# 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
Using this new grid position as the starting point for the propagation of the next time frame simplifies the _simple layer_ integral to the very same (46) solution with the CFL number evaluated at those points, with the corresponding local grid step in its definition. This is the starting point for the well-known Roe-like or Gudonov-like numerical schemes for solving advective equations, often applied to the Euler equations and perturbations over them towards the Navier-Stokes system [66]. Furthermore, in a similar manner, Lagrangian particle methods use this perspective of the information propagation to define the trajectories of their particles or grid points at which the rest of the solution is solved, see for example SPH methods [16]. We present an example of this simple test case for the diffusive and advective problems, for which it is possible to obtain an analytic solution--as a matter of fact, from the integral scheme--defined over a ring for different stencils and CFLs. In this basic form, the algorithm is equivalent to a piecewise homogeneously interpolated partial differential equation, i.e., a finite difference scheme. Moreover, combining this homogeneous discretization with the Taylor series definition of the Green function we recover a numerical scheme that is equivalent explicit Euler scheme, where the time derivative has been expanded up to the order marked by the stencil. The initial condition chosen for this problem is a sine that will advect and diffuse at a _speed_ according to its frequency,  
\[u(x,t_{0})=\frac{1}{2}+\frac{1}{2}\text{sin}(2\pi x)\to \begin{cases}\text{ First}-\text{order}:u_{1}(x,t)=\frac{1}{2}+\frac{1}{2}\text{sin}[2\pi x+(t-t_{0})a_{1}(2 \pi)]\\ \text{First}-\text{order}:u_{2}(x,t)=\frac{1}{2}+\frac{1}{2}e^{-(t-t_{0})a_{2}( 2\pi)^{2}}\text{sin}(2\pi x)\end{cases} . \tag{48}\]  
We will have then for the first-order equation a sine that travels through the ring periodically for the advection equation, and, for the diffusion equation a vanishing sine whose steady state is equal to \(1/2\). The relative integrated error, \(\varepsilon_{\text{Adv}},\varepsilon_{\text{Diff}}\), between the exact solution, Equation (48) and the one produced by the algorithm are presented for the advective and diffusive test cases on Figures 5 and 6, respectively. This relative error is computed by numerically integrate the differences between the analytic and numerical solutions and refering it to the maximum value of the analytic solution at the corresponding time frame, \(t\). We compare the performance of the algorithm for several stencil sizes, defined by the parameter \(q\) and different CFL numbers, that are selected by means of the total number of points of the grid, \(N\) in order to perform an equivalent time-evolving comparison. From these results we can extract the behavior of the linear propagation of the algorithm, showing a fast convergence for the stencil size and a well behaved time evolving error, where, once the numerical precision is reached for a certain stencil. Due to the pseudospectra of the exponential matrix, which will be an approximation of that of the linear operators, the error accumulates with each time iteration, yet it does at a logarithm rate, hence it can be assumed to be controlled and so, the algorithm will produce a solution consistent with the exact one. As the advective problem is a displacing sine, the established solution will be periodic, hence the propagation of the error will keep going, as it can be seen in Figure 5, even when it is the truncation error the one being propagated. In the same perspective we can see that the influence of the CFL, as it is always kept low enough not to affect the stability of the solution, can only be appreciated for the diffusive problem, where the highest CFL (dashed lines) produce a significantly higher error than the rest while still remaining bounded.