Lie point symmetries of difference equations and lattices

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations

Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.Comment: Minor changes and journal-re

Oscillations and stability of numerical solutions of the heat conduction equation

The mathematical model and results of numerical solutions are given for the one dimensional problem when the linear equations are written in a rectangular coordinate system. All the computations are easily realizable for two and three dimensional problems when the equations are written in any coordinate system. Explicit and implicit schemes are shown in tabular form for stability and oscillations criteria; the initial temperature distribution is considered uniform

Difference schemes with point symmetries and their numerical tests

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure

Symmetries of differential-difference dynamical systems in a two-dimensional lattice

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in \Gamma}\big)$ are considered according to their Lie point symmetry groups. The set $\Gamma$ represents the point $(n,m)$ and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12-dimensional for abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.Comment: 24 pages, 1 figur

Marginal cost-based pricing of distribution: a case study

This paper presents results of a software development project carried out by the “Electricity North West” (ENW) and “TNEI” to find economic use-of-system charges for the extra high-voltage (EHV) network. Several cost-based charging models which satisfy principles set by the Regulator, such as cost reflectivity, predictability, stability and transparency were developed. In this paper, the emphasis is put on the developed software and the comparison of nodal marginal charges obtained from the proposed pricing models

QCDF90: Lattice QCD with Fortran 90

We have used Fortran 90 to implement lattice QCD. We have designed a set of machine independent modules that define fields (gauge, fermions, scalars, etc...) and overloaded operators for all possible operations between fields, matrices and numbers. With these modules it is very simple to write high-level efficient programs for QCD simulations. To increase performances our modules also implements assignments that do not require temporaries, and a machine independent precision definition. We have also created a useful compression procedure for storing the lattice configurations, and a parallel implementation of the random generators. We have widely tested our program and modules on several parallel and single processor supercomputers obtaining excellent performances.Comment: LaTeX file, 8 pages, no figures. More information available at: http://hep.bu.edu/~leviar/qcdf90.htm

The Taming of QCD by Fortran 90

We implement lattice QCD using the Fortran 90 language. We have designed machine independent modules that define fields (gauge, fermions, scalars, etc...) and have defined overloaded operators for all possible operations between fields, matrices and numbers. With these modules it is very simple to write QCD programs. We have also created a useful compression standard for storing the lattice configurations, a parallel implementation of the random generators, an assignment that does not require temporaries, and a machine independent precision definition. We have tested our program on parallel and single processor supercomputers obtaining excellent performances.Comment: Talk presented at LATTICE96 (algorithms) 3 pages, no figures, LATEX file with ESPCRC2 style. More information available at: http://hep.bu.edu/~leviar/qcdf90.htm