Explicit Runge-Kutta methods for parabolic partial differential equations

Numerical methods for parabolic PDEs have been studied for many years. A great deal of the research focuses on the stability problem in the time integration of the systems of ODEs which result from the spatial discretization. These systems often are stiff and highly expensive to solve due to a huge number of components, in particular for multi-space dimensional problems. The combination of stiffness and problem size has led to an interesting variety of special purpose time integration methods. In this paper we review such a class of methods, viz. explicit Runge-Kutta methods possessing extended real stability intervals.