Accuracy of Euler-Type Methods

It has been stated that the Euler and backward Euler methods are 0 h) and that the trapezoidal method is 0 h ). These results can be shown more convincingly for the exponential test problem given by Equation 5.8, the exact solution of which is the negative exponential. This can be written in the following ino-anental form  

This equation, when compared to the exact series expansion, is accurate to the first two terms, and the term involving h is the magnitude of the error. So, once again, each step is 0(h ) while the global error is 0(h). 

These are among the most popular methods for solving ODE-IVPs. They are explicit and are based on the idea of using intermediate points in each major time step (note even though the independent variable often is time, it can be distance, volume, or 

RK methods are derived by applying a numerical integration method to the integral on the right-hand side. 

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