Scientific Computing Seminar
Abstract: We discuss explicit Runge-Kutta methods. Most of well known explicit methods has small stability domain with step size is bounded by stability condition. This restriction makes explicit methods useless for stiff equations. We propose explicit embedded integration schemes with large stability domains. The construction of these methods is achieved in two steps: firstly we compute stability polynomials of given order with optimal stability domains, i.e., possessing a Chebyshev alternation; secondly we realize a corresponding explicit Runge-Kutta method with the help of the theory of Runge-Kutta composition methods.
Stability domain of the method increases as square of the degree of the optimal stability polynomial. To construct the stability polynomial of the large degree we use asymptotic formula for polynomial of the least deviation from zero with weight function. This is very stable procedure of computation of roots of large degree stability polynomial.
For example, our computer program with 3 order explicit Runge-Kutta method uses degree 's' of the polynomial between 3 and 432. That can perform stable computations approximately 432 times faster than explicit Euler method. Actually, sped up of the method order 'p' can be up to bp*s, where b1=2, b2=0.81, b3=0.49, and b4=0.35.
The large stability domains allow some reasonable stiffness, the explicitness enables the method to treat very large problems, often space discretiztion of parabolic PDEs. The high order produces accurate results and the embedded formulas permit an efficient stepsize control.
PDE Seminar
<--- 1999 Index