Mathematical problems

Mathematical abstractions of toroidal plasmas are based on equations in partial derivatives, integral equations, ordinary differential equations and many appropriate numerical approximations.

Formulations of mathematical problems include initial, boundary, mixed initial-boundary, free boundary, eigenvalue, direct, inverse problems, problems with non-local conditions and etc. Most of the problems are non-linear. Many formulations are relatively new to traditional mathematical theory and require new analytical estimations and numerical methods for their solution.

An important direction of fusion research is development of high quality software for numerical modeling. Several specific to fusion features exist in software production and usage:

  • Time required for development of a code suitable for adequate plasma modeling is measured with years.
  • Codes are characterized with large volume of programs, implementation of high-end science theory, methods and technologies, high dynamics of software evolution.
  • Even well-developed codes require permanent modernization due to inclusion of new physical effects, implementation of new methods, increasing generality, accuracy and etc.
  • Multi-model and multi-variant programming and computations.
  • Absence of well-defined data structures and programming standards.
  • Extended computer resources: CPU, RAM, HD.
  • In many cases valuable software becomes lost when development personnel changes.

All this emphasizes the importance and urgency of implementation of the project described in the home page of this site.

Examples of software for solutions of problems in fusion are given on page "Software".