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  "body": "\n## Definition\n\nLet $a \\in \\R$, and consider the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$.\nFix $\\Delta x \u003e 0$ and $\\Delta t \u003e 0$, put $\\nu = a \\Delta t / \\Delta x$, and let $u_j^n$ approximate $u(j \\Delta x, n \\Delta t)$.\nThe first-order *upwind scheme* is the explicit [[finite difference|X7HYEBA5SA]] scheme\n$$u_j^{n+1} = \\begin{cases} u_j^n - \\nu \\left( u_j^n - u_{j-1}^n \\right) \u0026 \\text{if } a \\ge 0, \\\\ u_j^n - \\nu \\left( u_{j+1}^n - u_j^n \\right) \u0026 \\text{if } a \u003c 0. \\end{cases}$$\nEquivalently, with $a^+ = \\max(a, 0)$ and $a^- = \\min(a, 0)$,\n$$u_j^{n+1} = u_j^n - \\frac{\\Delta t}{\\Delta x} \\left( a^+ (u_j^n - u_{j-1}^n) + a^- (u_{j+1}^n - u_j^n) \\right).$$\n\n## Remarks\n\nExample: let $a \u003e 0$ and $0 \\le \\nu \\le 1$.\nThen $u_j^{n+1} = (1 - \\nu) u_j^n + \\nu u_{j-1}^n$ is a convex combination, so $\\min_j u_j^n \\le u_j^{n+1} \\le \\max_j u_j^n$, and the scheme is stable in the maximum norm.\nThe condition $\\nu \\le 1$ is the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]]: the numerical domain of dependence must contain the characteristic through the point.\nNon-example: for $a \u003e 0$, the downwind scheme $u_j^{n+1} = u_j^n - \\nu (u_{j+1}^n - u_j^n)$ uses the wrong side.\nIt multiplies the Fourier mode $e^{i j \\xi}$ by $g = 1 + \\nu - \\nu e^{i\\xi}$, with $\\abs{g}^2 = 1 + 2\\nu(1 + \\nu)(1 - \\cos \\xi) \u003e 1$ for $\\xi \\ne 0$, so it is unstable for every $\\nu \u003e 0$.\nFor a scalar conservation law $u_t + f(u)_x = 0$, upwinding takes the difference on the side given by the sign of the characteristic speed $f'(u)$.\nThe scheme is first-order accurate in $\\Delta x$ for fixed $\\nu$.\n\n## Explanations\n\n### L1 human\n\nThe difference is taken towards the side the wave comes from.\nThis simple rule is what makes the scheme stable.\n",
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