The non-linear Poisson equation in one-dimension,$$0 =  \frac{\partial^2u}{\partial x^2} - f(u)$$can be discretised as to give,$$u_{j-1} -2u_{j} + u_{j+1} = h^2 f(u_j)$$where $h$ is the step size of the mesh.Is there any advantage (in general or not) to write the non-linear source term as the average value of the solution variable $\bar{u_j}$ over the neighbouring mesh points?For example,$$f(u_j) \rightarrow f(\bar{u_j})$$where, $$\bar{u_j} = \frac{1}{2}\left( u_{j-1} + u_{j+1} \right)$$I have noticed in practice that this sometimes improves the solution stability when using relaxation methods.
For a non-linear PDEs should the source term be discretised at $u_j$ or averaged over $(u_{j+1} + u_{j-1})/2$?
numerical analysis;nonlinear equations;poisson
For your example equation, taking the average approach, the local consistency error $$\frac{1}{h^2}[u(x-h) - 2 u(x) + u(x+h)]-f(\frac{1}{2}[u(x-h)+u(x+h)]) = \frac{1}{2}f_uu_{xx}h^2 + hot.$$will be of order $2$ (instead of order $3$). ($hot.$ means higher order terms)Therefore, if your overall approximation is of order $1$, e.g. if you use upwind somewhere, then it doesn't matter. If you use higher order schemes, then this approach will limit of your convergence rate.