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@ -34,10 +34,7 @@ Description
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Cash, J.R.,
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Karp, A.H.
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ACM Transactions on Mathematical Software, vol. 16, 1990, pp. 201–222.
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\endverbatim
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Based on code from:
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\verbatim
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"Solving Ordinary Differential Equations I: Nonstiff Problems,
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second edition",
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Hairer, E.,
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@ -34,10 +34,7 @@ Description
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Prince, P. J.,
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Journal of Computational and Applied Mathematics,
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6 (1), 1980: pp. 19-26.
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\endverbatim
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Based on code from:
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\verbatim
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"Solving Ordinary Differential Equations I: Nonstiff Problems,
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second edition",
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Hairer, E.,
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@ -34,8 +34,6 @@ Description
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Fehlberg, E.,
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NASA Technical Report 315, 1969.
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Based on code from:
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\verbatim
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"Solving Ordinary Differential Equations I: Nonstiff Problems,
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second edition",
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Hairer, E.,
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@ -48,31 +46,6 @@ Description
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and allows to perform an adapdive step-size control using these two order
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without the need of re-evaluation.
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\f{align}{
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k_1 &= h f(t_k, y_k) \\
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k_2 &= h f(t_k + \frac{1}{4}h, y_k + \frac{1}{4}k_1) \\
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k_3 &= h f(t_k + \frac{3}{8}h, y_k + \frac{3}{32}k_1 + \frac{9}{32}k_2) \\
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k_4 &= h f(t_k + \frac{12}{13}h, y_k + \frac{1932}{2197}k_1
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\frac{7200}{2197}k_2 + \frac{7296}{2197}k_3) - \\
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k_5 &= h f(t_k + h, y_k + \frac{439}{216}k_1 - 8k_2 + \frac{3680}{513}k_3 -
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\frac{845}{4104}k_4) \\
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k_6 &= h f(t_k + \frac{1}{2}h, y_k - \frac{8}{27}k_1 + 2k_2 -
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\frac{3544}{2565}k_3 + \frac{1859}{4104}k_4 - \frac{11}{40}k_5) \\
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\f}
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Which yields the following update-steps for the 4th and 5th order:
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\f{align}{
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\Delta_4 &= \frac{25}{216}k_1 + \frac{1408}{2565}k_3 +
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\frac{2197}{4101}k_4 - \frac{1}{5}k_5 \\
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\Delta_5 &= \frac{16}{135}k_1 + \frac{6656}{12825}k_3 +
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\frac{9}{50}k_5 + \frac{2}{55}k_6
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\f}
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The difference between the 5th and 4th order (\f$\epsilon = \left|\Delta_5
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- \Delta_4\right|\f$) can be used to determine if the stepsize \f$h\f$ needs
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to be adjusted.
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SourceFiles
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RKF45.C
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@ -27,7 +27,6 @@ Class
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Description
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L-stable embedded Rosenbrock ODE solver of order (3)4.
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Based on code from:
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\verbatim
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"Solving Ordinary Differential Equations II: Stiff
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and Differential-Algebraic Problems, second edition",
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@ -37,7 +36,7 @@ Description
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Springer-Verlag, Berlin. 1996.
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\endverbatim
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Also provided are the constants from:
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The default constants are from:
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\verbatim
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"Implementation of Rosenbrock Methods"
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Shampine, L. F.,
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@ -46,6 +45,8 @@ Description
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with which the scheme is more accurate than with the L-Stable coefficients
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for small step-size but less stable for large step-size.
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The L-Stable scheme constants are provided commented-out in Rosenbrock43.C
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SourceFiles
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Rosenbrock43.C
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