Analytical solutions for the damped oscillations of a liquid in a parabolic container have been derived by Sampson et al [25, 14]. Figure 85 illustrates the numerical and analytical solutions at t = 1500 seconds. Wetting and drying occur at the two moving contact points and hydrostatic balance is approached as time passes.
Figure 86 gives the analytical and numerical solutions for the horizontal component of velocity (which is spatially constant). Figures 87 and 89 give the relative errors in surface elevation and horizontal velocity respectively, as functions of spatial resolution. Figures 88 and 90 give the corresponding convergence orders.
The errors are generally small, however convergence is not reached with increasing resolution. This is due to errors in velocity at the contact points.
Figure 86: Time evolution of the (spatially constant) horizontal velocity. Seven levels of refinement. 
