators. Numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability, the Forchheimer coefficient and the mesh size. The case of several possibly intersecting fractures in a heterogeneous porous medium is also discussed., Francisco J. Gaspar has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 705402, POROSOS. The work of Carmen Rodrigo is supported in part by the FEDER / MINECO, Spain project MTM2016-75139-R. The work of Andrés Arrarás and Laura Portero is supported in part by the FEDER / MINECO, Spain projects MTM2014-52859-P and MTM2016-75139-R.

#### Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems

• Arrarás Ventura, Andrés
• Portero Egea, Laura
In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements. If these elements are considered as subdomains, we can formulate a non-overlapping domain decomposition method based on the lowest-order Raviart–Thomas elements, properly enhanced with Lagrange multipliers on the boundaries of each subdomain (excluding the Dirichlet edges). A suitable choice of mixed finite element spaces and quadrature rules yields a cell-centered scheme for the pressures with a local 10-point stencil. The resulting system of differential-algebraic equations is integrated in time by the Crank–Nicolson method, which is known to be a stiffly accurate scheme. As a result, we obtain independent subdomain linear systems that can be solved in parallel. The behavior of the algorithm is illustrated on a variety of numerical experiments., This work was partially supported by MINECO grant MTM2014-52859.

#### Convergent expansions of the Bessel functions in terms of elementary functions

• López García, José Luis
• Pagola Martínez, Pedro Jesús
This is a post-peer-review, pre-copyedit version of an article published in Advances in Computational Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s10444-017-9543-y, We consider the Bessel functions Jν (z) and Yν (z) for ν > −1/2 and
z ≥ 0. We derive a convergent expansion of Jν (z) in terms of the derivatives of
(sin z)/z, and a convergent expansion of Yν (z) in terms of derivatives of (1−cos z)/z,
derivatives of (1 − e−z)/z and (2ν, z). Both expansions hold uniformly in z in any
fixed horizontal strip and are accompanied by error bounds. The accuracy of the
approximations is illustrated with some numerical experiments., This research was supported by the Spanish Ministry of "Economía y Competitividad",
project MTM2014-52859-P. The Universidad Pública de Navarra is acknowledged by its
financial support.

#### High order Nyström methods for transmission problems for Helmholtz equation

• Domínguez Baguena, Víctor
• Turc, Catalin
We present super-algebraic compatible Nyström discretizations for the four Helmholtz boundary operators of Calderón’s calculus on smooth closed curves in 2D. These discretizations are based on appropriate splitting of the kernels combined with very accurate product-quadrature rules for the different singularities that such kernels present. A Fourier based analysis shows that the four discrete operators converge to the continuous ones in appropriate Sobolev norms. This proves that Nyström discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually super-algebraic for smooth solutions., Catalin Turc gratefully acknowledge support from NSF through contract DMS-1312169. Víctor Domínguez is partially supported by Ministerio de Economía y Competitividad, through the grant MTM2014-52859.
This research was partially supported by Spanish MINECO grants MTM2011-22741 and MTM2014-54388.

#### Analytic approximations of integral transforms in terms of elementary functions: application to special functions

• Palacios Herrero, Pablo
This thesis focuses on the study of new analytical methods for the approximation of integral transforms and, in particular, of special functions that admit an integral representation. The importance of these functions lies in the fact that they are solutions to a great variety of functional equations that model speciﬁc physical phenomena. Moreover, they play an important role in pure and applied mathematics, as well as in other branches of science such as chemistry, statistics or economics. Usually, the integrals deﬁning these special functions depend on various parameters that have a speciﬁc physical meaning. For this reason, it is important to have analytical techniques that allow their computation in a quick and easy manner. The most commonly used analytical methods are based on series expansions of local validity: Taylor series and asymptotic (divergent) expansions that are, respectively, valid for small or large values of the physically relevant variable. However, neither of them is, in general, simultaneously valid for large and small values of the variable. In this thesis we seek new methods for the computation of analytic expansions of integral transforms satisfying the following three properties: (a) The expansions are uniformly valid in a large region of the complex plane. Ideally, these regions should be unbounded and contain the point 0 in their interior. (b) The expansions are convergent. Therefore, it is not necessary to obtain error bounds or to study the optimal term to truncate the expansion: the more terms considered, the smaller the error committed.
(c) The expansions are given in terms of elementary functions. We develop a theory of uniform expansions that shows the necessary and suﬃcient conditions to obtain expansions of integral transforms fulﬁlling the three conditions (a),(b) and (c) above. This theory is applied to obtain new series approximations satisfying (a), (b) and (c) of a large number of special functions. The new expansions are compared with other known representations that we may ﬁnd in the literature to show their ad-vantages and drawbacks. In contrast to the Taylor and asymptotic expansions, the main beneﬁt of the uniform expansions is that they are valid in a large region of the complex plane. For this reason, they may be used to replace the function they approximate (which is often diﬃcult to work with) when it appears in certain calculations, such as a factor of an integral or in a diﬀerential equation. Since these developments are also given in terms of elementary functions, such calculations may be carried out easily. Next, we consider a particularly important case: when the kernel of the integral transform is given by an exponential. We develop a new Laplace’s method for integrals that produces asymptotic and convergent expansions, in contrast to the classical Laplace method which produces divergent developments. The expansions obtained with this new method satisfy (a) and (b) but not (c), since the asymptotic sequence is given in terms of incomplete beta functions. Finally, we develop a new uniform asymptotic method 'saddle point near an end point' which does not satisfy (b) and (c) but, unlike the classical 'saddle point near an end point' method, allows us to calculate the coeﬃcients of the expansion by means of a simple and systematic formula., This work was supported by the Ministerio de Economa y Competitividad (MTM2014-52859-P and MTM2017-83490-P) and Universidad Publica de Navarra PRO-UPNA (6158)., Programa de Doctorado en Matemáticas y Estadística (RD 99/2011), Matematikako eta Estatistikako Doktoretza Programa (ED 99/2011)

