METHOD TO SOLVE THIRD ORDER POLYNOMIALS
                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 explicitmethods has small stability domains with a step size limited by a stabilitycondition. This restriction makes explicit methods useless for stiffequations. We propose explicit embedded integration schemes with largestability domains.Firstly, we compute stability polynomials of a given order with optimalstabilitydomains, i.e. possessing a Chebyshev alternation; secondly, we construct acorresponding explicit Runge-Kutta method using the theory ofRunge-Kutta composition methods.   Stability domain of the method increases as a square of the degree of theoptimal stability polynomial. To construct the stability polynomial oflarge degrees, we use an asymptotic formula for polynomials of the leastdeviationfrom zero with a weight function. This gives us a very stable procedure forlarge degrees.   For example, our computer program with a third order explicit Runge-Kuttamethoduses polynomials of degree between 3 and 432. That provides stablecomputations 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 thedegree of the polynomial used  Large stability domains allow some reasonable stiffness; theexplicitness makes possible to solve very large problems, e.g., spacediscretization of parabolic PDE's. The high order produces accurate results andthe embedded formulas permit an efficient stepsize control.