# On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems

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## Abstract

**:**

## 1. Introduction

- compare different discretizations. In particular, we consider what happens when the space is discretized by an upwind method or by central finite differences. We also describe different time discretizations, considering the $\theta $-method and the IMEX schemes;
- study numerically what happens in some particularly critical cases, where some smoothness assumptions are not satisfied. This is the case of boundary layer and blow-up solutions.

## 2. Discretization

#### 2.1. Space Discretization

**0**the backward difference quotients which approximate the first-order derivatives are simply replaced by forward difference quotients.

- Discretization by central finite differences (error $O\left({h}^{2}\right)$)$$\begin{array}{c}\frac{\mathrm{d}{u}_{i,j}\left(t\right)}{\mathrm{d}t}={\Delta}_{x}\left[\sigma ({x}_{i},{y}_{j},{u}_{i,j}\left(t\right)){\nabla}_{x}{u}_{i,j}\left(t\right)\right]+{\Delta}_{y}\left[\sigma ({x}_{i},{y}_{j},{u}_{i,j}\left(t\right)){\nabla}_{y}{u}_{i,j}\left(t\right)\right]-\hfill \\ -{\tilde{v}}_{1}{\delta}_{x}{u}_{i,j}\left(t\right)-{\tilde{v}}_{2}{\delta}_{y}{u}_{i,j}\left(t\right)-{\alpha}_{i,j}{u}_{i,j}\left(t\right)-g({x}_{i},{y}_{j},{u}_{i,j}\left(t\right))+s({x}_{i},{y}_{j},t).\hfill \end{array}$$
- Discretization by upwind scheme (error $O\left(h\right)$), case $\tilde{\mathit{v}}$ >
**0**$$\begin{array}{c}\frac{\mathrm{d}{u}_{i,j}\left(t\right)}{\mathrm{d}t}={\Delta}_{x}\left[{\sigma}_{i,j}\left({u}_{i,j}\left(t\right)\right){\nabla}_{x}{u}_{i,j}\left(t\right)\right]+{\Delta}_{y}\left[{\sigma}_{i,j}\left({u}_{i,j}\left(t\right)\right){\nabla}_{y}{u}_{i,j}\left(t\right)\right]-\hfill \\ -{\tilde{v}}_{1}{\nabla}_{x}{u}_{i,j}\left(t\right)-{\tilde{v}}_{2}{\nabla}_{y}{u}_{i,j}\left(t\right)-{\alpha}_{i,j}{u}_{i,j}\left(t\right)-{g}_{i,j}\left({u}_{i,j}\left(t\right)\right)+{s}_{i,j}\left(t\right).\hfill \end{array}$$

- when we use the upwind scheme, the order of the approximation is $O\left(h\right)$; when we use central FD, the order of the approximation is $O\left({h}^{2}\right)$;
- when we use the upwind scheme, $A\left(\mathit{u}\right(t\left)\right)$ is irreducibly diagonally dominant [11] (p. 23) irrespective of the choice of h. Having also positive diagonal elements and non positive off-diagonal elements, it is always a non-singular M-matrix [11] (p. 91). With central FD, $A\left(\mathit{u}\right(t\left)\right)$ is irreducibly diagonally dominant (and a non-singular M-matrix) only if h satisfies a condition on the step-size. In particular, by Equation (4) if is easy to find that this condition corresponds to$$h<min\left(\right)open="\{"\; close="\}">\frac{2{\sigma}_{min}}{\left(\right)}$$

#### 2.2. Time Discretization

## 3. The Lagged Diffusivity Method

#### 3.1. Effect of the Discretization

#### 3.2. Monotonicity of the Finite-Difference Operator and Convergence Analysis

- the functions $\sigma $ and g are continuous in u and continuously differentiable in $(x,y)$. Moreover, the functions $\alpha $ and s are continuous in their variables;
- $\sigma $ is uniformly bounded in $(x,y)\in \mathrm{\Omega}$ and $u\in {L}^{\infty}\left(\mathrm{\Omega}\right)$: there exist two positive constants ${\sigma}_{min}$ and ${\sigma}_{max}$ such that $0<{\sigma}_{min}\le \sigma (x,y,u)\le {\sigma}_{max}$. Moreover, we also assume $\alpha (x,y)\ge {\alpha}_{min}\ge 0$ and that $\tilde{\mathit{v}}={({\tilde{v}}_{1},{\tilde{v}}_{2})}^{T}$ is constant;
- for fixed $(x,y)\in \mathrm{\Omega}$, the function $\sigma $ is Lipschitz continuous in u (uniformly in $x,y$) with constant $\Lambda >0$;
- for fixed $(x,y)\in \mathrm{\Omega}$, the function g is uniformly monotone in u (uniformly in $x,y$) with constant $c>0$ [15] (p. 114); moreover, it is continuously differentiable in u.

