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  "body": "\n## Summary\n\nA MUSCL scheme is a [[finite volume method|HGKN8E09RS]] for [[conservation laws|8RA7AMWSAF]] that replaces the constant cell values of [[Godunov's scheme|43HZJ7DRT0]] by limited linear reconstructions.\nThe name stands for Monotonic Upstream-centred Scheme for Conservation Laws, and the approach goes back to Bram van Leer.\nThe reconstruction gives second-order accuracy where the solution is smooth.\nA [[flux limiter|NZ8XFNCKQW]] switches the slopes off near jumps, so shocks are captured without spurious oscillations.\nSee the [[definition|SB1TM11S5Q]].\n\n## Explanations\n\n### L1 human\n\nGodunov's scheme pretends that the solution is flat inside each cell, which smears out every feature.\nA MUSCL scheme draws a short straight line in each cell instead, with a slope taken from the neighbouring cells.\nNear a jump such a line could overshoot, so a limiter flattens the slope there.\nFor data on a straight line, such as $u_j = j$, the reconstruction hits the exact value at every cell edge.\nFor a step from $0$ to $1$, the limiter sets the slopes to zero next to the step, and no new maximum or minimum appears.\n",
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