Stability polynomials for explicit Runge–Kutta methods: optimal and damped

Explicit methods for numerical integration of ordinary differential equations are very appealing from the viewpoint of computations: they are cost efficient and ideally suit for massive parallel computers. As usual, one has to pay for the advantages: stability properties of explicit methods are rather poor and the Courant time steps which guarantee the stability may become terribly small especially for the ODEs obtained from the spatial discretization of PDEs. Possible solution is to make variable time steps which brings us to a series of nonclassical optimization problems for polynomials.