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maikease新蟲 (初入文壇)
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numerical stiffness 怎么理解？ 已有1人參與
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| The Navier-Stokes equations as expressed in Equation XXX become (numerically) very stiff at low Mach number due to the disparity between the fluid velocity and the acoustic speed (speed of sound). This is also true for incompressible flows, regardless of the fluid velocity, because acoustic waves travel infinitely fast in an incompressible fluid (speed of sound is infinite). The numerical stiffness of the equations under these conditions results in poor convergence rates |
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xiegangmai: 金幣+2, 謝謝參與 2012-12-09 10:42:38
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xiegangmai: 金幣+2, 謝謝參與 2012-12-09 10:42:38
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In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proved difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. Sometimes the step size is forced down to an unacceptably small level in a region where the solution curve is very smooth. The phenomenon being exhibited here is known as stiffness. In some cases we may have two different problems with the same solution, yet problem one is not stiff and problem two is stiff. Clearly the phenomenon cannot be a property of the exact solution, since this is the same for both problems, and must be a property of the differential system itself. It is thus appropriate to speak of stiff systems. |
2樓2012-12-09 10:38:04
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