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Derivation of the third-order Runge-Kutta method in general formation J.H. Kang and J.K. Lee P lasma A pplication Modeling, POSTECH

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P lasma A pplication Modeling, POSTECH The general form of R-K solution where C j is just a weight, and k j is a function of f and the previous k i expressions. Derivation of the third-order R-K method in general formation The general form of third-order R-K method where k 1 is obtained from the explicit Euler method (with ) the subsequent k j are evaluated at various locations within the interval, t n t t n+1, with corresponding values of, where is some combination of the earlier k i values. (1) (2) (3a) (3b) Taylor expansion for where and are the partial derivatives, and everything is evaluated at

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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 P lasma A pplication Modeling, POSTECH

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Substitute Eq.(3a)(3b) and (3c) into Eq.(2) (3c) P lasma A pplication Modeling, POSTECH

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

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