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  "body": "\n## Definition\n\nLet $x_0, \\ldots, x_{N-1}$ be collocation points in a domain $\\Omega$, and let $V_N$ be a space of global functions with an interpolation property.\nFor every vector of values $(v_j)$ there is a unique $p \\in V_N$ with $p(x_j) = v_j$ for all $j$.\nWrite $I_N v$ for this interpolant.\nA *pseudo-spectral method* for an equation $u_t = \\mathcal{N}(u)$, with $\\mathcal{N}$ a differential operator, is the following scheme:\n\n1. the unknowns are the values $u_j(t) \\approx u(x_j, t)$;\n2. each derivative $\\partial^m u$ is replaced by $\\partial^m (I_N u)$, evaluated at the points $x_j$;\n3. products and other nonlinear functions of $u$ and its derivatives are evaluated pointwise at the $x_j$;\n4. the equation is required to hold at each point $x_j$.\n\nIt is a [[spectral method|MYV657YRDW]] of collocation type.\n\n## Remarks\n\nPeriodic case: $\\Omega = [0, 2\\pi)$, $x_j = 2\\pi j / N$, and $V_N$ the [[trigonometric interpolants|SVABG134WE]] of degree about $N/2$.\nDerivatives are then computed with the [[discrete Fourier transform|TST0GH66A5]]: transform, multiply the coefficient of $e^{ikx}$ by $ik$, and transform back.\nWith the [[fast Fourier transform|S966EZX3ZT]] this costs $O(N \\log N)$ operations.\nOn an interval one often uses Chebyshev points $x_j = \\cos(\\pi j / (N - 1))$ and the [[Chebyshev polynomials|M6EAR1JHPP]].\n\nExample: for $N = 8$ and $u_j = \\sin x_j$, the transform has only the coefficients of $e^{\\pm ix}$.\nMultiplying them by $\\pm i$ and transforming back gives $\\cos x_j$ exactly.\nFor Burgers' equation $u_t + u u_x = 0$, one forms $u_x$ this way and then multiplies by $u_j$ at each point.\n\nNon-example: a spectral Galerkin method computes the product $u u_x$ through its exact Fourier coefficients, a convolution of coefficient sequences.\nIt does not use pointwise evaluation, so clause 3 fails.\n",
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