Runge Kutta methods
Runge Kutta (RK) methods are an important class of methods for integrating initial value problems formed by ODEs. Runge Kutta methods encompass a wide selection of numerical methods and some commonly used methods such as Explicit or Implicit Euler's Method, the implicit midpoint rule and the trapezoidal rule are actually simplified versions of a general RK method.
For the ODE,
the basic idea is to build a series of "stages", that approximate the solution at various points using samples of from other stages. Finally, the numerical solution is constructed from a linear combination of and all the precomputed stages.
Since the computation of one stage may involve other stages the right hand side is evaluated in a complicated nonlinear way. The most famous classical RK scheme is described below.
Fourth order Runge-Kutta method
The fourth order Runge-Kutta method could be summarized as:
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