{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nStiffness is not a single precise condition; the definition below is the standard working one.\nLet $I$ be an interval and consider an [[ordinary differential equation|0P97V20RW7]] $y' = f(t, y)$ with $f$ continuously differentiable, on $I$.\nThe equation is *stiff* on $I$ if explicit methods such as the [[Euler method|0JARTXX4XP]] need much smaller steps than accuracy requires.\nThe reason is that larger steps make the numerical solution unstable.\nConsider a linear system $y' = A y + g(t)$ with a constant $n \\times n$ matrix $A$.\nAssume its [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1, \\dots, \\lambda_n$ all have $\\operatorname{Re} \\lambda_i \u003c 0$.\nA common measure is then the stiffness ratio\n$$\\frac{\\max_i \\abs{\\operatorname{Re} \\lambda_i}}{\\min_i \\abs{\\operatorname{Re} \\lambda_i}},$$\nand the system is called stiff when this ratio is large.\n\n## Remarks\n\nExample: $y' = -1000\\,(y - \\cos t)$.\nOn the test equation $y' = \\lambda y$, the Euler method gives $y_{k+1} = (1 + h\\lambda)\\, y_k$, which stays bounded only if $\\abs{1 + h\\lambda} \\le 1$.\nFor $\\lambda = -1000$ this forces $h \\le 0.002$, although after a short time every solution is close to the smooth particular solution $\\frac{10^6}{10^6 + 1} \\cos t + \\frac{10^3}{10^6 + 1} \\sin t$, which is nearly $\\cos t$.\nThe implicit Euler method $y_{k+1} = (1 - h\\lambda)^{-1} y_k$ is stable for all $h \u003e 0$ when $\\lambda \u003c 0$.\nNon-example: $y' = -y$ is not stiff, since its stability bound $h \\le 2$ is far larger than the steps needed for accuracy.\nFor nonlinear equations one uses the eigenvalues of the [[Jacobian matrix|047NDTEPCV]] $\\partial f / \\partial y$ along the solution.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "RR5FCG92PN",
    "kind": "definition",
    "msc": [
      "65L04",
      "34A99"
    ],
    "name": "Stiff equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stiff_equation"
      }
    ],
    "tag": "AXND5JRQAZ",
    "uses": [
      "0P97V20RW7",
      "0JARTXX4XP",
      "0BQAA2BK3Q",
      "047NDTEPCV"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/AXND5JRQAZ/",
  "links": [
    {
      "field": "defines",
      "target": "RR5FCG92PN"
    },
    {
      "field": "uses",
      "target": "0P97V20RW7"
    },
    {
      "field": "uses",
      "target": "0JARTXX4XP"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "047NDTEPCV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AXND5JRQAZ.md",
  "msc": [
    "65L04",
    "34A99"
  ],
  "name": "Stiff equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AXN/AXND5JRQAZ.md",
  "status": null,
  "tag": "AXND5JRQAZ",
  "uri": "m2://AXND5JRQAZ",
  "url": "https://math2.tamnd.com/wiki/stiff-equation-definition/",
  "writing": {
    "tag": "AXND5JRQAZ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}