# 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  
In the situation where the transport coefficient, \(a_{0_{2}}\), depends either on time or space we are forced to use the Taylor Series expansion of the Green function in order to introduce the correction terms, \(\psi_{0_{2}}\). We use \(\psi_{0_{2}}\) as the correction term that include the initial condition of the Green function, i.e., the Dirac delta, while in the previous section we use capital \(\
abla\) to refer to the correction terms without being applied to the initial condition. While it might be possible to obtain a closed Green function expression for certain Transport Coefficients, and it might be useful to understand the behavior of the PDE under study, the general situation will require for the Taylor series expression. Starting from the second-order operator, we can introduce into the time marching scheme the correction terms from the expansion, c.f. Equation (30), in the following manner,  
\[\psi_{2}\big{(}x,x^{\prime}\big{)} =\Bigg{[}\Bigg{(}\frac{da_{2}}{dt}\bigg{|}_{t=t_{0}}+a_{2}\frac{ d^{2}a_{2}}{dx^{2}}\Bigg{)}\frac{d^{2}}{dx^{2}}+\bigg{(}2a_{2}\frac{da_{2}}{dx} \bigg{)}\frac{d^{3}}{dx^{3}}\Bigg{]}\delta\big{(}x-x^{\prime}\big{)} \tag{49}\] \[\frac{1}{2}\tau^{2}\chi_{2,j} =\frac{1}{2}\tau^{2}\int_{y_{i}}^{y_{i}+1}\psi_{2}\big{(}x,x^{ \prime}\big{)}J_{j}\big{(}x^{\prime}-x_{i}\big{)}dx^{\prime}\] \[=\frac{1}{2}\Bigg{\{}\Bigg{[}\Pi_{2}^{2}(x_{i})\left.\frac{da_{2} }{dt}\right|_{t=t_{0}}+\Pi_{2}^{4}(x_{i})\frac{d^{2}a_{2}}{d\zeta^{2}}\Bigg{]} \frac{d^{2}}{d\zeta^{2}}\Big{[}J_{j}(\zeta)\big{]}+\bigg{[}2\Pi_{2}^{4}(x_{i}) \frac{d\theta_{2}}{d\zeta}\bigg{]}\frac{d^{3}}{d\zeta^{3}}\big{[}J_{j}(\zeta) \big{]}\Bigg{\}}(\zeta=0),\]  
where we have introduced a _non-dimensional_ set of variables to recover the CFL number evaluated at point \(x_{i}\), taking profit that in general, both physically and mathematically, the transport coefficient for a purely diffusive equation is different from zero (see some examples of exceptions for this rule in [42]),  
\[\Pi_{2}(x_{i})=\frac{\sqrt{\tau a_{2}(x_{i},t_{0})}}{\Delta x},\ \ \delta_{2}(x,t)=\frac{a_{2}(x,t)}{a_{2}(x_{i},t_{0})},\ \ \mathfrak{t}=\frac{t}{\tau},\ \ \xi=\frac{1}{\Delta x}(x-x_{i}). \tag{50}\]  
Recall that the while the correction terms \(\psi_{2}\) are evaluated in the initial time frame as they are part of the Taylor series expansion of the Green function, they include also the temporal variation of the transport coefficient, that will be, in general, as small as the time step, making this term almost negligible under the short-time approximation. As a result, the propagation of the unknown function due to the correction terms will be,  
\[u_{cor}(x_{i},t_{0}+\tau)=\overbrace{\bigg{(}\Pi_{2}^{2}\frac{d \theta_{2}}{dt}+\Pi_{2}^{4}\frac{d^{2}\theta_{2}}{d\zeta^{2}}\bigg{)}}\delta_{2 +1}^{2}[u](x=x_{i},t=t_{0})+ \tag{51}\] \[\overbrace{\bigg{(}6\Pi_{2}^{4}\frac{d\theta_{2}}{d\zeta}\bigg{)} }\delta_{2+1}^{3}[u](x=x_{i},t=t_{0}),\]  
and, consequently, they can be introduced at the general scheme, c.f., Equation (45), as additional terms at the propagation. Therefore, each time step the three derivatives \(\frac{d\theta_{2}}{dt}\), \(\frac{d\theta_{2}}{d\zeta}\), \(\frac{d\theta_{2}}{d\zeta^{2}}\) must be computed before the time propagation occurs. While originally these corrections were of order \(\tau^{2}\), once the discretization appears this order of magnitude is piloted by both the CFL and the \(\delta_{2t+1}\) operator. For a sufficiently small grid and a sufficiently well behaved unknown, as the \(\delta_{2t+1}\) operator does not include the grid step in the definition we are using, the values produced by this operator will be of the order of its modulus of continuity [67]or any other measure of its regularity.