  We apply the concept of effective order to strong stability preserving (SSP)
explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods,
methods with an effective order of accuracy are designed to satisfy a relaxed
set of order conditions, but yield higher order accuracy when composed with
special starting and stopping methods. We show that this allows the
construction of four-stage SSP methods with effective order four (such methods
cannot have classical order four). However, we also prove that effective order
five methods - like classical order five methods - require the use of
non-positive weights and so cannot be SSP. By numerical optimization, we
construct explicit SSP Runge-Kutta methods up to effective order four and
establish the optimality of many of them. Numerical experiments demonstrate the
validity of these methods in practice.
