Which discretization scheme to use for elliptic PDE?

Question

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.

Also if initial position of wire is near horizontal (far from equilibrium), then the algorithm becomes unstable after some time, are there ways to improve stability? The problem seems to be that when the free part of wire gets too curved the forces raise and the triangular matrix from elliptic equation becomes near to singular.

Answer 1

I can't give you a discretization solution that WILL work, but you might consider implicit in the space dimensions rather than explicit. Also from what I can tell you can likely use von neumann analysis to determine the steps sizes such that you will have a stable solution with the explicit space dimension steps.