UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 626 UCD Building, 1250 14th St., Denver

TIME: 2 pm (Refreshments served at 1:45 pm)

DATE: April 5, 1999

                Alexei Medovikov, Russian Academy of Sciences 

        High order explicit methods for stiff ordinary differential equations.

   We discuss explicit Runge-Kutta methods. Most of well known explicit
methods has small stability domains with a step size limited by a stability
condition. This restriction makes explicit methods useless for stiff
equations. We propose explicit embedded integration schemes with large
stability domains.
Firstly, we compute stability polynomials of a given order with optimal
stability
domains, i.e. possessing a Chebyshev alternation; secondly, we construct a
corresponding explicit Runge-Kutta method using the theory of
Runge-Kutta composition methods.

   Stability domain of the method increases as a square of the degree of the
optimal stability polynomial. To construct the stability polynomial of
large degrees, we use an asymptotic formula for polynomials of the least
deviation
from zero with a weight function. This gives us a very stable procedure for
large degrees.

   For example, our computer program with a third order explicit Runge-Kutta
method
uses polynomials of degree between 3 and 432. That provides stable
computations approximately 432 times faster than the explicit Euler method.
An explicit Runge-Kutta method of order $p$ can be  accellarated up to
$\beta_p*s$ times, where

$\beta_1=2$, $\beta_2=0.81$, $\beta_3=0.49$ and $\beta_4=0.35$ and $s$ is the
degree of the polynomial used

  Large stability domains allow some reasonable stiffness; the
explicitness makes possible to solve very large problems, e.g., space
discretization of parabolic PDE's. The high order produces accurate results and
the embedded formulas permit an efficient stepsize control.