Approximate Taylor methods for ODEs

Abstract A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge–Kutta schemes. Compared to Runge–Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansio…

Discrete multiresolution based on hermite interpolation: computing derivatives

Abstract Harten’s framework for multiresolution representation of data has been extended by Warming and Beam in [SIAM J. Sci. Comp. 22 (2000) 1269] to include Hermite interpolation. It needs the point-values of the derivative, which are usually unavailable, so they have to be approximated. In this work we show that the way in which the derivatives are approximated is crucial for the success of the method, and we present a new way to compute them that makes the scheme adequate for non-smooth data.

TRITIUM - A Real-Time Tritium Monitor System for Water Quality Surveillance

In this work the development results of the TRITIUM project is presented. The main objective of the project is the construction of a near real-time monitor for low activity tritium in water, aimed at in-situ surveillance and radiological protection of river water in the vicinity of nuclear power plants. The European Council Directive 2013/51/Euratom requires that the maximum level of tritium in water for human consumption to be lower than 100 Bq/L. Tritium levels in the cooling water of nuclear power plants in normal operation are much higher than the levels caused by the natural and cosmogenic components, and may easily surmount the limit required by the Directive. The current liquid-scint…

High Order Extrapolation Techniques for WENO Finite-Difference Schemes Applied to NACA Airfoil Profiles

Finite-difference WENO schemes are capable of approximating accurately and efficiently weak solutions of hyperbolic conservation laws. In this context high order numerical boundary conditions have been proven to increase significantly the resolution of the numerical solutions. In this paper a finite-difference WENO scheme is combined with a high order boundary extrapolation technique at ghost cells to solve problems involving NACA airfoil profiles. The results obtained are comparable with those obtained through other techniques involving unstructured meshes.

A numerical treatment of wet/dry zones in well-balanced hybrid schemes for shallow water flow

The flux-limiting technology that leads to hybrid, high resolution shock capturing schemes for homogeneous conservation laws has been successfully adapted to the non-homogeneous case by the second and third authors. In dealing with balance laws, a key issue is that of well-balancing, which can be achieved in a rather systematic way by considering the 'homogeneous form' of the balance law.The application of these techniques to the shallow water system requires also an appropriate numerical treatment for the wetting/drying interfaces that appear initially or as a result of the flow evolution. In this paper we propose a numerical treatment for wet/dry interfaces that is specifically designed f…

Mathematical Methods in Image Processing and Computer Vision

Image processing and computer vision are growing research fields that take advantage of the increasing power or modern computers linked with sophisticated techniques coming from many fields of expertise and in particular from mathematics. We present an introduction to some problems in computer vision and image processing and to some mathematical techniques and concepts that are nowadays routinely used to approach them.

Highly Accurate Conservative Finite Difference Schemes and Adaptive Mesh Refinement Techniques for Hyperbolic Systems of Conservation Laws

We review a conservative finite difference shock capturing scheme that has been used by our research team over the last years for the numerical simulations of complex flows [3, 6]. This scheme is based on Shu and Osher’s technique [9] for the design of highly accurate finite difference schemes obtained by flux reconstruction procedures (ENO, WENO) on Cartesian meshes and Donat-Marquina’s flux splitting [4]. We then motivate the need for mesh adaptivity to tackle realistic hydrodynamic simulations on two and three dimensions and describe some details of our Adaptive Mesh Refinement (AMR) ([2, 7]) implementation of the former finite difference scheme [1]. We finish the work with some numerica…

Cell-Average Multiwavelets Based on Hermite Interpolation

Isotopic Effects and Surface Absorption in $^{35-37}$Cl+$^{24}$Mg Interactions

Abstract The few-nucleon transfer is found to play an important role in the isotopic effects observed in absorption. This conclusion is obtained by measuring elastic scattering and quasielastic reactions and by analysing elastic data with both phenomenological and microscopic models. The sensitivity domain is found to be different for imaginary and real potentials. The implication for the validity of the dispersion relation for phenomenological potentials at the real sensitivity radius, when transfers are important, is discussed.

