Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while an implicit method finds it by solving an equation involving both the current state of the system and the later one. Mathematically, if is the current system state and is the state at the later time ( is a small time step), then, for an explicit method
It is clear that implicit methods require an extra computation (solving the above equation), and they can be much harder to implement. Implicit methods are used because many problems arising in real life are stiff, for which the use of an explicit method requires impractically small time steps to keep the error in the result bounded (see numerical stability). For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
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.
The forward Euler method
With the backward Euler method
one finds the implicit 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.