13.1 Difference quotients and consistency

On a grid \(x_j = j\Delta x\), \(t_n = n\Delta t\), write \(u_j^n\) for the computed approximation to \(u(x_j, t_n)\). Taylor’s theorem gives, for smooth \(u\), \[ \frac {u(x + h) - u(x)}{h} = u' + \frac h2u'' + O(h^2), \qquad \frac {u(x + h) - 2u(x) + u(x - h)}{h^2} = u'' + \frac {h^2}{12}u'''' + O(h^4) . \] A scheme is consistent if its truncation error, the residual left when the exact solution is substituted into it, tends to \(0\) with \(\Delta x\) and \(\Delta t\); its order is the power with which it does so. The simplest scheme for \(u_t = \kappa u_{xx}\) is the explicit forward-time centred-space (FTCS) scheme \[ u_j^{n + 1} = u_j^n + r\bigl (u_{j + 1}^n - 2u_j^n + u_{j - 1}^n\bigr ), \qquad r = \frac {\kappa \,\Delta t}{\Delta x^2} , \] with truncation error \(O(\Delta t + \Delta x^2)\).

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