On a new centered strategy to control the accuracy of weighted essentially non oscillatory algorithm for conservation laws close to discontinuities

Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations

The numerical simulation of physical phenomena represented by non-linear hyperbolic systems of conservation laws presents specific difficulties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming difficulties. In this work we present a …

High Order in Space and Time Schemes Through an Approximate Lax-Wendroff Procedure

This paper deals with the scheme proposed by the authors in Zorio, Baeza and Mulet (J Sci Comput 71(1):246–273, 2017). This scheme is an alternative to the techniques proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185–2198, 2003) to obtain high-order accurate schemes using Weighted Essentially Non Oscillatory finite differences and approximating the flux derivatives required by the Cauchy-Kovalevskaya procedure by simple centered finite differences. We analyse how errors in first-order terms near discontinuities propagate through both versions of the Cauchy-Kovalevskaya procedure. We propose a fluctuation control, for which the approximation of the first-order derivative to be used in th…

Weighted Extrapolation Techniques for Finite Difference Methods on Complex Domains with Cartesian Meshes

The design of numerical boundary conditions in high order schemes is a challenging problem that has been tackled in different ways depending on the nature of the problem and the scheme used to solve it numerically. In this paper we propose a technique to extrapolate the information from the computational domain to ghost cells for schemes with structured Cartesian Meshes on complex domains. This technique is based on the application of Lagrange interpolation with weighted filters for the detection of discontinuities that permits a data dependent extrapolation, with high order at smooth regions and essentially non oscillatory properties near discontinuities. This paper is a sequel of Baeza et…

Adaptation based on interpolation errors for high order mesh refinement methods applied to conservation laws

Adaptive mesh refinement is nowadays a widely used tool in the numerical solution of hyperbolic partial differential equations. The algorithm is based on the numerical approximation of the solution of the equations on a hierarchical set of meshes with different resolutions. Among the different parts that compose an adaptive mesh refinement algorithm, the decision of which level of resolution is adequate for each part of the domain, i.e., the design of a refinement criterion, is crucial for the performance of the algorithm. In this work we analyze a refinement strategy based on interpolation errors, as a building block of a high order adaptive mesh refinement algorithm. We show that this tec…

Reprint of: Approximate Taylor methods for ODEs

Abstract A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge–Kutta schemes. Compared to Runge–Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansio…

Natural and artificial radioactivity levels in Livingston Island (Antarctic regions).

Radioactive contamination of the sea and land is due, on the one hand, to fallout from atmospheric atomic explosions since 1945, and, on the other, to emissions produced by nuclear and radioactive facilities. Given its geographic position far distant from the aforementioned main sources of radioactive contamination, Antarctica should have the lowest levels that can be measured on the Earth of artificial radionuclides in the various receptor media which are characteristic of the trophic chain. In the case of Antarctica, these are melt-water, sea-water, mosses, algae, and lichens. With the aim of contributing basic information on the radiation levels present in the Antarctic ecosystem, we hav…

Monotone cubic spline interpolation for functions with a strong gradient

Abstract Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. Hence, some additional techniques have to be incorporated so as to ensure monotonicity. The final interpolator is not actually a spline as C 2 regularity and monotonicity are not ensured at the same time. In this paper, we study sufficient conditions to obtain monot…

Short and medium effects on the environment of Valencia, Spain, of the Chernobyl nuclear plant accident.

As a consequence of the 26 April 1986 accident at the Chernobyl nuclear plant, a large amount of radioactivity was released into the atmosphere. The radioactive plume formed could be detected in practically the whole of the Northern Hemisphere a few days later. The zone most affected by the radioactive cloud over Spain was that of the Mediterranean coast and the Balearic Islands. In this paper, the authors examine the level of the radioactive contamination reached in various receptive media in Valencia, such as air, dry-fallout, water, soil, grass and milk samples collected in Valencia immediately after the accident. The activity levels are compared with those found during 1964 and 1965 due…

Elastic scattering of 35Cl and 37Cl on 24Mg

Abstract Elastic scattering of 35Cl and 37Cl on 24Mg was measured at two c.m. energies. Optical model analysis with different potentials are compared. Isotopics effects on absorption are observed. The closure approximation model is found to give a good reproduction of experimental data.

Recent evolution of the multi-isotopic radioactive content in ice of Livingston Island, Antarctica.

The temporal arrangement of the ice layers that are produced in ecosystems with perpetual snows form situations that greatly favour the study of the temporal evolution of the radioactive fallout that occurs in the said zones, whether this fallout is natural or artificial in origin. This allows one to investigate the causes of the fallout and the mechanisms transporting the radionuclides involved from their source point to the study zone, as well as their subsequent behaviour in that zone. There are special difficulties involved in this type of study in Antarctica. Some are of a general character deriving from the conditions of extreme climate and isolation which complicate the processes of …