#### An efficient numerical method for singularly perturbed time dependent parabolic 2D convection-diffusion systems

• Clavero, Carmelo
• Jorge Ulecia, Juan Carlos
In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown., This research was partially supported by the project MTM2014-52859-P and by the Aragón Government and European Social Fund, Spain (group E24–17R ).

#### Modified Douglas splitting methods for reaction–diffusion equations

• Arrarás Ventura, Andrés
• Portero Egea, Laura
• Hout, K. J. in ’t
• Hundsdorfer, W.
We present modifications of the second-order Douglas stabilizing corrections method, which is a splitting method based on the implicit trapezoidal rule. Inclusion of an explicit term in a forward Euler way is straightforward, but this will lower the order of convergence. In the modifications considered here, explicit terms are included in a second-order fashion. For these modified methods, results on linear
stability and convergence are derived. Stability holds for important classes of reaction–diffusion equations, and for such problems the modified Douglas methods are seen to be often more efficient than related methods from the literature., The work of A. Arrarás and L. Portero was partially supported by MINECO grant MTM2014-52859.

#### Orthogonal basis with a conicoid first mode for shape specification of optical surfaces: reply

• Ferreira González, Chelo
• López García, José Luis
• Pérez Sinusía, Ester
• Navarro, Rafael
We present some comments to the paper 'Orthogonal basis with a conicoid first mode for shape specification of optical surfaces: comment'., This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union MTM2014-52859 and FIS2014-58303.

#### New fractional step Runge-Kutta-Nyström methods up to order three

• Bujanda Cirauqui, Blanca
• Jorge Ulecia, Juan Carlos
• Moreta, M. Jesús
Fractional Step Runge–Kutta–Nyströ (FSRKN) methods have been revealed to be an excellent option to integrate numerically many multidimensional evolution models governed by second order in time partial differential equations. These methods, combined with suitable spatial discretizations, lead to strong computational cost reductions respect to many classical implicit time integrators. In this paper, we present the construction process of several implicit FSRKN methods of two and three levels which attain orders up to three and satisfy adequate stability properties. We have also performed some numerical experiments in order to show the unconditionally convergent behavior of these schemes as well as their computational advantages., B. Bujanda had financial support from VOLTEADOR 0011-1365-2018-000178, PERISEIS 4.0 PC054-055 and IoTrain RTI2018-095499-B-C31. M. J. Moreta had financial support from MTM 2015-66837-P. and PGC2018-101443-B-I00 J. C. Jorge had financial support from MTM2014-52859-P and MTM2017-83490-P.

#### Zernike-like systems in polygons and polygonal facets

##### Digital.CSIC. Repositorio Institucional del CSIC
• Ferreira, Chelo
• López, José L.
• Navarro, Rafael
• Pérez Sinusía, Ester
arXiv:1506.07396v1, Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Opt. Lett. 32, 74 (2007)] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both the general form and the explicit expressions for a typical example of telescope optical aperture are provided., European Union (FIS2011-22496); Government of Aragón (E99); Spanish Ministry of Economía y Competitividad (FIS2014-58303), (MTM2014-52859); State University of Navarra., Peer Reviewed

#### Orthogonal basis for the optical transfer function

##### Digital.CSIC. Repositorio Institucional del CSIC
• Ferreira, Chelo
• López, José L.
• Navarro, Rafael
• Pérez Sinusía, Ester
We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 = OTFperfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed., Ministerio de Economía y Competitividad (MINECO) (MTM2014-52859-P, FIS2014-58303-P)., Peer Reviewed

#### Orthogonal basis with a conicoid first mode for shape specification of optical surfaces

##### Digital.CSIC. Repositorio Institucional del CSIC
• Ferreira, Chelo
• López, José L.
• Navarro, Rafael
• Pérez Sinusía, Ester
Open Access., A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed., This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union MTM2014-52859 and FIS2014-58303., Peer Reviewed

#### Orthogonal basis for the optical transfer function

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza
• Ferreira, C.
• López, J.L.
• Navarro, R.
• Sinusa, E.P.
We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 = OTFperfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed.

#### The use of two-point Taylor expansions in singular one-dimensional boundary value problems I

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza
• Ferreira, C.
• López, J.L.
• Pérez Sinusía, E.
We consider the second-order linear differential equation (x+1)y¿+f(x)y'+g(x)y=h(x) in the interval (-1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x=±1 containing the interval [-1, 1]. Then, the end point of the interval x=-1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.

#### An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza
• Clavero, C.
• Jorge, J. C.
In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.