Method of lines

Remarks

Example: for the heat equation $u_t = u_{xx}$ on $(0, 1)$ with zero boundary values and grid $x_j = j \Delta x$, $\Delta x = 1/(N + 1)$, step 1 gives $U_j' = (U_{j+1} - 2U_j + U_{j-1}) / \Delta x^2$ for $1 \le j \le N$, with $U_0 = U_{N+1} = 0$. Step 2 with the Euler method and step size $\Delta t$ gives the explicit scheme $U_j^{n+1} = U_j^n + r (U_{j+1}^n - 2U_j^n + U_{j-1}^n)$ with $r = \Delta t / \Delta x^2$. The eigenvalues of the semi-discrete matrix lie in $(-4/\Delta x^2, 0)$, and the Euler method is stable for $\Delta t \lambda \in [-2, 0]$, so the scheme is stable when $r \le 1/2$. Non-example: Rothe's method discretises time first, which gives a sequence of elliptic boundary value problems in $x$; it fails clause 1, since no system of ODEs is formed. The semi-discrete systems are often stiff, because their eigenvalues grow like $\Delta x^{-2}$ for diffusion, so implicit or stabilised time integrators are used.