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  "body": "\n## Statement\n\nConsider the [[heat equation|VP2HN86VAG]] $u_t = u_{xx} + u_{yy}$ on the square $(0, 1)^2$, with $u = 0$ on the boundary and a given initial value.\nLet $M \\ge 1$, $h = 1/(M + 1)$, $\\Delta t \u003e 0$, and let $u^n_{jk}$ approximate $u(jh, kh, n \\Delta t)$ for $1 \\le j, k \\le M$, with zero values on the boundary.\nLet $\\delta_x^2$ and $\\delta_y^2$ be the second differences $(\\delta_x^2 v)_{jk} = (v_{j+1,k} - 2 v_{jk} + v_{j-1,k}) / h^2$ and $(\\delta_y^2 v)_{jk} = (v_{j,k+1} - 2 v_{jk} + v_{j,k-1}) / h^2$.\nThe Peaceman–Rachford alternating-direction implicit method computes\n$$\\left(I - \\tfrac{\\Delta t}{2} \\delta_x^2\\right) u^{*} = \\left(I + \\tfrac{\\Delta t}{2} \\delta_y^2\\right) u^n, \\qquad \\left(I - \\tfrac{\\Delta t}{2} \\delta_y^2\\right) u^{n+1} = \\left(I + \\tfrac{\\Delta t}{2} \\delta_x^2\\right) u^{*}.$$\n\n1. Each half step consists of independent tridiagonal systems, one for each grid line in the $x$ direction or in the $y$ direction.\n2. The method is unconditionally stable: $\\norm{u^{n+1}}_2 \\le \\norm{u^n}_2$ for all $\\Delta t \u003e 0$ and $h$, where $\\norm{v}_2^2 = h^2 \\sum_{j,k} v_{jk}^2$.\n3. For smooth solutions, the method is second-order accurate: its local truncation error is $O(\\Delta t^2 + h^2)$.\n\n## Remarks\n\nThe matrices $\\delta_x^2$ and $\\delta_y^2$ are symmetric, negative definite and commute, with common eigenvectors $\\sin(p \\pi j h) \\sin(q \\pi k h)$.\nOn such an eigenvector, with eigenvalues $-\\alpha$ and $-\\beta$ ($\\alpha, \\beta \u003e 0$), one step multiplies by\n$$g = \\frac{1 - \\frac{\\Delta t}{2} \\alpha}{1 + \\frac{\\Delta t}{2} \\alpha} \\cdot \\frac{1 - \\frac{\\Delta t}{2} \\beta}{1 + \\frac{\\Delta t}{2} \\beta},$$\nand each factor has absolute value less than $1$; this gives clause 2.\nEliminating $u^*$ shows that the method equals the two-dimensional [[Crank–Nicolson method|M92S0Y15N1]] plus the term $\\frac{\\Delta t^2}{4} \\delta_x^2 \\delta_y^2 (u^{n+1} - u^n)$, which is $O(\\Delta t^3)$ for smooth solutions; this gives clause 3.\nEach tridiagonal system is solved in linear time by the [[tridiagonal matrix algorithm|F1QWA1PEQQ]], so a step costs $O(M^2)$ operations, against a sparse system with $M^2$ unknowns for Crank–Nicolson.\n\n## Explanations\n\n### L1 human\n\nImplicit methods for the heat equation in two dimensions lead to a large linear system at every step.\nThe alternating-direction implicit method splits each step in two halves.\nIn the first half it is implicit only along the rows of the grid, in the second only along the columns.\nSo it solves many small tridiagonal systems instead of one large one, and it keeps the stability and the second-order accuracy of Crank–Nicolson.\n\n### L2 human\n\nThe method is an operator splitting of $\\delta_x^2 + \\delta_y^2$, and its analysis uses that the two pieces commute; for variable coefficients or non-rectangular domains this fails and the analysis is harder.\nThe same iteration, used with varying parameters, is an iterative solver for the discrete Poisson equation and for Lyapunov and Sylvester matrix equations.\nIn three dimensions one usually uses the Douglas form of the method instead.\n",
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