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)).(42)Since we want to construct a second-order method, we start with the Taylor expansiony(t+h) =y(t) +hy (t) +h22y (t) +O(h3).The first derivative can be replaced by the right-hand side of the differential equation(42), and the second derivative is obtained by differentiating (42), ,y (t) =ft(t,y) +fy(t,y)y (t)=ft(t,y) +fy(t,y)f(t,y),with Jacobianfy.

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 …

Loading..

Tags:

  Methods, Runge, Kutta, 3 runge kutta methods

Information