Improved Euler s method

The Runge-Kutta method is widely used as a numerical method to solve differential equations. This method is more accurate than the improved Euler s method. This method computes the solution of the initial value problem. 

This brief discussion of Euler s method raises two important issues in all methods for solving the initial value problem in order to improve the approximate solution to the initial value problem, we need to pay close attention to how the solution curve is modeled (straight line vs. polynomials, etc.) and the size of the step to use h). How well these issues are resolved by a particular method is measured by a comparison with the Taylor series for the expected solution F t). 

Different choices of b yield different second-order Runge-Kutta methods. If 6 = the method is called the improved Euler s or Heun s method. If 6 = 1, the method is called the improved polygon or modified Euler s method. As demonstrated by this development, Runge-Kutta methods are not unique since they involve the choice of an arbitrary constant. All second-order methods involve the evaluation of two slopes ki and A)2 (equation (3.1.38) and equation (3.1.39)) and the value of is a weighted average of these two slopes (equations (3.1.35) and (3.1.36)). 

The accuracy of Euler s method is improved by introducing the trapezoidal rule ... 

The formula is known both as the improved Euler formula and as Heun s method [170]. This method has roughly the stability of the explicit Euler method. Unfortunately, the stability may not be improved by iterating the corrector because this iteration procedure converges to the trapezoid rule solution only if At is small enough. 

---