# 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.

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.

• Are you certain that it's elliptic for the regime you're interested in? Since it's nonlinear, its character may depend on the solution. For instance, if $F=1$, the first equation is hyperbolic (though I'm uncertain what to make of the second). – David Ketcheson Jul 13 '12 at 3:07