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  "body": "\n## Summary\n\nThe finite difference method solves differential equations approximately on a grid of points.\nIt replaces each derivative by a difference quotient of the values at nearby grid points, which turns the equation into a system of algebraic equations.\nIt is the oldest and simplest numerical method for [[partial differential equations|GC5BDZ6QBC]].\nIts convergence rests on consistency and stability, as in the [[Lax equivalence theorem|D5M7TRD3MY]].\nFor time-dependent problems, stability often requires a step restriction such as the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]].\n\n## Explanations\n\n### L1 human\n\nA derivative is a limit of slopes $(u(x + h) - u(x))/h$ as $h \\to 0$.\nThe finite difference method simply stops before the limit and uses a small fixed $h$.\nPut points $0, h, 2h, \\ldots$ on the line and keep track only of the values of the unknown function there.\nFor example, $u''(x)$ is replaced by $(u(x + h) - 2u(x) + u(x - h))/h^2$, which is exact for quadratic polynomials.\nThe equation then becomes one linear equation per grid point, which a computer can solve.\n",
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