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Explicit Numerical Integrators - Revision history
2024-03-28T12:48:38Z
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Jules: Created page with "== Modified Euler Method == The modified Euler (or Heun's) method is a two-stage [http://en.wikipedia.org/wiki/Predictor%E2%80%93corrector_method predictor-corrector method]:..."
2020-11-22T08:14:03Z
<p>Created page with "== Modified Euler Method == The modified Euler (or Heun's) method is a two-stage [http://en.wikipedia.org/wiki/Predictor%E2%80%93corrector_method predictor-corrector method]:..."</p>
<p><b>New page</b></p><div>== Modified Euler Method ==<br />
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The modified Euler (or Heun's) method is a two-stage [http://en.wikipedia.org/wiki/Predictor%E2%80%93corrector_method predictor-corrector method]:<br />
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Predictor stage:<br />
: <math> \boldsymbol{\tilde{x}}(t + \Delta t) = \boldsymbol{x}(t) + \Delta t \boldsymbol{f}(\boldsymbol{x}(t), t) \, </math><br />
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Corrector stage:<br />
: <math> \boldsymbol{x}(t + \Delta t) = \boldsymbol{x}(t) + \frac{\Delta t}{2} \left[ \boldsymbol{f}(\boldsymbol{x}(t), t) + \boldsymbol{f}(\boldsymbol{\tilde{x}}(t + \Delta t), t) \right] \, </math><br />
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== 4th-Order Runge Kutta Method ==<br />
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The 4th-order Runge-Kutta algorithm is one of the most popular numerical integration methods for power systems. <br />
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: <math>\boldsymbol{k}_{1} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t), t) \, </math><br />
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: <math>\boldsymbol{k}_{2} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \frac{\boldsymbol{k}_{1}}{2}, t + \frac{\Delta t}{2}) \, </math><br />
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: <math>\boldsymbol{k}_{3} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \frac{\boldsymbol{k}_{2}}{2}, t + \frac{\Delta t}{2}) \, </math><br />
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: <math>\boldsymbol{k}_{4} = \Delta t \boldsymbol{f}(\boldsymbol{x}(t) + \boldsymbol{k}_{3}, t + \Delta t) \, </math><br />
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: <math> \boldsymbol{x}(t + \Delta t) = \boldsymbol{x}(t) + \frac{1}{6} \left( \boldsymbol{k}_{1} + 2 \boldsymbol{k}_{2} + 2 \boldsymbol{k}_{3} + \boldsymbol{k}_{4} \right) \, </math><br />
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[[Category:Modelling / Analysis]]</div>
Jules