Which discretization scheme to use for elliptic PDE?
While simulating motion of nonlinear inelastic wire one meets the following equations \begin{align} &{\partial^2\varphi\over\partial t^2}=2{\partial F\over\partial s}{\partial\varphi\over\partial s}+F{\partial^2\varphi\over\partial s^2} \ &{\partial^2F\over\partial s^2}+\left({\partial\varphi\over\partial t}\right)^2-F\left({\partial\varphi\over\partial s}\right)^2=0\;. \end{align} Here coordinates of wire are given by \begin{equation} {\partial x\over\partial s}=\cos\varphi\qquad\qquad{\partial y\over\partial s}=\sin\varphi \;. \end{equation} What is the best discretization scheme to use for this elliptic equation with regard to this specific application? How can this be determined?
I tried the following finite difference schemes on nonstaggered grid \begin{align} {\partial^2\varphi\over\partial t^2} &\approx \frac{\varphi(t+\Delta t) - 2 \varphi(t) + \varphi(t-\Delta t)}{{\Delta t}^{2}}+\mathcal{O}\left(\Delta t^2\right)\ {\partial^2\varphi\over\partial s^2} &\approx \frac{\varphi(s+\Delta s) - 2 \varphi(s) + \varphi(s-\Delta s)}{{\Delta s}^{2}}+\mathcal{O}\left(\Delta s^2\right) \ {\partial^2F\over\partial s^2} &\approx \frac{F(s+\Delta s) - 2 F(s) + F(s-\Delta s)}{{\Delta s}^{2}}+\mathcal{O}\left(\Delta s^2\right) \ {\partial \varphi\over\partial s} &\approx \frac{\varphi(s+\Delta s) - \varphi(s-\Delta s)}{2{\Delta s}}+\mathcal{O}\left(\Delta s^2\right) \ {\partial \varphi\over\partial t} &\approx \frac{3\varphi(t) - 4\varphi(t-\Delta t) + \varphi(t-2\Delta t)}{2{\Delta t}}+\mathcal{O}\left(\Delta t^2\right) \ {\partial F\over\partial s} &\approx \frac{F(s+\Delta s) - F(s-\Delta s)}{2{\Delta s}}+\mathcal{O}\left(\Delta s^2\right) \end{align} From here the following equations are obtained, for elliptic equation \begin{equation} F_{m+1}+F_{m}\left[-2-{1\over4}\left(\varphi_{m+1,n}-\varphi_{m-1,n}\right)^2\right]+F_{m-1}=-\left({\Delta s\over2\Delta t}\right)^2\left(3\varphi_{m,n}-4\varphi_{m,n-1}+\varphi_{m,n-2}\right)^2 \end{equation} and for explicit timestep \begin{equation} \varphi_{m,n+1}=2\varphi_{m,n}-\varphi_{m,n-1}+\left({\Delta t\over\Delta s}\right)^2\bigg[{1\over2}\left(F_{m+1}-F_{m-1}\right)\left(\varphi_{m+1,n}-\varphi_{m-1,n}\right)+F_{m}\left(\varphi_{m+1,n}-2\varphi_{m,n}+\varphi_{m-1,n}\right)\bigg]\;. \end{equation} Here index $m$ denotes variable $s$ and $n$ stands for different times, we thus need to keep track of $\varphi$ at three different times. 935