## 4 Results
### 1D Case
#### 4.1.2 Performance Study of the Preconditioners  
In this section, we include an efficiency study to monitor the performance of the new preconditioners. We ran the code with a fixed \(\Delta t=10^{-1}\) and \(M=2048\) grid points until the final time \(t_{final}=5.0\). As we pointed out earlier, the time accuracy of the numerical scheme affects the convergence speed of the JFNK method. In other words, more accurate time schemes result in better conditioned systems (the condition number is improved). This automatically accelerates the convergence of the linear solver (Krylov block) and consequently may decrease the number of nonlinear iterations (Newton block). Therefore, we expect that the JFNK method based on the Crank-Nicolson and the backward Euler time schemes behave differently. Indeed, we observe this behavior in Figure 5. Without the preconditioners, the average numbers of nonlinear and linear iterations are significantly different for the two time schemes. In fact, the JFNK method with the backward Euler scheme takes almost _twice_ the number linear and nonlinear iterations to achieve the same final time run. A truly meaningful acceleration manifests when we turn the preconditioners on (Figure 5). The average number of Newton iterations are from _four_ to _six_, which are very reasonable, but the average number of GMRES iterations (linear iterations) significantly drops whenever we activate the preconditioners. Notice that when the time progresses, the sharp structure smears and the solution profile becomes smoother (Figure 3). Then, the JFNK method with the preconditioners converges effortlessly, requiring only _one_ GMRES iteration (second raw of Figure 5).  
Figure 4: Comparison of the results of [19] versus ours for time \(t=1,5,10,50\).  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & \multicolumn{2}{c}{**Spatial Convergence Analysis with \(E=||c^{\Delta\pi\pi^{2-1}}-c^{\Delta\pi\pi^{2}}||_{p}\) in Columns 1, 3, 5**} \\ \hline \(p=1,k=1,\cdots,8\) & **Conv. Rate** & \(p=2,k=1,\cdots,8\) & **Conv. Rate** & \(p=\infty,k=1,\cdots,8\) & **Conv. Rate** \\ \hline \(6.67\times 10^{-4}\) & 1.9975 & \(1.93\times 10^{-3}\) & 1.9987 & \(9.70\times 10^{-3}\) & 1.9224 \\ \(1.67\times 10^{-4}\) & 2.0006 & \(4.83\times 10^{-4}\) & 1.9985 & \(2.56\times 10^{-3}\) & 1.9972 \\ \(4.18\times 10^{-5}\) & 2.0001 & \(1.21\times 10^{-4}\) & 1.9999 & \(6.41\times 10^{-4}\) & 1.9994 \\ \(1.04\times 10^{-5}\) & 2.0002 & \(3.02\times 10^{-5}\) & 1.9950 & \(1.60\times 10^{-4}\) & 1.9847 \\ \(2.61\times 10^{-6}\) & 2.0015 & \(7.57\times 10^{-6}\) & 2.0064 & \(4.05\times 10^{-5}\) & 2.0220 \\ \(6.52\times 10^{-7}\) & 1.9639 & \(1.88\times 10^{-6}\) & 2.0005 & \(9.97\times 10^{-6}\) & 2.0647 \\ \(1.67\times 10^{-7}\) & 1.9678 & \(4.71\times 10^{-7}\) & 2.0402 & \(2.38\times 10^{-6}\) & 2.2178 \\ \(4.27\times 10^{-8}\) & & \(1.14\times 10^{-7}\) & & \(5.12\times 10^{-7}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Spatial errors and Conv. rate between sequentially refined grid solutions with \(\Delta t~{}=~{}10^{-5}\).  
Another important aspect of our algorithm is the convergence rate of the nonlinear block. In the literature, it is reported that the JFNK method has the super-linearly convergence rate property [20; 41]. We computationally verify this behavior in Figure 6. Figure 6 shows the convergence rate as a function of nonlinear iterations. To produce this figure, we use \(M=2048\) grid points and a fixed time step \(\Delta t=10^{-1}\). The code runs until the final time \(t_{final}=1.0\) with the second-order version (Crank-Nicolson-based) of the preconditioner (Section 3.4.2). Figure 6 clearly indicates that the rate of convergence of our algorithm with the new preconditioner is super linear.