**Theorem**

**1.**

**Theorem**

**2.**

## 4. Solution Procedure

#### 4.1. Solution of the Inner Systems

#### 4.2. Starting Vectors and Stopping Criteria

#### 4.3. Summary of the Solution Method

Algorithm 1 Lagged diffusivity procedure | |

Require: initial condition ${\mathit{u}|}_{0}$ at $t=0$; a tolerance $\overline{\u03f5}$ | |

1: for $n=1,2,\dots $ do | Time level |

2: Initialize solution vector for lagged iteration: ${\mathit{u}}^{\left(0\right)}{=\mathit{u}|}_{n-1}$ | |

3: Initialize lagged tol.: ${\u03f5}_{1}={\tilde{\u03f5}}_{0}\parallel \mathit{F}\left({\mathit{u}}^{\left(0\right)}\right)\parallel $ | |

4: for $\nu =0,1,\dots $ do | Lagged iteration |

5: Initialize linear solver tol.: ${\widehat{\u03f5}}_{1}=max(\widehat{\eta}\parallel {\mathit{F}}_{\nu}\left({\mathit{u}}^{\left(\nu \right)}\right)\parallel ,\widehat{\eta}{\u03f5}_{\nu +1})$ | |

6: for $k=0,1,\dots $ do | Simpl. Newton iteration |

7: for $j=0,1,\dots $ do | Linear solver iteration |

8: Compute $(j+1)$-th iterate ${\mathit{u}}^{(\nu +1,k+1,j+1)}$ | |

9: Compute residual ${\mathit{r}}_{j+1}$ of the linear system | |

10: if $\parallel {\mathit{r}}_{j+1}\parallel \le $ then break | |

11: j = j+1 | |

12: end for | |

13: Compute Newton residual ${\mathit{F}}_{\nu}\left({\mathit{u}}^{(\nu +1,k+1)}\right)$ | |

14: if $\parallel {\mathit{F}}_{\nu}\left({\mathit{u}}^{(\nu +1,k+1)}\right)\parallel \le {\u03f5}_{\nu +1}$ then break | |

15: Update linear solver tol.: ${\widehat{\u03f5}}_{k+1}=\widehat{\eta}\parallel {\mathit{F}}_{\nu}\left({\mathit{u}}^{(\nu +1,k+1)}\right)\parallel $ | |

16: k = k+1 | |

17: end for | |

18: Update vectors and matrices: find ${\mathit{F}}_{\nu +1}\left({\mathit{u}}^{(\nu +1)}\right)$ | |

19: $\nu =\nu +1$ | |

20: ${\u03f5}_{\nu +1}=0.5{\u03f5}_{\nu}$ | |

21: if ${\u03f5}_{\nu +1}\le \overline{\u03f5}$ then return | |

22: end for | |

23: $n=n+1$ | |

24: end for |

**break**means that we exit the current loop and go back to the outer one. On the other hand,

**return**means that we exit from the entire procedure and the algorithm returns the final result. If the smoothness assumptions in Section 3.2 are satisfied (and if the inner linear solver converges), then the convergence of Algorithm 1 is ensured by Theorem 2 or by analogous theorems, depending on the chosen discretization.

## 5. Numerical Experiments

#### 5.1. Effect of the Discretization Scheme

#### 5.2. Blow-Up

#### 5.2.1. Preliminary Results and Analysis

#### 5.2.2. Effect of Refining the Time Grid

- 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 $\mathit{u}$ is always smaller than the maximum of $\mathit{u}$ at boundary points when $t\sim T$;
- as a consequence of the previous point, the smaller the number of grid points, the smaller $max\left(\mathit{u}\right)$. Indeed, we get farther from the boundary and, thus, from the singularity. This is also shown in the next subsection, where we observe that $max\left(\mathit{u}\right)$ increases when we increase the number of points of the space grid;
- the singularity occurs abruptly when $t=T$ is reached, so the solution is still quite small also at times quite close to T. Next, we show that by further reducing $\Delta t$, we are able to get nearer to T, leading to an increase of $max\left(\mathit{u}\right)$.

