Transcription of 3 Runge-Kutta Methods - IIT
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3 Runge-Kutta Methods - IIT
3 Runge-Kutta MethodsIn contrast to the multistep Methods of the previous section, Runge-Kutta methodsare single-step Methods however, with multiplestagesper step. They are motivatedby the dependence of the Taylor Methods on the specific IVP. These new Methods donot require derivatives of the right-hand side functionfin the code, and are thereforegeneral-purpose initial value problem solvers. Runge-Kutta Methods are among themost popular ODE solvers. They were first studied by Carle Runge and Martin Kuttaaround 1900. Modern developments are mostly due to John Butcher in the Second-Order Runge-Kutta MethodsAs always we consider the general first-order ODE systemy (t) =f(t,y(t)).
predictor for the (implicit) trapezoidal rule. We obtain general explicit second-order Runge-Kutta methods by assuming y(t+h) = y(t)+h h b 1k˜ 1 +b 2k˜ 2 i +O(h3) (45) with k˜ 1 = f(t,y) k˜ 2 = f(t+c 2h,y +ha 21k˜ 1). Clearly, this is a generalization of the classical Runge-Kutta method since the choice b 1 = b 2 = 1 2 and c 2 = a 21 = 1 ...
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