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METHOD TO SOLVE THIRD ORDER POLYNOMIALS
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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 RungeKutta 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 RungeKutta method using the theory ofRungeKutta 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 RungeKuttamethoduses polynomials of degree between 3 and 432. That provides stablecomputations approximately 432 times faster than the explicit Euler method.An explicit RungeKutta 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.