#### 5.2.3. Effect of Refining the Space Discretization

#### 5.3. Boundary Layer

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Summary of the nonlinear terms present at each time level and at each lagging iteration for different time discretizations. (

**a**) $\theta -$method; (

**b**) IMEX scheme with explicit treatment of $\mathit{G}\left(\mathit{u}\right)$; (

**c**) IMEX scheme with non-constant velocity term $\tilde{\mathit{v}}\left(\mathit{u}\right)$ treated explicitly. Here ${A}_{1}\left(\mathit{u}\right)$ represents the part of A dependent on $\sigma \left(\mathit{u}\right)$ and $\tilde{A}\left(\mathit{u}\right)$ represents the part of A dependent on $\tilde{\mathit{v}}\left(\mathit{u}\right)$.

**Figure 3.**Systems arising from the discretization of the initial PDE and from the used iterative methods.

**Figure 5.**Evolution of the computed solution as t approaches T in the cases ${u}^{*}={u}_{2}^{*}$, ${u}^{*}={u}_{4}^{*}$ and ${u}^{*}={u}_{5}^{*}$.

**Figure 6.**Last computed solution for the problems in Figure 4 for $\Delta t={10}^{-3}$. (

**a**) ${u}_{2}$, ${t}_{f}=0.993$; (

**b**) ${u}_{3}$, ${t}_{f}=0.972$; (

**c**) ${u}_{4}$, ${t}_{f}=0.789$; (

**d**) ${u}_{5}$, ${t}_{f}=0.806$.

$\tilde{\mathit{v}}$ | Discretization | Lin. Solver | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | ${\mathit{j}}^{*}$ | ${\mathit{t}}_{\mathit{tot}}$ |
---|---|---|---|---|---|---|---|---|

${(0,0)}^{T}$ | Central FD | AM | 91,812 | $1.37\times {10}^{-5}$ | $4.16\times {10}^{-5}$ | $1.97\times {10}^{-5}$ | 7581 | $341.9$ |

BiCG(1) | 91,812 | $1.36\times {10}^{-5}$ | $4.16\times {10}^{-5}$ | $1.97\times {10}^{-5}$ | 1199 | $57.5$ | ||

Upwind | AM | $91,812$ | $1.37\times {10}^{-5}$ | $4.16\times {10}^{-5}$ | $1.97\times {10}^{-5}$ | 7581 | $356.5$ | |

BiCG(1) | $91,812$ | $1.36\times {10}^{-5}$ | $4.16\times {10}^{-5}$ | $1.97\times {10}^{-5}$ | 1199 | $64.0$ | ||

${(10,10)}^{T}$ | Central FD | AM | $91,814$ | $1.37\times {10}^{-5}$ | $3.63\times {10}^{-5}$ | $1.71\times {10}^{-5}$ | 6608 | $311.0$ |

BiCG(1) | $91,814$ | $1.33\times {10}^{-5}$ | $3.61\times {10}^{-5}$ | $1.71\times {10}^{-5}$ | 979 | $63.6$ | ||

Upwind | AM | $92,397$ | $1.38\times {10}^{-5}$ | $2.25\times {10}^{-3}$ | $1.07\times {10}^{-3}$ | 6710 | $407.9$ | |

BiCG(1) | $92,397$ | $1.33\times {10}^{-5}$ | $2.25\times {10}^{-3}$ | $1.07\times {10}^{-3}$ | 967 | $72.5$ | ||

${(300,300)}^{T}$ | Central FD | AM | $93,532$ | $1.38\times {10}^{-5}$ | $1.21\times {10}^{-5}$ | $5.71\times {10}^{-6}$ | 594 | $37.5$ |

BiCG(1) | - | - | - | - | - | - | ||

Upwind | AM | $110,834$ | $1.62\times {10}^{-5}$ | $9.07\times {10}^{-3}$ | $4.30\times {10}^{-3}$ | 748 | $41.7$ | |

BiCG(1) | $110,834$ | $1.57\times {10}^{-5}$ | $9.08\times {10}^{-3}$ | $4.30\times {10}^{-3}$ | 1534 | $79.7$ | ||

