# On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems
## 3 The Lagged Diffusivity Method
### Monotonicity of the Finite-Difference Operator and Convergence Analysis  
Let us assume that the following smoothness properties hold true:  
1. 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;
2. \(\sigma\) is uniformly bounded in \((x,y)\in\Omega\) and \(u\in L^{\infty}(\Omega)\): there exist two positive constants \(\sigma_{\min}\) and \(\sigma_{\max}\) such that \(0<\sigma_{\min}\leq\sigma(x,y,u)\leq\sigma_{\max}\). Moreover, we also assume \(\alpha(x,y)\geq\alpha_{\min}\geq 0\) and that \(\tilde{v}=(\tilde{\sigma}_{1},\tilde{\sigma}_{2})^{T}\) is constant;
3. for fixed \((x,y)\in\Omega\), the function \(\sigma\) is Lipschitz continuous in \(u\) (uniformly in \(x,y\)) with constant \(\Lambda>0\);
4. for fixed \((x,y)\in\Omega\), 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 \(u\).  
Under these smoothness assumptions, the monotonicity of the finite different operator and the convergence of the algorithm are assured for grid functions \(u_{i,j}\) belonging to a set \(\mathcal{B}_{\rho,\beta}\), where \(\rho\) is a bound on the discrete \(l_{2}(\Omega_{h})\) norm \(\|\mathbf{u}\|_{h}:=(h^{2}\sum_{i=1}^{N}\sum_{j=1}^{M}u_{i,j}^{2})^{\frac{2}{2}}\) and \(\beta\) is a bound over the absolute value of the backward difference quotients of \(u_{i,j}\).  
Thus, considering for instance the discretization by central finite differences and time integration by the \(\theta-\)method, the following theorems hold true  
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 \(\mathbf{G}(\mathbf{u})\); (**c**) IMEX scheme with non-constant velocity term \(\tilde{v}(\mathbf{u})\) treated explicitly. Here \(A_{1}(\mathbf{u})\) represents the part of \(A\) dependent on \(\sigma(\mathbf{u})\) and \(\tilde{A}(\mathbf{u})\) represents the part of \(A\) dependent on \(\tilde{v}(\mathbf{u})\).  
**Theorem 1**.: _Let \(\mathbf{u}^{*}\in\mathcal{B}_{\rho,\beta}\) be a solution of the nonlinear system \(\mathbf{F}(\mathbf{u})=\mathbf{0}\). If_  
\[\alpha_{\min}+\frac{1}{\tau}+c>\frac{\beta^{2}\Lambda^{2}}{2\sigma_{\min}} \tag{13}\]  
_then \(\mathbf{F}(\mathbf{u})\) is uniformly monotone in \(\mathcal{B}_{\rho,\beta}\) and the solution \(\mathbf{u}^{*}\) of Equation (9) is unique._  
**Theorem 2**.: _Let \(\mathbf{u}^{*}\in\mathcal{B}_{\rho,\beta}\) be the solution of the nonlinear system \(\mathbf{F}(\mathbf{u})=\mathbf{0}\) defined in Equation (9) with \(A(\mathbf{u})\) non-singular M-matrix and \(\mathbf{G}(\mathbf{u})\) diagonal mapping. We assume the smoothness Properties 1-4 to be satisfied and that condition Equation (13) on the monotonicity of \(\mathbf{F}(\mathbf{u})\) is satisfied._  
_Starting from an arbitrary starting vector \(\mathbf{u}^{(0)}\), let then \(\mathbf{u}^{(v+1)}\) be the solution of system Equation (10) with residual \(\mathbf{r}^{(v+1)}\) satisfying Equation (12), with \(\epsilon_{v}\to 0\) for \(v\to\infty\). Finally, \(\mathbf{u}^{(v)}\) belongs to \(\mathcal{B}_{\rho_{v},\beta_{v}}\), \(v=0,1,...\). Then, the sequence \(\{\mathbf{u}^{(v)}\}\) converges to \(\mathbf{u}^{*}\)._  
The proofs of these theorems and more information on \(\mathcal{B}_{\rho,\beta}\) can be found in [16]. In the literature it is also possible to find proofs of the convergence of the procedure for less general cases (e.g., see [2] for steady state reaction diffusion problems). Proceeding analogously, similar results can be easily found also for the other discretizations.