On the practical importance of the SSP property for Runge-Kutta time integrators for some common Godunov-type schemes

We investigate through analysis and computational experiment explicit second and third-order strong-stability preserving (SSP) Runge-Kutta time discretization methods in order to gain perspective on the practical necessity of the SSP property. We consider general theoretical SSP limits for these schemes and present a new optimal third-order low-storage SSP method that is SSP at a CFL number of 0.838. We compare results of practical preservation of the TVD property using SSP and non-SSP time integrators to integrate a class of semi-discrete Godunov-type spatial discretizations. Our examples involve numerical solutions to Burgers' equation and the Euler equations. We observe that 'well-designed' non-SSP and non-optimal SSP schemes with SSP coefficients less than one provide comparable stability when used with time steps below the standard CFL limit. Results using a third-order non-TVD CWENO scheme are also presented. We verify that the documented SSP methods with the number of stages greater than the order provide a useful enhanced stability region. We show by analysis and by numerical experiment that the non-oscillatory third-order reconstructions used in (Liu and Tadmor Numer. Math. 1998; 79:397-425, Kurganov and Petrova Numer. Math. 2001; 88:683-729) are in general only second-and first-order accurate, respectively.