Stabilized explicit Runge-Kutta methods are methods with large stability domains along the real-negative axis. They are of great importance for large stiff problems, originating mainly from parabolic equations. 1) We will show, using the theory of order stars, that the optimal stability polynomial of degree s and order p, possesses exactly p complex roots, if p is even, and p-1 complex roots if p is odd. Furthermore, the error constant is always positive, and for a given order, it decreases as the degree increases. 2) These polynomials can be approximated by orthogonal polynomials, whose recurrence relations allow us to construct a new type of stabilized explicit Runge-Kutta methods. These can be seen as a combination of Van der Houwen-Sommeijer-type methods and Lebedev-type methods.

MINISIMPOSIUM SESSION: 2. Numerical methods for Partial Differential Equations