{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7M6MQ2HB1X"
    },
    {
      "field": "uses",
      "target": "F7H4H711MY"
    },
    {
      "field": "uses",
      "target": "M7XKVNATFA"
    },
    {
      "field": "uses",
      "target": "YY6P3RSDWC"
    },
    {
      "field": "defines",
      "target": "GJQVTDHKM2"
    },
    {
      "field": "about",
      "target": "M7XKVNATFA"
    }
  ],
  "body": "\n## Summary\n\nNumerical methods for ordinary differential equations compute approximate values of the solution of an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$, at a finite set of times.\nMost [[ordinary differential equations|0NT64JV96A]] have no solution in closed form, so these methods are the main way to solve them in science and engineering.\nThe two big families are one-step methods, such as the [[Euler method|0JARTXX4XP]] and the [[Runge–Kutta methods|7DBPJKYC6C]], and [[linear multistep methods|BX4D85A79P]].\nTheir theory studies consistency, measured by the [[local truncation error|T220HF0562]], together with stability and convergence.\nIt pays special attention to [[stiff equations|RR5FCG92PN]] and to [[adaptive step size|6MREWBZR7R]] control.\n\n## Explanations\n\n### L1 human\n\nA differential equation tells how fast a quantity changes, given its current value.\nA numerical method starts from the known initial value and takes small time steps, using the rate of change to predict the next value each time.\nFor $y' = y$, $y(0) = 1$, steps of length $0.1$ with the rule \"new value equals old value times $1.1$\" give $1.1^{10} \\approx 2.594$ at time $1$, close to the true value $e \\approx 2.718$.\nBetter methods sample the rate of change more cleverly and get much closer with the same number of steps.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "65L05",
      "65L06",
      "65L20"
    ],
    "name": "Numerical methods for ordinary differential equations",
    "same_as": {
      "wikidata": "Q541182",
      "wikipedia": "Numerical methods for ordinary differential equations"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations"
      }
    ],
    "tag": "558XADG64M"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/558XADG64M/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/558XADG64M.md",
  "msc": [
    "65L05",
    "65L06",
    "65L20"
  ],
  "name": "Numerical methods for ordinary differential equations",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/558/558XADG64M.md",
  "status": null,
  "tag": "558XADG64M",
  "uri": "m2://558XADG64M",
  "url": "https://tamnd.github.io/math2/wiki/numerical-methods-for-ordinary-differential-equations/",
  "writing": {
    "tag": "558XADG64M",
    "kind": "concept",
    "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
      }
    ]
  }
}