# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 3 Numerical Implementation
### The Numerical Transposition of the Integral Scheme
#### 3.2.1 Basic Time Evolving Scheme
\begin{split}\int_{y_{i}}^{y_{i+1}}\mathcal{G}_{2}\big{(}x-x^{ \prime},\tau\big{)}l_{j}\big{(}x^{\prime}-x_{i}\big{)}dx^{\prime}=l_{j}(0)+ \Pi_{2}^{2}\frac{d^{2}}{d\zeta^{2}}\big{[}l_{j}(\zeta)\big{]}_{\zeta=0}+...+ \frac{1}{q!}\Pi_{2}^{2q}\frac{d^{2q}}{d\zeta^{2q}}\big{[}l_{j}(\zeta)\big{]}_{ \zeta=0}.\end{split}  \tag{44}\]  
With the resulting final expression of the time evolving solution defined as,  
\[ \begin{split} u(x_{i},t_{0}+\tau)&=u(x_{i},t_{0})+ \sum_{k=1}^{q}\Pi_{2}^{2k}\frac{(2k)!}{(k)!}\delta_{2q+1}^{2k}[u](x=x_{i})\end{split}  \tag{45}\]  
Comparing the results obtained from the two expressions of the Green function, we appreciate the effect the \(\Pi_{2}\) number has on the computation of the solution and a perspective about the limit of the CFL different from its usual stability definition [64]. Wecan see this comparison in Figure 4, where the Logarithmic difference between the two possible weights is presented. Equation (45) does not take into account the interaction between the boundaries of the domain of validity \(\Omega_{i}\) and the Green function, resulting into a persistent error that will accumulate through time iterations. Furthermore, depending on the numerical scheme used, making the global error of the computed function divergent. Unfortunately, we cannot obtain a closed expression in general of the open domain Green function when we have a spatial dependent transport coefficient, therefore we will need to consider this limitation of the CFL in order not to introduce extra error into the time-evolving solution. First-order operator follows the same logic we have introduced: the two possible expressions for the Green function are valid as long as the CFL number is bounded correctly. Yet, the fact that the first-order Green function behaves as a Dirac delta modifies the problem, breaking the integral (40) in the following manner,  
\[ \begin{split} u(x_{i},t_{0}+\tau)&=\sum_{j=-q}^{q} u\big{(}x_{i+j},t_{0}\big{)}\int_{y_{i}}^{y_{i+1}}\mathcal{G}_{1}\big{(}x-x^{ \prime},\tau\big{)}l_{j}\big{(}x^{\prime}-x_{i}\big{)}dx^{\prime}\\ &=u(x_{i},t_{0})+\sum_{k=1}^{q}\Big{\{}\Pi_{1}^{2k}\delta_{2q+1} ^{2k}[u](x=x_{i})+\Pi_{1}^{2k-1}\delta_{2q+1}^{2k-1}[u](x=x_{i})\Big{\}}.\end{split}  \tag{46}\]  
Which is the same expression we recover using the Taylor series expansion, being \(\Pi_{1}=a_{1}\tau/\Delta x\) the usual CFL number. There are two details to point out, regarding this last expression:  
* The CFL number for a constant advection simply marks the frontier after which the integral term (46) is zero, i.e., the information is propagated outside of the domain of validity, i.e., outside of the corresponding \(\Omega_{i}\). * In the case of non-constant advection the situation is more complicated, while the Green function is still a Dirac delta, its dependencies affects the integral, making it become a _simple layer_ integral [65]. Therefore, using the exact, open domain Green function will produce a different result from the one obtained by the Taylor series, and  
Figure 4: Tendency of the difference between the 2 time-evolving weights as a function of the CFL number for diffusive problems, the ones obtained exactly over a subdomain of validity and the ones obtained through the Taylor Series Expansion. their difference will be defined by the value of the CFL in a similar manner as in the second-order. Having a non-constant transport coefficient will affect also to the Taylor series expansion, yet in a simpler manner: the values of \(\Pi_{1}\) will be time and spatial dependent in general, yet evaluated at time frame \(t_{0}\) and at \(x_{i}\),  
\[\Pi_{1}(x_{i})=\frac{a_{1}(x_{i},t_{0})\tau}{\Delta x} \tag{47}\]  
On the other hand, using the exact Green function will have the transport coefficient evaluated at the advected point, \(x_{i}+a_{1}(x_{i},t_{0})\tau\), therefore defining a new grid position that requires evaluation.