# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 1 Introduction  
In this work, we explore the potential of the weak solution of a general partial differential equation (PDE) as an efficient, high-order numerical scheme. In particular, we are interested in PDEs that commonly arise in the modeling of continuum and physical media, such as balance-like differential equations. These equations relate the time variation of a physical magnitude to the spatial variation of its flux and the production or destruction terms--which tend to be nonlinear. This introductory approach serves as the first step towards a numerical methodology to solve such nonlinear PDEs with arbitrary boundary conditions by means of their weak or integral solution, defined by the Green function of the problem. Therefore, we first focus on the mathematical description of the integral solution, demonstrating how the Green function defines its properties and affects the initial and boundary conditions, before addressing the peculiarities of its numerical transposition into a time-evolving scheme. Finally, we demonstrate and validate the potential of the methodology through simple yet illustrative toy models. Consequently, we have organized the paper into two main sections: a description of the mathematical foundations of the method and its translation into a numerical algorithm to solve the time-evolving problem. In the first section, we begin with the generalintegral expression of the solution and then separate each of its contributions, identifying a quasilinear term, a nonlinear term, and a boundary term, the details of which will be introduced and analyzed. The only strong assumption required for this methodology is the short-time approximation, i.e., that the time intervals at which we obtain the solution are separated by a short time-step, \(\tau\ll 1\). This approximation is necessary for handling the integral terms numerically and, more importantly, for obtaining the Green function of the problem. Indeed, under this hypothesis, the Green function can be understood as the integral kernel of the solution, propagating or _filtering_ the initial condition from one time frame to the next while satisfying the PDE and its boundary conditions. The nonlinear terms are then included by means of the variation of parameters technique, also through integration involving the Green function. Thus, this function arises as the key element for obtaining the solution to the problem, as it encapsulates the quasilinear part of the PDE. Consequently, we derive its closed analytical form for the PDEs of interest and provide a general series expansion in time for its appropriate implementation. The numerical computation of the integral solution is consequently organized to follow the gradual levels of complexity inherent to the time-evolving solution: the quasilinear propagation of the initial condition, the integration of the nonlinear terms, and the boundary conditions. As can be deduced, the accuracy and convergence of the integral scheme depend on the appropriate implementation of the Green function; thus, each of these steps revolves around its interaction with the function. The first step involves defining the numerical Green function, which is deduced from its combination with a proper spatial representation of the discrete unknown function. We, therefore, introduce an efficient discretization that produces a numerical translation that is both accurate and straightforward, adapting a collocation scheme to ensure convergence and consistency. In this process, the usual Courant-Friedrichs-Lewy (CFL) number naturally arises as a non-dimensional group of the problem, showing the potential of the method for operating outside the usual limitations. Then, we study the nonlinear terms, requiring a numerical investigation of their integration over the time-step. As before, we introduce an interpolation methodology into the integral scheme. In particular, we adapt a numerical quadrature based on Runge-Kutta schemes.