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  "body": "\n## Summary\n\nThe method of lines solves a time-dependent partial differential equation in two stages.\nFirst only the space variables are discretised, by [[finite differences|X7HYEBA5SA]], finite volumes or finite elements.\nThis turns the equation into a large system of ordinary differential equations in time.\nThen this system is solved with a numerical method for [[initial value problems|0FXDW992DS]].\nThis separation lets one reuse mature ODE solvers, with their error control and their methods for stiff problems.\n\n## Explanations\n\n### L1 human\n\nPicture the space-time plane with a few vertical lines, one for each grid point in space.\nAlong each line the unknown is a function of time only, and the PDE couples each line to its neighbours.\nSo the PDE becomes a system of ODEs, one per line, which a standard ODE solver can handle.\nFor the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$, replacing $u_{xx}$ at $x_j$ by $(u_{j+1} - 2u_j + u_{j-1})/\\Delta x^2$ gives the system $u_j'(t) = (u_{j+1}(t) - 2u_j(t) + u_{j-1}(t))/\\Delta x^2$.\n",
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