${(500,500)}^{T}$ | Central FD | AM | - | - | - | - | - | - |

BiCG(1) | - | - | - | - | - | - | ||

Upwind | 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 | $94.0$ |

**Table 2.**Comparison between discretization by finite differences and by upwind scheme with high $\tilde{\mathit{v}}$.

$\tilde{\mathit{v}}$ | Discretization | Lin. Solver | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | ${\mathit{j}}^{*}$ | ${\mathit{t}}_{\mathit{tot}}$ |
---|---|---|---|---|---|---|---|---|

${(500,500)}^{T}$ | 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 | $96.0$ | ||

Upwind | 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 | $96.8$ | ||

${(2000,2000)}^{T}$ | Central FD | AM | - | - | - | - | - | - |

BiCG(1) | - | - | - | - | - | - | ||

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 | $89.7$ | ||

Upwind | AM | $241,926$ | $1.49\times {10}^{-5}$ | $9.81\times {10}^{-3}$ | $4.65\times {10}^{-3}$ | 370 | $36.7$ | |

BiCG(1) | - | - | - | - | - | - | ||

BiCG(2) | $241,926$ | $1.37\times {10}^{-5}$ | $9.81\times {10}^{-3}$ | $4.65\times {10}^{-3}$ | 514 | $73.2$ | ||

BiCG(4) | $241,926$ | $1.74\times {10}^{-5}$ | $9.81\times {10}^{-3}$ | $4.65\times {10}^{-3}$ | 247 | $86.0$ | ||

${(50000,50000)}^{T}$ | Central FD | AM | - | - | - | - | - | - |

BiCG(1) | - | - | - | - | - | - | ||

BiCG(2) | $2,976,435$ | $1.36\times {10}^{-5}$ | $1.01\times {10}^{-6}$ | $4.79\times {10}^{-7}$ | 1077 | $162.4$ | ||

BiCG(4) | $2,976,435$ | $1.35\times {10}^{-5}$ | $1.01\times {10}^{-6}$ | $4.79\times {10}^{-7}$ | 507 | $197.7$ | ||

Upwind | AM | $4,300,370$ | $9.19\times {10}^{-6}$ | $9.96\times {10}^{-3}$ | $4.71\times {10}^{-3}$ | 322 | $33.5$ | |

BiCG(1) | - | - | - | - | - | - | ||

BiCG(2) | $4,300,370$ | $2.39\times {10}^{-6}$ | $9.96\times {10}^{-3}$ | $4.71\times {10}^{-3}$ | 476 | $69.0$ | ||

BiCG(4) | $4,300,370$ | $8.79\times {10}^{-6}$ | $9.95\times {10}^{-3}$ | $4.71\times {10}^{-3}$ | 233 | $86.6$ |

**Table 3.**Error and data of the lagged diffusivity procedure for ${u}_{2}^{*}$, ${u}_{4}^{*}$ and ${u}_{5}^{*}$ as t approaches T.

t | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | $max\mathit{u}$ | $min\mathit{u}$ | ${\mathit{k}}_{\mathit{t}}$ | ${\mathit{j}}_{\mathit{t}}$ |
---|---|---|---|---|---|---|---|---|

${u}^{*}={u}_{2}^{*}$, $T=1.0$ | ||||||||

$0.3$ | 369 | $1.35\times {10}^{-5}$ | $1.74\times {10}^{-5}$ | $1.86\times {10}^{-5}$ | $1.403$ | $0.594$ | 19 | 185 |

$0.6$ | 1209 | $8.66\times {10}^{-6}$ | $6.89\times {10}^{-5}$ | $6.27\times {10}^{-5}$ | $2.405$ | $0.721$ | 21 | 202 |

$0.9$ | $15,619$ | $1.57\times {10}^{-5}$ | $1.04\times {10}^{-3}$ | $6.88\times {10}^{-4}$ | $8.402$ | $0.918$ | 25 | 254 |

${u}^{*}={u}_{4}^{*}$, $T=0.8$ | ||||||||

$0.3$ | 592 | $1.03\times {10}^{-5}$ | $1.84\times {10}^{-3}$ | $1.88\times {10}^{-3}$ | $1.930$ | $0.505$ | 20 | 186 |

$0.5$ | 1828 | $1.68\times {10}^{-5}$ | $1.76\times {10}^{-3}$ | $1.56\times {10}^{-3}$ | $2.369$ | $0.558$ | 21 | 208 |

$0.7$ | $22,484$ | $1.08\times {10}^{-5}$ | $1.54\times {10}^{-2}$ | $7.66\times {10}^{-3}$ | $5.851$ | $0.772$ | 25 | 263 |

