Difference methods

Finite difference methods play one of the most important roles in obtaining quantitative results about plasma behavior. Application of such methods requires careful treatment of many properties of the original differential problem.

For example, at development and application of numerical for the kinetic equations with Coulomb collisional operator methods it is necessary to take into account the following singularities of differential problems:

  • Problems are formulated in unlimited area in space of velocities.
  • Coefficients of the operator of Coulomb collisions behave essentially differently in various points of the phase space and aspire to zero at infinite velocity, and with a different velocity.
  • Use of a curvilinear coordinate system reduces in emerging singular points, in which coefficients in differential operators are degenerated.
  • Owing to application of an averaging method, inside area separatrix layers are available, on which additional conditions are put.
  • In a number of statements the problem is nonlinear and integro-differential. The collisional coefficients in the kinetic equation may depend on a solution integrally.
  • Some operators in the equation are not selfajoint, of fixed sign and do not satisfy to a principle of maximum.
  • The Coefficients of the equation can sharply vary at passage through the boundary between passing and trapped particles.
  • Coefficients at mixed derivatives can be rather large.
  • The dependence of distribution function on velocity has an exponential character and can vary on several orders.

Specific difficulties are met in numerical modeling of plasma evolution based on the Ohm law:

  • Loss of accuracy due to summation of large and small quantities.
  • Close to a constant to the 3rd to 5th digits functions with a strongly varying derivatives.
  • Very localized in space externally driven currents.
  • Stiff behavior of transport coefficients in time and space.

The enumerated singularities should be discounted at passage to discrete problems and in numerical methods of its solution. Many details are given in the book “F.S. Zaitsev. Mathematical modelling of toroidal plasma evolution. - Moscow: MAX Press Publishing Co., 2005, 524 p. (in Russian).”