Explicit and Implicit Methods - Illustration Using The Forward and Backward Euler Methods

Illustration Using The Forward and Backward Euler Methods

Consider the ordinary differential equation

with the initial condition Consider a grid for 0≤k≤n, that is, the time step is and denote for each . Discretize this equation using the simplest explicit and implicit methods, which are the forward Euler and backward Euler methods (see numerical ordinary differential equations) and compare the obtained schemes.

Forward Euler method

The forward Euler method

yields

for each This is an explicit formula for .

Backward Euler method

With the backward Euler method

one finds the implicit equation

for (compare this with formula (3) where was given explicitly rather than as an unknown in an equation).

This is a quadratic equation, having one negative and one positive root. The positive root is picked because in the original equation the initial condition is positive, and then at the next time step is given by

In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no exact solution exists. Then one uses root-finding algorithms, such as Newton's method.