High order explicit methods for parabolic equations

This paper discusses explicit embedded integration methods with large stability domains of order 3 and 4. The high order produces accurate results, the large stability domains allow some reasonable stiffness, the explicitness enables the method to treat very large problems, often space discretization of parabolic PDEs, and the embedded formulas permit an efficient stepsize control. The construction of these methods is achieved in two steps: firstly me compute stability polynomials of a 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 composition methods.