Presentation on theme: "Derivation of the third-order"— Presentation transcript:
1 Derivation of the third-order Plasma ApplicationModeling, POSTECHDerivation of the third-orderRunge-Kutta methodin general formationJ.H. Kang and J.K. Lee
2 The general form of R-K solution Plasma ApplicationModeling, POSTECHThe general form of R-K solutionwhere 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 formationTaylor expansion for(1)where and are the partial derivatives, and everything is evaluated atThe 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)
3 Plasma ApplicationModeling, POSTECHWe 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
4 Substitute Eq.(3a)(3b) and (3c) into Eq.(2) Plasma ApplicationModeling, POSTECH(3c)Substitute Eq.(3a)(3b) and (3c) into Eq.(2)
5 The exact value of in the Taylor series Plasma ApplicationModeling, 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
Your consent to our cookies if you continue to use this website.