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Derivation of the third-order Runge-Kutta method in general formation

P lasma A pplication M odeling, POSTECH. Derivation of the third-order Runge-Kutta method in general formation. J.H. Kang and J.K. Lee. 2005. 09. 27. Taylor expansion for. (1). where and are the partial derivatives, and everything is evaluated at. (2).

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Derivation of the third-order Runge-Kutta method in general formation

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  1. Plasma Application Modeling, POSTECH Derivation of the third-order Runge-Kutta method in general formation J.H. Kang and J.K. Lee 2005. 09. 27

  2. Taylor expansion for (1) where and are the partial derivatives, and everything is evaluated at (2) where k1 is obtained from the explicit Euler method (with ) the subsequent kj are evaluated at various locations within the interval, tn t tn+1, with corresponding values of , where is some combination of the earlier ki values. (3b) Plasma Application Modeling, POSTECH • The general form of R-K solution where Cj is just a weight, and kj is a function of f and the previous ki expressions. • Derivation of the third-order R-K method in general formation • The general form of third-order R-K method (3a)

  3. We are looking for a solution to third order in h. So we only need second order here. Thus, we can take a low order approximation of Plasma Application Modeling, POSTECH

  4. (3c) Plasma Application Modeling, POSTECH Substitute Eq.(3a)(3b) and (3c) into Eq.(2)

  5. (4) • The exact value of in the Taylor series (5) Plasma Application Modeling, POSTECH • Now we just choose coefficients which make equal Eq.(4) and Eq.(5) • One example of the third order R-K method

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