Explicit and implicit methods are approaches used in
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
for obtaining numerical approximations to the solutions of time-dependent
ordinary and
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s, as is required in
computer simulation
Computer simulation is the running of a mathematical model on a computer, the model being designed to represent the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determin ...
s of
physical processes. ''Explicit methods'' calculate the state of a system at a later time from the state of the system at the current time, while ''implicit methods'' find a solution 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
:
while for an implicit method one solves an equation
:
to find
Computation
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 practice 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
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context: one important context is numerical linear algebra, and ...
). 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.
Since the implicit method cannot be carried out for each kind of differential operator, it is sometimes advisable to make use of the so called operator splitting method, which means that the differential operator is rewritten as the sum of two complementary operators
:
while one is treated explicitly and the other implicitly.
For usual applications the implicit term is chosen to be linear while the explicit term can be nonlinear. This combination of the former method is called ''Implicit-Explicit Method'' (short IMEX,
[Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo: ''Implicit-Explicit Methods for Evolutionary Partial Differential Equations'', SIAM, ISBN 978-1-61197-819-3 (2024).]).
Illustration using the forward and backward Euler methods
Consider the
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
:
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
Numerical methods for ordinary differential equations are methods used to find Numerical analysis, numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although ...
) and compare the obtained schemes.
;Forward Euler method:

The forward
Euler method
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical analysis, numerical procedure for solving ordinary differential equations (ODEs) with a given Initial value problem, in ...
:
yields
:
for each
This is an explicit formula for
.
;Backward Euler method:
With the
backward Euler method
In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for ordinary differential equations, numerical methods for the solution of ordinary differential equatio ...
:
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
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
, having one negative and one positive
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
. 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 analytical solution exists. Then one uses
root-finding algorithm
In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor ...
s, such as
Newton's method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...
, to find the numerical solution.
;Crank-Nicolson method:
With the
Crank-Nicolson 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 can be numerically solved using
root-finding algorithm
In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor ...
s, such as
Newton's method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...
, to obtain
.
Crank-Nicolson can be viewed as a form of more general IMEX (''Im''plicit-''Ex''plicit) schemes.
;Forward-Backward Euler method:

In order to apply the IMEX-scheme, consider a slightly different differential equation:
:
It follows that
:
See also
*
Courant–Friedrichs–Lewy condition
*
SIMPLE algorithm, a semi-implicit method for pressure-linked equations
Sources
{{DEFAULTSORT:Explicit And Implicit Methods
Numerical differential equations