N. Mohankumar

On the evaluation of H and X functions

Abstract We indicate a Double Exponential Formula based numerical integration method for the evaluation of the Ambarzumian–Chandrasekhar H function and the X function of neutron transport for the single speed and isotropic case. This method is significantly more economical than our earlier scheme, which was based on IMT quadrature. For c 5 , the present method converges faster than our earlier IMT scheme. This will be adequate for all radiative transport and transport theory applications. These findings are supported by appropriate error analysis. Unlike the IMT method, the DE quadrature nodes are generated by a simple algebraic expression which is a great advantage.

An accurate method for the generalized Fermi-Dirac integral

Abstract We present an accurate quadrature method for the evaluation of the generalized Fermi–Dirac integral, which is based on splitting the range of integration, Gauss–Legendre integration in the split intervals and correction for the poles of the integration in the interval containing the real part of the poles of the integrand. The present method will work for any values of the parameters k , θ and η of the integrand.

Two new series for the Fermi–Dirac integral

Abstract We derive two new series expressions for the Fermi–Dirac integral F j ( x ) based on the Mittag–Leffler expansion of sech ( z ) . The first series is exact for all values of x with j > ( − 1 ) . The second series is exact for all values of x and for half-integral values of j ⩾ ( 1 / 2 ) . These expressions will be of use while setting up approximate short expressions for the integral. They also will be useful to interpret the results of Fermi–Dirac distribution in physical phenomena.

On the evaluation of Fermi–Dirac integral and its derivatives by IMT and DE quadrature methods

Abstract The numerical values of the derivatives of the Fermi–Dirac integral, up to the third order, with respect to its parameters are evaluated by both the IMT and the Double Exponential integration methods. These integrals find extensive use in astrophysical problems. It is found that both the IMT and the DE methods are very efficient, especially for handling the end point integrable singularities. Also, the DE method is slightly superior to the IMT method. Unlike the IMT or Gauss methods, the nodes of DE method can be evaluated in terms of built in elementary functions, which is a big advantage.

A note on the numerical evaluation of the H function

Abstract Exploiting the relation between the Ambarzumian-Chandrasekhar H function and the X function of the isotropic and single speed neutron transport, an accurate numerical method is presented to evaluate these X and H functions. The method uses the IMT quadrature scheme. The theoretical error estimate is compared with actual errors.

Milne, a routine for the numerical solution of Milne's problem

The routine Milne provides accurate numerical values for the classical Milnes problem of neutron transport for the planar one speed and isotropic scattering case. The solution is based on the Case eigen-function formalism. The relevant X functions are evaluated accurately by the Double Exponential quadrature. The calculated quantities are the extrapolation distance and the scalar and the angular fluxes. Also, the H function needed in astrophysical calculations is evaluated as a byproduct.

A new approach for determining the time step when propagating with the Lanczos algorithm

A new criterion for choosing the time step used when numerically solving time-dependent Schroedinger equation with the Lanczos method is presented. Following Saad, Stewart and Leyk, an explicit expression for the time step is obtained from the remainder of the Chebyshev series of the matrix exponential.

On time-step bounds in unitary quantum evolution using the Lanczos method

To solve the non-relativistic time dependent Schrodinger equation using the Lanczos method, Park and Light have provided an approximate expression for the time step for a given accuracy. We provide an exact expression for the time step in terms of the eigenvalues and eigenvectors of the resulting tridiagonal matrix. For two test problems, the values of the time step provided by Park and Light differ significantly from the exact values provided by the present method. We also indicate upper and lower bounds for the time step in terms of the maximum and minimum eigenvalues. These bounds indicate the possibility of using a new time step given by the geometric mean of the eigenvalues of the tridiagonal matrix.

The accurate evaluation of a particular Fermi-Dirac integral

Abstract In this paper, we provide a simple and very accurate numerical method to evaluate a particular two-dimensional Fermi-Dirac integral that arises in the plasma transport theory. The proposed method involves suitable coordinate and variable transformations which eliminate variable limits of integration. The resulting inner integral is evaluated by a trapezoidal integration with pole correction. The outer integral is evaluated by the IMT scheme. We provide reasonable error bounds. The results agree very well with the results of Fullerton and Rinker, who employ a combination of quadrature, asymptotic methods and Chebyshev fitting.

Migpore, a code package for the estimation of migration of radioactive species in a porous medium☆

Abstract In the event of an accidental leakage of high level radioactive waste buried deep in repositories surrounded by rock, the build up of the concentration of the radioactive species within the rock needs to be assessed. Towards this, we follow the model of Chen and Li and provide a numerical code to solve the relevant partial differential equations using a compact finite difference scheme. Program summary Program title: Migpore Catalogue identifier: AEQI_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEQI_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5396 No. of bytes in distributed program, including test data, etc.: 26483 Distribution format: tar.gz Programming language: Fortran 77. Computer: PC under Linux or Windows. Operating system: Ubuntu 10.04 (Kernel version 2.6.32-47-generic), Windows-XP. Classification: 4.3, 4.12, 13. External routines: We included the following Lapack subroutines in migpor.for program.: DCOPY, DGBTF2, DGBTRF, DGBTRS, DGEMM, DGEMV, DGER, DLASWP, DCAL, DSWAP, DTBSV, DTRSM, IDAMAX, IEEECK, ILAENV, IPARMQ, LSAME, XERBLA. Nature of problem: We consider the case of an accidental leakage of high level radioactive waste into the surrounding rock (a porous medium). Solution method: We follow the model of Chen and Li to estimate the buildup of radioactivity due to migration from the source. This amounts to the solution of two coupled partial differential equations. The numerical algorithm that we provide solves these equations by a higher order compact finite difference scheme. Running time: The Migpore code package needs approximately 50 min to run the test inputs. For other cases the run time will depend on the species, the distance and time step.