# On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems
## 5 Numerical Experiments
### Blow-Up
#### 5.2.2 Effect of Refining the Time Grid  
Taking into account the previous results, let us set \(\Delta t=10^{-3}\). We now start our analysis from \(t_{0}=T-0.1\), so that we do not need an extremely large number of iterations in order to get close to the blow-up zone. In Figure 6 we show how the computed solution looks like at the last computable time step before the blow-up; we call \(t_{f}\) this final time. In Table 4 we report some data on the lagged iterative procedure at \(t=t_{f}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(u^{*}\) & \(t_{f}\) & \(res_{0}\) & \(res\) & \(err_{h}\) & \(err_{2}\) rel. & \(\max u\) & \(\min u\) & \(\max u^{*}\) & \(\min u^{*}\) \\ \hline \(u_{2}^{*}\) & 0.993 & \(3.48\times 10^{6}\) & \(1.04\times 10^{-5}\) & \(4.74\times 10^{-3}\) & \(2.35\times 10^{-3}\) & 37.545 & 1.000 & 37.545 & 1.000 \\ \(u_{3}^{*}\) & 0.972 & \(1.28\times 10^{10}\) & \(1.03\times 10^{-5}\) & \(2.90\times 10^{-2}\) & \(3.54\times 10^{-3}\) & 52.508 & 0.992 & 52.508 & 0.992 \\ \(u_{4}^{*}\) & 0.789 & 353,703 & \(1.29\times 10^{-5}\) & \(3.54\times 10^{-1}\) & \(6.65\times 10^{-2}\) & 30.314 & 0.976 & 30.314 & 0.976 \\ \(u_{5}^{*}\) & 0.806 1 & 8301 & \(1.89\times 10^{-5}\) & \(9.77\times 10^{-2}\) & \(7.38\times 10^{-2}\) & 24.049 & 0.505 & 23.067 & 0.505 \\ \hline \end{tabular}  
* In this case we slightly exceed the limit \(T=0.8\). This is due to the discretization: also when we have \(t>T\), the singularity is still sufficiently far from the grid points and the algorithm keeps working for a while. However, after a few other time steps, the algorithm stalls as expected. The role of the discretization can be proven by increasing the number of elements (e.g., see Table 5, where the algorithms stalls at \(t=0.795\)).  
\end{table} 
Table 4: Error and data of the lagged diffusivity procedure for the last computable solution for the problems in Figure 4 for \(\Delta t=10^{-3}\). The first thing we notice is the extremely large values of \(res_{0}\), which are even more remarkable considering that \(\Delta t\) has been reduced. This can be explained as in the previous paragraph by considering that the solution changes more and more rapidly as we approach \(T\). A gradual increase of \(res_{0}\) with \(t\) is indeed what can be observed by considering the value of \(res_{0}\) at different time levels. The analysis of the plots reveals that the lagged diffusivity procedure is able to reproduce the behavior of all the solutions near their respective blow-up zones. On the other hand, the global errors gradually increase, but they generally remain in the order of \(10^{-3}\div 10^{-2}\). Indeed, only for \(u_{4}^{*}\) they exceed \(10^{-1}\) near \(T\). Notwithstanding this, maximum and minimum values of the solution vector \(u\) correspond to the maximum and minimum values of \(u^{*}\) (computed in the inner grid points) to at least the second decimal digit in all the cases. It is to be noticed that the maximum of \(u\) is the point where the solution is nearest to the blow-up one; a good precision in evaluating the solution in this point is thus important for accurately reproducing the blow-up behavior. Finally, we also notice that these maxima and minima of \(u\) are not so large after all: we are near to the blow-up time, but the maximum of the solution never exceeds values \(\sim 30\), nonetheless. This cannot be explained by numerical errors in the solution procedure, since we said that the maximum of the computed \(u\) is really similar to the value of \(u^{*}\) in the grid point where the maximum occurs. Different causes concur to this:  
* the blow-up occurs on the boundary, where the solution is given by the Dirichlet boundary conditions. Thus, at inner points, where we compute the solution vector, the maximum of \(u\) is always smaller than the maximum of \(u\) at boundary points when \(t\sim T\);
* as a consequence of the previous point, the smaller the number of grid points, the smaller \(\max(u)\). Indeed, we get farther from the boundary and, thus, from the singularity.