${u}^{*}={u}_{5}^{*}$, $T=0.8$ | ||||||||

$0.3$ | 112 | $1.53\times {10}^{-5}$ | $3.51\times {10}^{-4}$ | $7.59\times {10}^{-4}$ | $0.941$ | $0.333$ | 17 | 120 |

$0.5$ | 346 | $1.17\times {10}^{-5}$ | $3.29\times {10}^{-4}$ | $6.42\times {10}^{-4}$ | $1.148$ | $0.356$ | 19 | 138 |

$0.7$ | 3077 | $1.45\times {10}^{-5}$ | $1.36\times {10}^{-3}$ | $1.85\times {10}^{-3}$ | $2.694$ | $0.433$ | 22 | 175 |

**Table 4.**Error and data of the lagged diffusivity procedure for the last computable solution for the problems in Figure 4 for $\Delta t={10}^{-3}$.

${\mathit{u}}^{*}$ | ${\mathit{t}}_{\mathit{f}}$ | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | $max\mathit{u}$ | $min\mathit{u}$ | $max{\mathit{u}}^{*}$ | $min{\mathit{u}}^{*}$ |
---|---|---|---|---|---|---|---|---|---|

${u}_{2}^{*}$ | $0.993$ | $3.48\times {10}^{6}$ | $1.04\times {10}^{-5}$ | $4.74\times {10}^{-3}$ | $2.35\times {10}^{-3}$ | $37.545$ | $1.000$ | $37.545$ | $1.000$ |

${u}_{3}^{*}$ | $0.972$ | $1.28\times {10}^{10}$ | $1.03\times {10}^{-5}$ | $2.90\times {10}^{-2}$ | $3.54\times {10}^{-3}$ | $52.508$ | $0.992$ | $52.508$ | $0.992$ |

${u}_{4}^{*}$ | $0.789$ | $353,703$ | $1.29\times {10}^{-5}$ | $3.54\times {10}^{-1}$ | $6.65\times {10}^{-2}$ | $30.314$ | $0.976$ | $30.314$ | $0.976$ |

${u}_{5}^{*}$ | $0.806{\phantom{\rule{0.166667em}{0ex}}}^{1}$ | 8301 | $1.89\times {10}^{-5}$ | $9.77\times {10}^{-2}$ | $7.38\times {10}^{-2}$ | $24.049$ | $0.505$ | $23.067$ | $0.505$ |

**Table 5.**Error and data of the lagged diffusivity procedure for the last computable solution for the problems in Figure 2 for $\Delta t={10}^{-3}$ and $N=250$.

${\mathit{u}}^{*}$ | ${\mathit{t}}_{\mathit{f}}$ | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | $max\mathit{u}$ | $min\mathit{u}$ | $max{\mathit{u}}^{*}$ | $min{\mathit{u}}^{*}$ |
---|---|---|---|---|---|---|---|---|---|

${u}_{2}^{*}$ | $0.981$ | $2.93\times {10}^{6}$ | $1.11\times {10}^{-5}$ | $9.68\times {10}^{-4}$ | $4.97\times {10}^{-4}$ | $37.207$ | $0.985$ | $37.207$ | $0.985$ |

${u}_{3}^{*}$ | $0.965$ | $9.86\times {10}^{9}$ | $1.95\times {10}^{-5}$ | $1.04\times {10}^{-2}$ | $1.34\times {10}^{-3}$ | $51.201$ | $0.974$ | $51.201$ | $0.974$ |

${u}_{4}^{*}$ | $0.783$ | $544,443$ | $8.42\times {10}^{-6}$ | $1.53\times {10}^{-1}$ | $3.09\times {10}^{-2}$ | $30.287$ | $0.953$ | $30.290$ | $0.953$ |

${u}_{5}^{*}$ | $0.795$ | $14,438$ | $1.20\times {10}^{-5}$ | $1.82\times {10}^{-2}$ | $1.42\times {10}^{-2}$ | $24.430$ | $0.496$ | $24.546$ | $0.486$ |

**Table 6.**Error and data of the lagged diffusivity procedure for the last computable solution for the problems in Figure 4 for $\Delta t={10}^{-5}$, $N=250$.

