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  "body": "\n## Definition\n\nConsider an evolution equation $u_t = \\mathcal{A}(u)$ for an unknown $u(x, t)$, $x \\in \\Omega \\subseteq \\R^m$, $t \\ge 0$.\nHere $\\mathcal{A}$ is a differential operator in the space variable $x$.\nBoundary conditions and an initial value $u(\\cdot, 0) = u_0$ are given.\nThe *method of lines* computes an approximate solution in two steps.\n\n1. Semi-discretisation: choose a spatial discretisation with $N$ unknowns, such as [[finite differences|X7HYEBA5SA]] on a grid. Choose an operator $A_N \\colon \\R^N \\to \\R^N$ that approximates $\\mathcal{A}$ with the boundary conditions built in. This gives the system of ordinary differential equations $U'(t) = A_N(U(t))$, with $U(0)$ an approximation of $u_0$.\n2. Time integration: solve this [[initial value problem|0FXDW992DS]] with a [[numerical method for ordinary differential equations|558XADG64M]].\n\n## Remarks\n\nExample: for the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ 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 $U_0 = U_{N+1} = 0$.\nStep 2 with the [[Euler method|0JARTXX4XP]] 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 $r = \\Delta t / \\Delta x^2$.\nThe 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 $r \\le 1/2$.\nNon-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.\nThe semi-discrete systems are often stiff, because their eigenvalues grow like $\\Delta x^{-2}$ for diffusion, so implicit or stabilised time integrators are used.\n\n## Explanations\n\n### L1 human\n\nDiscretise space, keep time continuous, and hand the resulting ODE system to an ODE solver.\nThe two choices, in space and in time, can then be made and analysed separately.\n",
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