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**Derivation of the third-order**

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

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**The general form of R-K solution**

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 Taylor expansion for (1) where and are the partial derivatives, and everything is evaluated at The general form of third-order R-K method (2) where k1 is obtained from the explicit Euler method (with ) (3a) 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)

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Plasma Application Modeling, POSTECH 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

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**Substitute Eq.(3a)(3b) and (3c) into Eq.(2)**

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

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**The exact value of in the Taylor series**

Plasma Application Modeling, POSTECH (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

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