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  "body": "\n## Summary\n\nA stiff equation is an ordinary differential equation on which explicit numerical methods must take extremely small steps to stay stable.\nThis happens even where the solution is smooth, for example with the [[Euler method|0JARTXX4XP]].\nStiffness typically comes from processes that decay on very different time scales.\nStiff equations are solved with implicit methods, which remain stable with large steps.\nSee the [[definition|AXND5JRQAZ]].\n\n## Explanations\n\n### L1 human\n\nImagine a system with one part that settles in a millisecond and another part that changes over hours.\nAfter the first millisecond the fast part is at rest, and the solution looks slow and smooth.\nStill, a simple step-by-step method must keep tiny steps, or its errors in the fast part blow up.\nFor $y' = -1000\\,(y - \\cos t)$ the solution quickly follows $\\cos t$, yet the explicit Euler method needs steps below $0.002$.\n",
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