# On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems
## 5 Numerical Experiments
### Effect of the Discretization Scheme
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\bar{\sigma}\) & **Discretization** & **Lin. Solver** & \(\mathbf{res_{0}}\) & \(\mathbf{res}\) & \(\mathbf{err_{h}}\) & \(\mathbf{err_{2}}\)** rel.** & \(\mathbf{j^{*}}\) & \(\mathbf{t_{tot}}\) \\ \hline \multirow{8}{*}{\((500,500)^{T}\)} & \multirow{4}{*}{Central FD} & AM & - & - & - & - & - & - \\  & & BiCG(1) & - & - & - & - & - & - \\  & & BiCG(2) & 96,513 & \(1.14\times 10^{-5}\) & \(9.45\times 10^{-6}\) & \(4.47\times 10^{-6}\) & 807 & 99.1 \\  & & BiCG(4) & 96,513 & \(1.37\times 10^{-5}\) & \(9.45\times 10^{-6}\) & \(4.47\times 10^{-6}\) & 382 & 96.0 \\  & & AM & 124,785 & \(1.71\times 10^{-5}\) & \(9.41\times 10^{-3}\) & \(4.46\times 10^{-3}\) & 552 & 42.2 \\  & & BiCG(1) & 124,785 & \(1.47\times 10^{-5}\) & \(9.41\times 10^{-3}\) & \(4.46\times 10^{-3}\) & 1398 & 94.0 \\  & & BiCG(2) & 124,785 & \(1.46\times 10^{-5}\) & \(9.41\times 10^{-3}\) & \(4.46\times 10^{-3}\) & 673 & 72.7 \\  & & BiCG(4) & 124,785 & \(5.36\times 10^{-6}\) & \(9.41\times 10^{-3}\) & \(4.46\times 10^{-3}\) & 299 & 96.8 \\ \hline \multirow{8}{*}{\((2000,200)^{T}\)} & \multirow{4}{*}{Central FD} & AM & - & - & - & - & - & - \\  & & BiCG(2) & 150,303 & \(8.03\times 10^{-6}\) & \(4.77\times 10^{-6}\) & \(2.26\times 10^{-6}\) & 629 & 69.6 \\  & & BiCG(4) & 150,303 & \(1.04\times 10^{-5}\) & \(4.77\times 10^{-6}\) & \(2.26\times 10^{-6}\) & 296 & 89.7 \\ \cline{1-1}  & & AM & 241,926 & \(1.49\times 10^{-5}\) & \(9.81\times 10^{-3}\) & \(4.65\times 10^{-3}\) & 370 & 36.7 \\ \cline{1-1}  & & BiCG(1) & - & - & - & - & - & - \\ \cline{1-1}  & & BiCG(2) & 241,926 & \(1.37\times 10^{-5}\) & \(9.81\times 10^{-3}\) & \(4.65\times 10^{-3}\) & 514 & 73.2 \\ \cline{1-1}  & & BiCG(4) & 241,926 & \(1.74\times 10^{-5}\) & \(9.81\times 10^{-3}\) & \(4.65\times 10^{-3}\) & 247 & 86.0 \\ \hline \multirow{8}{*}{\((5000,5000)^{T}\)} & \multirow{4}{*}{Central FD} & AM & - & - & - & - & - \\  & & BiCG(1) & - & - & - & - & - \\ \cline{1-1}  & & BiCG(2) & 2,976,435 & \(1.36\times 10^{-5}\) & \(1.01\times 10^{-6}\) & \(4.79\times 10^{-7}\) & 1077 & 162.4 \\ \cline{1-1}  & & BiCG(4) & 2,976,435 & \(1.35\times 10^{-5}\) & \(1.01\times 10^{-6}\) & \(4.79\times 10^{-7}\) & 507 & 197.7 \\ \cline{1-1}  & & AM & 4,300,370 & \(9.19\times 10^{-6}\) & \(9.96\times 10^{-3}\) & \(4.71\times 10^{-3}\) & 322 & 33.5 \\ \cline{1-1}  & & Upwind & BiCG(1) & - & - & - & - & - \\ \cline{1-1}  & & BiCG(2) & 4,300,370 & \(2.39\times 10^{-6}\) & \(9.96\times 10^{-3}\) & \(4.71\times 10^{-3}\) & 476 & 69.0 \\ \cline{1-1}  & & BiCG(4) & 4,300,370 & \(8.79\times 10^{-6}\) & \(9.95\times 10^{-3}\) & \(4.71\times 10^{-3}\) & 233 & 86.6 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison between discretization by finite differences and by upwind scheme with high \(\bar{\sigma}\). it is also to be noted that the upwind scheme is much faster, especially when we use the AM as inner solver. Thus, we may opt for central finite differences with, for instance, BiCGstab(2) as linear solver if we require smaller errors, while we may choose an upwind method with AM if we require lower computational cost.