Crank–Nicolson method

Statement

Let $\kappa > 0$ and consider the heat equation $u_t = \kappa u_{xx}$ on $0 < x < 1$, $t > 0$, with $u(0, t) = u(1, t) = 0$ and a given initial value $u(x, 0)$. Let $M \ge 1$, $\Delta x = 1/(M + 1)$, $x_j = j \Delta x$, $\Delta t > 0$, $t_n = n \Delta t$ and $r = \kappa \Delta t / \Delta x^2$. For a grid vector $v = (v_0, \ldots, v_{M+1})$ with $v_0 = v_{M+1} = 0$, put $(\delta^2 v)_j = (v_{j+1} - 2 v_j + v_{j-1}) / \Delta x^2$. The Crank–Nicolson method computes $u^{n+1}$ from $u^n$ by $$\frac{u_j^{n+1} - u_j^n}{\Delta t} = \frac{\kappa}{2} \left( (\delta^2 u^{n+1})_j + (\delta^2 u^n)_j \right), \qquad 1 \le j \le M.$$

1. Each step has a unique solution $u^{n+1}$, given by a tridiagonal linear system.
2. The method is unconditionally stable: $\norm{u^{n+1}}_{2} \le \norm{u^n}_2$ for every $r > 0$, where $\norm{v}_2^2 = \Delta x \sum_{j=1}^M v_j^2$.
3. Assume the exact solution has bounded derivatives $u_{ttt}$, $u_{xxtt}$ and $u_{xxxx}$. Then the local truncation error is $O(\Delta t^2 + \Delta x^2)$. On each interval $[0, T]$ the error $\max_{n \Delta t \le T} \norm{u(\cdot, t_n) - u^n}_2$ is also $O(\Delta t^2 + \Delta x^2)$.