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  "body": "\n## Summary\n\nThe Lax–Friedrichs method is an explicit [[finite difference method|X7HYEBA5SA]] for hyperbolic equations such as [[conservation laws|8RA7AMWSAF]].\nIt replaces the value at the current point by the average of its two neighbours and then applies a centred difference of the flux.\nThe averaging adds numerical viscosity, which makes the scheme stable under the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]].\nIt is first-order accurate and smears shocks, but it is simple and robust.\nSee the [[definition|53H8EVRC1J]].\n\n## Explanations\n\n### L1 human\n\nTo update a value on a grid, a naive scheme uses the old value plus a centred estimate of the change, and this blows up.\nThe Lax–Friedrichs method uses the average of the two neighbouring old values instead of the old value itself.\nThis small change acts like a little diffusion, and it damps the growing oscillations.\nFor the equation $u_t + a u_x = 0$, the new value is a weighted average of the two neighbours when $\\abs{a} \\Delta t \\le \\Delta x$.\n",
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