[lzhaok6]

I'm solving a 1D linear acoustic wave equation using Finite Element Method for spatial discretization (linear element) and second order Finite Difference Method (explicit Newmark scheme) for temporal discretization. Specifically, I want to propagate an exponential decay shock wave in a 1D shock tube (use boundary condition to imposing the wave profile). The governing equation and boundary condition are attached.

As we know, the dispersion is an inherent property of FEM when solving wave propagation problem and using CFL number equals to one seems to give the best solution since the numerical wave speed then equals the physical wave speed.

My question is:
How to examine the "observed order of accuracy" of temporal discretization in this problem? When I used CFL=1, there seems to be no dispersion in the solution and the wave profile after a certain time of propagation agrees well with the theoretical profile. Then, I tried to fix the mesh size and decrease the time step, which means CFL number is reduced and smaller than 1. However, right after the CFL number drops below 1, I start to see spurious oscillation right after the wave front which increases the overall error in the solution. Thus, the accuracy is even degraded when the time step size is decreased. As a result, I cannot retrieve the theoretical order of accuracy of time discretization.

I saw people recommend to keep CFL number fixed while investigating the time integration accuracy, which means mesh size is decreased as time step is decreased. Is this a reasonable way to examine the temporal order of accuracy? Did I misunderstand something?