# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 3 Numerical Implementation
### The Numerical Transposition of the Integral Scheme
#### 3.2.2 Non-Constant Transport Coefficients
Even in a situation where it produces a jump of order of the unity, these corrections will be still dominated by the order of magnitude of the CFL and, more important, the variation of the transport coefficient, that again, if it is smooth enough will produce bounded and correspondingly small corrections. Moreover, in a situation where we had a study a discontinuous transport coefficient that produce at any \(x_{i}\) a jump on its derivatives, we could introduce analytically its variation at the moment we compute the integral, obtaining numerically controllable extra terms for each of the discontinuities introduced. We also point out that, if the transport coefficient \(a_{2}\) satisfies a diffusion equation being its own diffusion coefficient, the even correction term will be zero. The final propagation of the unknown function, including the corrections, for a second-order operator with non-constant transport coefficients can be written as,  
\[ \begin{split} u(x_{i},t_{0}+\tau)=u(x_{i},t_{0})&+ \sum_{k=1}^{q}\Pi_{2}^{2k}(x_{i},t_{0})\bigg{[}\frac{(2k)!}{k!}+f_{ev,k}(x_{i}, t_{0})\bigg{]}\delta_{2q+1}^{2k}[u](x=x_{i})\\ &+\sum_{k=1}^{q}\Pi_{2}^{2k}(x_{i},t_{0})f_{odd,k}(x_{i},t_{0}) \delta_{2q+1}^{2k-1}[u](x=x_{i}),\end{split}  \tag{52}\]  
where the terms \(f_{k}\) will be, respectively, the odd and even correction terms, that will be composed of the derivatives of the transport coefficient evaluated at the point where we are computing the solution. The correction terms for the first-order operator behave similarly, being piloted by the derivatives of the advection coefficient and the CFL number. Yet, as the transport coefficient of the first-order operator might be zero in some regions, we cannot choose \(a_{1}(x_{i},t_{0})\) as the characteristic advection to make the expression _non-dimensional_. We therefore select a value, \(a_{1,0}\), that will be considered as representative of the order of magnitude of the advection coefficient, resulting in the following correction expression,  
\[\frac{1}{2}\tau^{2}\chi_{1,j}=\frac{1}{2}\Bigg{\{}\bigg{(}\Pi_{1}\left.\frac{d \hat{a}_{1}}{dt}\right|_{t=t_{0}}+\frac{1}{2}\Pi_{1}^{2}\frac{d\hat{a}_{1}^{2} }{d\zeta}\bigg{)}\frac{d}{d\zeta}\left[l_{j}(\zeta)\right]\bigg{\}}(\zeta=0), \leavevmode\
obreak\ \leavevmode\
obreak\ \Pi_{1}=\frac{\tau a_{1,0}}{\Delta x} \tag{53}\]  
where we see again that the corrections will be zero if the advection coefficient satisfied the original PDE with a transport coefficient equal to itself, i.e., an inviscid Burgers equation. Under this non-dimensional grouping, the CFL is no longer a representation of _how far_ the information is advected locally at every \(x_{i}\), nonetheless, it still preserves the measure of how much the Taylor series expansion differs from the exact solution and, therefore, the stability of the global numerical method as in its classical definition. The propagation of the unknown function due to the correction terms will be,  
\[u_{cor}(x_{i},t_{0}+\tau)=\Bigg{(}\frac{1}{2}\Pi_{1}\frac{d\hat{a}_{1}}{dt}+ \frac{1}{4}\Pi_{1}^{2}\frac{d\hat{a}_{1}^{2}}{d\zeta}\bigg{)}\delta_{2q+1}^{ 1}[u](x=x_{i},t=t_{0}). \tag{54}\]  
To which all the considerations regarding of their order of magnitude being related to the regularity of the function we introduced for the second-order operator are applicable. Despite being in general of low order of magnitude, as the presence of advective fronts that present steep slopes are common in mathematical modeling these corrections will be of higher importance than the diffusive ones. Moreover, being the first-order operator simpler to operate with, higher order corrections are easier to derive and implement too in case of such requirement. We present an example of application for both an advective and a diffusive problem, with transport coefficients presenting steep variations so to highlight the correction terms.