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dopri45_solver [2015/08/13 15:00]
dopri45_solver [2015/08/13 15:56]
erik created
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 +Dormand-Prince
 +
 +Explicit one-step solver based on Runge-Kutta (4,5) pair of Dormand and Prince.
 +
 +Applicable for **non-stiff** problems of **medium** accuracy.
 +
 +The [[Butcher tableau]] for the adaptive Dormand–Prince method is:
 +
 +<table class="​Butcher_tableau">​
 +<tr>
 +<td width="​20px"></​td>​
 +<td class="​right">​0</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​right">​1/​5</​td>​
 +<​td>​1/​5</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​right">​3/​10</​td>​
 +<​td>​3/​40</​td>​
 +<​td>​9/​40</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​right">​4/​5</​td>​
 +<​td>​44/​45</​td>​
 +<​td>​-56/​15</​td>​
 +<​td>​32/​9</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​right">​8/​9</​td>​
 +<​td>​19372/​6561</​td>​
 +<​td>​-25360/​2187</​td>​
 +<​td>​64448/​6561</​td>​
 +<​td>​-212/​729</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​right">​1</​td>​
 +<​td>​9017/​3168</​td>​
 +<​td>​-355/​33</​td>​
 +<​td>​46732/​5247</​td>​
 +<​td>​49/​176</​td>​
 +<​td>​-5103/​18656</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​bottomright">​1</​td>​
 +<td class="​bottom">​35/​384</​td>​
 +<td class="​bottom">​0</​td>​
 +<td class="​bottom">​500/​1113</​td>​
 +<td class="​bottom">​125/​192</​td>​
 +<td class="​bottom">​-2187/​6784</​td>​
 +<td class="​bottom">​11/​84</​td>​
 +<td class="​bottom"></​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td style="​border-right:​1px solid;"></​td>​
 +<​td>​5179/​57600</​td>​
 +<​td>​0</​td>​
 +<​td>​7571/​16695</​td>​
 +<​td>​393/​640</​td>​
 +<​td>​-92097/​339200</​td>​
 +<​td>​187/​2100</​td>​
 +<​td>​1/​40</​td>​
 +</tr>
 +<tr>
 +<​td></​td>​
 +<td class="​right"></​td>​
 +<​td>​35/​384</​td>​
 +<​td>​0</​td>​
 +<​td>​500/​1113</​td>​
 +<​td>​125/​192</​td>​
 +<​td>​-2187/​6784</​td>​
 +<​td>​11/​84</​td>​
 +<​td>​0</​td>​
 +</tr>
 +</​table>​
 +
 +Applicable [[Solver settings|solver settings]]:
 +  * Absolute tolerance
 +  * Relative tolerance
 +  * Initial step size
 +  * Max step size 
 +  * Min step size 
 +  * Refine  ​
 +  * Limit data points to last
 +  * Norm Control
 +  * Allowed step size violations
 +  * Enable saturation
 +
 +====Reference====
 +  * Dormand, J. R. and P. J. Prince, “A family of embedded Runge-Kutta formulae,​” _J. Comp. Appl. Math._, Vol. 6, 1980, pp 19-26.
  
dopri45_solver.txt · Last modified: 2015/08/13 15:00 (external edit)