Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems

We analyze explicit Runge-Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs type. For the time discretization, we consider explicit second- and thirdorder Runge-Kutta schemes. We identify a general set of properties on the space stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then we establish L2-norm error estimates with quasi-optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge-Kutta schemes and any polynomial degree in space and for second-order Runge- Kutta schemes and first-order polynomials in space. For second-order Runge-Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for smooth and rough solutions. The case of finite volumes is briefly discussed.