Finite-Difference Methods Consider the spatially-periodic initial value problem for \(v(x,t)\): \[ v_t=cv_x,\qquad \text {for }0\le x\le 1 \] with \[ v(x,0)=f(x),\qquad \text {for }0\le x\le 1. \] Here the function \(f\) has period 1, and \(c\) is a positive constant. Note that the exact solution of the problem is \(v(x,t)=f(x+ct)\).
Consider the following difference approximations to the PDE: \[ \begin {aligned} \text {i)}\quad &u_j^{n+1}=u_j^n+\frac {ck}{h}(u_{j+1}^n-u_j^n),\\ \text {ii)}\quad &u_j^{n+1}=u_j^n+\frac {ck}{h}(u_j^n-u_{j-1}^n),\\ \text {iii)}\quad &u_j^{n+1}=u_j^n+\frac {ck}{2h}(u_{j+1}^n-u_{j-1}^n). \end {aligned} \] In each case, \(Nh=1\), and the difference scheme is supplemented with the discrete initial conditions \(u_j^0=f(jh)\) for \(j=0,1,\ldots ,N\) and boundary conditions \(u_N^n=u_0^n\), \(u_{-1}^n=u_{N-1}^n\), and \(u_{N+1}^n=u_1^n\) for all time levels \(n\ge 0\). For each scheme, tell whether it is a reasonable scheme to use on the above problem, and justify your answers.