I recently coded a 1 stage and 3 stage optimal TVD-RK explicit scheme using eqn 3.3 here
on the equation Ux+Uy=0, where x and y are the independent variables with which U velocity is differentiated. It is a 1D problem. The space is discretized using the upwind scheme.
I am running it for 100 mesh size, and a constant declared CFL of 1. I see that as the solution progresses to t=1s for TVD-3, the solution disperses in both value and shape very fast, while for TVD-1 it gives perfect shift. My question is-is this supposed to happen? I do not think changing the time accuracy to 3 stages should necessarily affect the shape/values of the function to such a huge degree.
If I not, then I think I do not understand the implementation of TVD. I did it exactly as described in the paper. Except I am not sure if I should have CFL different for each stage, or if the CFL should be multiplied by (1/(2p+1)) where p is the order of the polynomial. Can you guys help me?