  Strong stability preserving (SSP) Runge-Kutta methods are often desired when
evolving in time problems that have two components that have very different
time scales. Where the SSP property is needed, it has been shown that implicit
and implicit-explicit methods have very restrictive time-steps and are
therefore not efficient. For this reason, SSP integrating factor methods may
offer an attractive alternative to traditional time-stepping methods for
problems with a linear component that is stiff and a nonlinear component that
is not. However, the strong stability properties of integrating factor
Runge-Kutta methods have not been established. In this work we show that it is
possible to define explicit integrating factor Runge-Kutta methods that
preserve the desired strong stability properties satisfied by each of the two
components when coupled with forward Euler time-stepping, or even given weaker
conditions. We define sufficient conditions for an explicit integrating factor
Runge--Kutta method to be SSP, namely that they are based on explicit SSP
Runge--Kutta methods with non-decreasing abscissas. We find such methods of up
to fourth order and up to ten stages, analyze their SSP coefficients, and prove
their optimality in a few cases. We test these methods to demonstrate their
convergence and to show that the SSP time-step predicted by the theory is
generally sharp, and that the non-decreasing abscissa condition is needed in
our test cases. Finally, we show that on typical total variation diminishing
linear and nonlinear test-cases our new explicit SSP integrating factor
Runge-Kutta methods out-perform the corresponding explicit SSP Runge-Kutta
methods, implicit-explicit SSP Runge--Kutta methods, and some well-known
exponential time differencing methods.
