If we employ the Method of Lines for discretization (separate time and space discretization) of hyperbolic PDEs we obtain after spatial discretization by our favorite numerical method (fx. Finite Volume Method) does it matter in practice which ODE solver we employ for the temporal discretization (TVD/SSP/etc)?
Some additional information added: The accuracy issue can be a problem for non-smooth problems. It is will-known that nonlinear hyperbolic PDEs can develop shocks in finite time despite the initial solution is smooth in which case accuracy can degrade to first order for high-order methods.
ODE Stability analysis is typically done based on linearization to obtain a linear semi-discrete system of ODEs of the form q_t = J q (with q a perturbation vector), where the eigenvalues of J should be scaled inside the absolute stability region of chosen time-stepping method. Alternative strategies is to use pseudospectra or possibly an energy method for stability analysis.
I understand that the motivation for TVD/SSP methods are to avoid spurious oscillations caused by the time-stepping methods which may result in unphysical behavior. Question is if experiences shows these types of time-stepping methods to be superior compared to, e.g., a classical work horse as explicit Runge-Kutta Method or others. Obviously, they should have better properties for classes of problem where solution may exhibit shocks. One could therefore argue that we should only employ these types of methods for time-integration.