The Semi-Implicit Euler's Method

As noted earlier, the solution of ODE’s using implicit methods (such as the Backward Euler’s method) requires an iterative procedure at each time step. Let’s take a second look at the Backward Euler’s method:

 

 Atkinson et al (Atkinson, Han & Stewart, Numerical Solution of ODE’s) recommend a two-step approach to solving the above equation in order to avoid iterations. The two-step process can be summarized as follows:

 

 

Atkinson et al also indicate that by solving it as a two-step process, the method is no longer absolutely stable. That is, while the original Backward-Euler’s method would converge at a solution for any step size, the two-step process would not.

Methods that are used to solve for implicit formulations without an iterative process are termed as “Semi-Implicit” methods. The computational effort for iterations at each time step is avoided in the semi-implicit method, while at the same time “reasonable” stability can be guaranteed. Press et al (Press, Teukolsky, Vetterling & Flannery, Numerical Recipes, The Art of Scientific Computing) state “It is not guaranteed to be stable, but it usually is…”.

Backward Euler - part 2

The figure below shows the solution to the ODE from Curtiss & Hirschfelder, previously presented here. The figure below, however, was solved using the Backward Euler's method. The Backward Euler clearly has better convergence and stability compared to the Forward Euler. Irrespective of the step size 'h', the solutions clearly converge. The increase in calculation time from the iteration at each time step is made up for by the superior stability characteristics of this method.



So, which of the above solutions is now the "correct" one?!