Accuracy of the Convective Operator

We tested the accuracy of the Galerkin convective operator using a third-order Adams-Bashforth temporal scheme, and a periodic domain as shown in figure 4.8. We started with the initial condition $u=sin(\pi cos(\pi x))$ and examined the $L_\infty$ error at t=0 and t=2. The convection velocity was constant. We chose a time step small enough so that the time-stepping error is small compared to the initial projection error for N < 16. We see that exponential convergence is maintained after one time period.

  

Figure 4.8: Exponential accuracy is achieved for the wave equation with $u=sin(\pi cos(\pi x))$ as initial condition.