# 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  
The most straightforward problem we can define, containing the _core_ parts of the methodology, is one where we do not have the non-homogeneous contribution, we are under a ring domain where the boundaries are connected, hence they do not contribute to the problem, and the transport coefficient is constant. We will start by the second-order  
Figure 3: Visual representation of the Hermanns Navarro Algorithm for optimizing Piecewise interpolation through the Chebyshev–Gauss–Lobatto collocation methodology. operator, showing how the Courant-Friedrichs-Lewy number appears even for this simple problem. Introducing the numerical discretization,  
\[ \begin{split} u(x,t_{0}+\tau)&=\int\mathcal{G}_{2} \big{(}x-x^{\prime},\tau\big{)}\sum_{i=-N}^{N}H(x,\Omega_{i})\sum_{j=-q}^{q}l_{ j}\big{(}x^{\prime}-x_{i}\big{)}u\big{(}x_{i+j},t_{0}\big{)}dx^{\prime}\\ &=\sum_{i=-N}^{N}\sum_{j=-q}^{q}u\big{(}x_{i+j},t_{0}\big{)}\int_{ y_{i}}^{y_{i+1}}\mathcal{G}_{2}\big{(}x-x^{\prime},\tau\big{)}l_{j}\big{(}x^{ \prime}-x_{i}\big{)}dx^{\prime}\\ &=\sum_{i=-N}^{N}\sum_{j=-q}^{q}u\big{(}x_{i+j},t_{0}\big{)}w_{i, j}(x)\end{split}  \tag{41}\]  
Now, there are two possibilities to derive the weights of the time-marching solution, each of them determined by which of the mathematical definition of the Green function we select: the Taylor series expansion or the closed expression. Starting from this last one, with the appropriate changes of variable we can recover,  
\[ \begin{split} w_{i,j}(x)&=\frac{1}{\sqrt{4\pi a_{ 2}\tau}}\int_{y_{i}}^{y_{i+1}}e^{-\frac{1}{4}\big{(}\frac{x-x^{\prime}}{4\pi^{ 2}}\big{)}^{2}}l_{j}\big{(}x^{\prime}-x_{i}\big{)}dx^{\prime}=\frac{1}{\sqrt{4 \pi}}\frac{1}{\Pi_{2}}\int_{-1/2}^{+1/2}e^{-\big{(}\frac{x-x^{\prime}}{4\Pi_{ 2}}\big{)}^{2}}l_{j}\big{(}\zeta^{\prime}\big{)}d\zeta^{\prime}\\ \zeta&=\frac{1}{\Delta x}(x-x_{i}),\;\;y_{i}=x_{i}- \frac{\Delta x}{2},\;\;y_{i+1}=x_{i}+\frac{\Delta x}{2},\;\;\Pi_{2}=\frac{ \sqrt{a_{2}\tau}}{\Delta x}\end{split}  \tag{42}\]  
where \(\Pi_{2}\) is the Courant-Friedrichs-Lewy (CFL) number for the diffusive problem, also known as the diffusion number. As it can be expected, with the appropriate change of variables, i.e., centering the interpolation over the discretization point \(x_{i}\), these weights are the same for the whole set of points and independent of \(i\). Hence, they can be pre-computed without affecting the iterative process. Yet, the higher this value is, the more important the boundary terms of the defining integral will be. Now, with the definition of these weights we see how the value of this number defines the width of the Green function as it appears as its standard deviation, hence a CFL lower than \(1/2\) will imply that the Gaussian is contained without being affected by the piecewise subdomains of validity. If the time-evolved solution is also studied over the same grid points, i.e., we are interested only over the points \(u(x_{i},t_{0}+\tau)\), therefore, we can simplify the integral as \(\zeta=0\), obtaining,  
\[ \begin{split} u(x_{i},t_{0}+\tau)&=\text{Erf}\bigg{(} \frac{1}{4}\frac{1}{\Pi_{2}}\bigg{)}u(x_{i},t_{0})+\sum_{k=1}^{q}(2\Pi_{2})^{2k }\frac{1}{\sqrt{\pi}}\gamma\bigg{(}\frac{1}{2}+k,\frac{1}{16\Pi_{2}^{2}}\bigg{)} \delta_{2q+1}^{2k}[u](x=x_{i}),\end{split}  \tag{43}\]  
where \(\gamma\) is the lower incomplete gamma function [62], Erf is the usual Error function and \(\delta_{2q+1}^{2k}[\cdot]\) is the centered finite differences operator of order \(2k\), defined over a stencil of \(2q+1\) at point \(x=x_{i}\)[63]. Now this definition is exact and takes into account all the possible interactions with the boundaries of the piecewise domain of validity as the integral is performed over the analytical expression of the Green function. If, on the contrary, we use the Taylor series expansion of the Green function, we recover a similar expression,  
\[