${\mathit{u}}^{*}$ | ${\mathit{t}}_{0}$ | ${\mathit{t}}_{\mathit{f}}$ | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | $max\mathit{u}$ | $min\mathit{u}$ | $max{\mathit{u}}^{*}$ | $min{\mathit{u}}^{*}$ |
---|---|---|---|---|---|---|---|---|---|---|

${u}_{2}^{*}$ | $0.981$ | $0.99193$ | $3.67\times {10}^{8}$ | $7.42\times {10}^{-7}$ | $6.02\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $62.539$ | $0.999$ | $62.539$ | $0.999$ |

${u}_{3}^{*}$ | $0.965$ | $0.97141$ | $2.75\times {10}^{8}$ | $7.48\times {10}^{-5}$ | $2.07\times {10}^{-3}$ | $2.41\times {10}^{-4}$ | $61.276$ | $0.980$ | $61.276$ | $0.980$ |

${u}_{4}^{*}$ | $0.783$ | $0.79343$ | $6.87\times {10}^{8}$ | $1.05\times {10}^{-6}$ | $4.33\times {10}^{-2}$ | $5.89\times {10}^{-3}$ | $60.203$ | $0.984$ | $60.203$ | $0.984$ |

${u}_{5}^{*}$ | $0.795$ | $0.80277{\phantom{\rule{0.166667em}{0ex}}}^{1}$ | $1.13\times {10}^{9}$ | $2.15\times {10}^{-7}$ | $1.38\times {10}^{-2}$ | $9.34\times {10}^{-3}$ | $61.238$ | $0.502$ | $61.238$ | $0.502$ |

**Table 7.**Analysis of a boundary layer example as $\left|\lambda \right|$ increases (boundary layer occurs for large $\left|\lambda \right|$).

$\tilde{\mathit{v}}$ | Lin. Solver | ${\mathit{res}}_{0}$ | $\mathit{res}$ | ${\mathit{err}}_{\mathit{h}}$ | ${\mathit{err}}_{2}$ rel. | ${\mathit{\nu}}^{*}$ | ${\mathit{j}}^{*}$ |
---|---|---|---|---|---|---|---|

$\lambda =-1$ | BiCG(1) | $1.6\times {10}^{6}$ | $1.48\times {10}^{-4}$ | $7.50\times {10}^{-8}$ | $5.84\times {10}^{-8}$ | 31 | 1925 |

BiCG(2) | $1.6\times {10}^{6}$ | $1.25\times {10}^{-4}$ | $7.26\times {10}^{-8}$ | $5.65\times {10}^{-8}$ | 31 | 977 | |

BiCG(4) | $1.6\times {10}^{6}$ | $1.32\times {10}^{-4}$ | $7.09\times {10}^{-8}$ | $5.53\times {10}^{-8}$ | 31 | 460 | |

$\lambda =-10$ | BiCG(1) | $1.7\times {10}^{6}$ | $1.41\times {10}^{-4}$ | $2.18\times {10}^{-5}$ | $6.43\times {10}^{-5}$ | 31 | 1389 |

BiCG(2) | $1.7\times {10}^{6}$ | $1.14\times {10}^{-4}$ | $2.18\times {10}^{-5}$ | $6.43\times {10}^{-5}$ | 31 | 708 | |

BiCG(4) | $1.7\times {10}^{6}$ | $1.53\times {10}^{-4}$ | $2.18\times {10}^{-5}$ | $6.43\times {10}^{-5}$ | 31 | 366 | |

$\lambda =-100$ | BiCG(1) | $1.8\times {10}^{6}$ | $1.36\times {10}^{-4}$ | $7.63\times {10}^{-4}$ | $9.36\times {10}^{-3}$ | 31 | 638 |

BiCG(2) | $1.8\times {10}^{6}$ | $1.41\times {10}^{-4}$ | $7.63\times {10}^{-4}$ | $9.36\times {10}^{-3}$ | 31 | 156 | |

BiCG(4) | $1.8\times {10}^{6}$ | $9.10\times {10}^{-5}$ | $7.63\times {10}^{-4}$ | $9.36\times {10}^{-3}$ | 31 | 314 |

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Mezzadri, F.; Galligani, E.
On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems. *Algorithms* **2017**, *10*, 88.
https://doi.org/10.3390/a10030088

**AMA Style**

Mezzadri F, Galligani E.
On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems. *Algorithms*. 2017; 10(3):88.
https://doi.org/10.3390/a10030088

**Chicago/Turabian Style**

Mezzadri, Francesco, and Emanuele Galligani.
2017. "On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems" *Algorithms* 10, no. 3: 88.
https://doi.org/10.3390/a10030088