Numerical Problems of Mathematical Physics

Organizers

• V.G. Ganzha (ganzha@hrz.uni-kassel.de)
  University of Kassel
  Kassel D-34127
  Germany
• E.V. Vorozhtsov (adm@itam.nsk.su)
  Institute of Theoretical and Applied Mechanics
  Russian Academy of Sciences
  Novosibirsk 630090
  Russia

Description

The computer algebra systems (CAS's) were previously developed mainly for ensuring a possibility for symbolic computations on a computer. However, the powerful CAS's developed in recent years have also a big set of built-in functions for high-precision numerical computations. In comparison with the conventional languages for numerical computations, such as FORTRAN and C, the CAS's have considerable advantages, since they enable one to perform numerical computations, necessary analytic investigations, and to visualize the obtained results within the framework of the same computer algebra system.

The main purpose of the proposed session is the demonstration of the capabilities of advanced CAS when solving numerically the mathematics physics problems, in particular, the problems governed by partial differential equations.

Talks

• The Use of Mathematica for Computation of Stability Regions with Guaranteed Accuracy
  Victor G. Ganzha K. Schaub and Evgenii V. Vorozhtsov
  Abstract: We present a symbolic-numerical algorithm for the stability investigation of difference schemes approximating the partial differential equations of hyperbolic or parabolic type. The use of a combination of rational arithmetic and the arithmetic of floating-point numbers available in the Mathematica system has enabled us to compute the stability region boundaries with guaranteed accuracy. Computational examples are given, in which we analyze a number of difference schemes for one- and two-dimensional partial differential equations.

• Curvilinear Grid Topology Effect on the Stability of Difference Schemes
  Victor G. Ganzha and Evgenii V. Vorozhtsov
  Abstract: A general vector-matrix algorithm has been proposed and implemented with Mathematica for obtaining the characteristic equation of a difference scheme on curvilinear grids approximating the Euler equations. All the known types (C,O,H) of curvilinear grids around the airfolis have been considered and it is shown how the grid topology affects the stability of difference schemes.
• Rapid Prototyping for the Construction of Higher Order Finite Element Methods on Sparse Grids
  Hans-Joachim Bungartz
  Abstract: For the development of new algorithmic concepts in the area of numerical analysis or, more generally, scientific computing, modern tools for an efficient rapid prototyping like computer algebra programs like Maple or Mathematica and shell-script-type interpreter languages like Perl gain more and more in importance. This is mainly due to the fact that the period of time from an idea to a first programmed version of the corresponding algorithm can be cut down significantly, since all the implementation and declaration overhead coming along with standard (numerical) programming languages can be avoided. As an example for that, we present a new unidirectional approach for $d$-dimensional finite element methods of higher order on sparse grids that enables us to deal with problems of an arbitrary number $d$ of dimensions and to use polynomial bases of an arbitrary degree $p$ with the same storage requirements as and only a little bit more of computational work than in the usual piecewise linear case. The code consists of two parts: a kind of setup phase which is done in Maple and the iterative solver programmed in Perl. We report on both numerical and implementational experiences and on interface problems resulting from the use of tools originating from different ``worlds''.

• Symbolic Computation of Linear and Nonlinear Modified (Partial Differential) Equations.
  Jean-Antoine Désidéri and Margarita Spiridonova
  Abstract: The so-called ``equivalent'' (Lerat, Peyret, 1974) or ``modified'' (Warming, Hyett, 1974) partial differential equation is a very useful, classical and basic tool of analysis of finite-difference schemes for time-dependent problems, mostly but not only, of hyperbolic type. Although limited in application to simple (usually scalar) model equations, it allows in particular a rigorous analysis of the truncation error, in which dispersive and diffusive errors are identified. Thus new schemes are systematically compared with well-known ones by this analysis applied to a test equation (often the wave or heat equation). Furthermore, the approach can be used constructively to build new schemes in which particular types of error are absent (to a certain order of accuracy).

  Originally, Warming and Hyett developed codes of symbolic computation of modified equations using, to our knowledge, the language MACSYMA. We have developed a collection of programs of the same nature for symbolic derivation of the modified equation associated with a given finite-difference equation defined in a rather general format using the computer algebra system MAPLE.

  This contribution intends to report on the implementation of these programs and their experimentation related to various finite-difference schemes. For the case of linear (hyperbolic or parabolic) partial differential equations, we have examined classical schemes and less classical examples in which equations in one or two space dimensions including a source term are approximated by upwind schemes. For the case of nonlinear equations, the MacCormack scheme applied to Burgers equation was studied as well as Runge-Kutta-type methods applied to general hyperbolic equations.