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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^m$ be open and let $F \\colon \\Omega \\to \\R^m$ be the $C^1$ flux of a [[hyperbolic conservation law|XN081NCPF5]] $u_t + F(u)_x = 0$.\nA *Riemann solver* for $F$ is a map $H \\colon \\Omega \\times \\Omega \\to \\R^m$.\nIts value $H(u_l, u_r)$ is the numerical flux at an interface between a left state $u_l$ and a right state $u_r$.\nIt is obtained from an exact or approximate solution of the [[Riemann problem|GJWKX22VVE]] with data $u_l, u_r$.\nIt is required to be:\n\n1. consistent: $H(u, u) = F(u)$ for all $u \\in \\Omega$;\n2. locally Lipschitz continuous in both arguments.\n\nThe *exact Riemann solver* is $H(u_l, u_r) = F(w(0; u_l, u_r))$, where $w(x/t; u_l, u_r)$ is the self-similar [[entropy solution|1A01A3EJRX]] of the Riemann problem, when it exists and is unique.\n\n## Remarks\n\nExample: the HLL solver uses two estimates $S_L \u003c S_R$ of the slowest and fastest wave speeds.\nIt sets $H = F(u_l)$ if $0 \\le S_L$, $H = F(u_r)$ if $S_R \\le 0$, and otherwise\n$$H(u_l, u_r) = \\frac{S_R F(u_l) - S_L F(u_r) + S_L S_R (u_r - u_l)}{S_R - S_L}.$$\nFor $u_l = u_r = u$ every case gives $F(u)$, so this solver is consistent.\n\nNon-example: $H(u_l, u_r) = F(u_l) + F(u_r)$ fails clause 1, since $H(u, u) = 2F(u)$.\nA scheme with such a flux approximates the wrong equation.\n\nFor the scalar equation $u_t + a u_x = 0$, the exact solver gives $H(u_l, u_r) = a u_l$ if $a \u003e 0$ and $a u_r$ if $a \u003c 0$, which is the upwind flux.\n",
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