4 Stability

There is a stability constraint that is common to many methods for obtaining numerical solutions of the wave equation. Issues relating to stability of numerical methods can be extremely complicated, but the following Key Point is enough for our purposes.

This is called the CFL condition (named after an acronym of three mathematicians Courant, Friedrichs and Lewy).

Exercises

1. Suppose that $u=u\left(x,t\right)$ satisfies the wave equation $u_{tt}=c^2{u}_{xx}$ in $t>0$ and $0<x<0.6$ . It is given that $u$ satisfies boundary conditions $u\left(0,t\right)=u\left(0.6,t\right)=0$ $\left(t>0\right)$ and initial conditions that need not be stated for the purposes of this question. The application is such that the wave speed $c=1.4$ .
   
   The numerical method

   $u_j^{n+1}=2u_j^n-u_j^{n+1}+{\mu }^2\left(u_{j+1}^n-2u_j^n+u_{j-1}^n\right)$

   where $\mu =c\delta t∕\delta x$ , is implemented using $\delta x=0.15$ and $\delta t=0.1$ .
   
   Suppose that, after 7 time-steps, the following data forms part of the numerical solution:

   $\begin{array}{ccccccc}\hfill u_0^6& \hfill =\hfill & 0.0000\hfill & \hfill \hfill & \hfill u_0^7& \hfill =\hfill & 0.0000\hfill \\ \hfill u_1^6& \hfill =\hfill & 0.1024\hfill & \hfill \hfill & \hfill u_1^7& \hfill =\hfill & 0.0997\hfill \\ \hfill u_2^6& \hfill =\hfill & 0.1986\hfill & \hfill \hfill & \hfill u_2^7& \hfill =\hfill & 0.1730\hfill \\ \hfill u_3^6& \hfill =\hfill & 0.2361\hfill & \hfill \hfill & \hfill u_3^7& \hfill =\hfill & 0.1169\hfill \\ \hfill u_4^6& \hfill =\hfill & 0.0000\hfill & \hfill \hfill & \hfill u_4^7& \hfill =\hfill & 0.0000\hfill \end{array}$

   Carry out the next time-step so as to find an approximation to $u$ at $t=8\delta t$ .

2. Suppose that $u=u\left(x,t\right)$ satisfies the wave equation $u_{tt}=c^2{u}_{xx}$ in $t>0$ and $0<x<1$ . It is given that $u$ satisfies boundary conditions $u\left(0,t\right)=u\left(1,t\right)=0$ $\left(t>0\right)$ . The initial elevation may be summarised as

   $\begin{array}{ccccccccccc}\hfill f_0& \hfill =\hfill & 0.0000\hfill & \hfill \hfill & \hfill f_1& \hfill =\hfill & 0.7812\hfill & \hfill \hfill & \hfill f_2& \hfill =\hfill & 0.2465\hfill \\ \hfill f_3& \hfill =\hfill & -0.1209\hfill & \hfill \hfill & \hfill f_4& \hfill =\hfill & 0.0000\hfill \end{array}$

   and the string is initially at rest (that is, $g\left(x\right)=0$ ). The application is such that the wave speed $c=1$ .
   
   Carry out the first two time-steps of the numerical method

   $u_j^{n+1}=2u_j^n-u_j^{n-1}+{\mu }^2\left(u_{j+1}^n-2u_j^n+u_{j-1}^n\right)$

   where $\mu =c\delta t∕\delta x$ in which $\delta x=0.25$ and $\delta t=0.2$ .
   
   

Answer