Euler–Cromer algorithm

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In mathematics, the Euler–Cromer algorithm or symplectic Euler method is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method.

1 Setting
2 The method
3 Example
4 References

[edit] Setting

The Euler–Cromer algorithm can be applied to a pair of differential equations of the form

${dx \over dt} = f(t,v)$

${dv \over dt} = g(t,x),$

where f and g are given functions. Here, x and v may be either scalars or vectors. The equations of motion in Hamiltonian mechanics take this form if the Hamiltonian is of the form

$H = T(t,v) + V(t,x). \,$

The differential equations are to be solved with the initial condition

$x(t_0) = x_0, \qquad v(t_0) = v_0.$

[edit] The method

The Euler–Cromer algorithm produces an approximate discrete solution by iterating

$v_{n+1} = v_n + f(t_n, x_n) \, \Delta t \quad$

$x_{n+1} = x_n + g(t_n, v_{n+1}) \, \Delta t \quad$

where $Δ t$ is the time step and $t n = t 0 + n Δ t$ is the time after n steps.

The difference with the standard Euler method is that the Euler–Cromer method uses $v n + 1$ in the equation for $x n + 1$ , while the Euler method uses $v n$ .

The Euler–Cromer method is a first-order integrator, just as the standard Euler method. This means that it commits a global error of the order of Δt. However, the Euler–Cromer method is a symplectic integrator, unlike the standard method. As a consequence, the Euler–Cromer method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.

[edit] Example

The motion of a spring satisfying Hooke's law is given by

${dx \over dt} = v(t)$

${dv \over dt} = -{k \over m}x. \quad$

The Euler–Cromer algorithm for this equation is

$v_{n+1} = v_n - {k \over m}x_n\Delta t \quad$

$x_{n+1} = x_n + v_{n+1} \Delta t. \quad$

[edit] References

Giordano, Nicholas J.; Hisao Nakanishi (July 2005). Computational Physics, 2nd edition, Benjamin Cummings. ISBN 0-1314-6990-8.
MacDonald, James. The Euler-Cromer method. University of Delaware. Retrieved on 2007-03-03.
Vesely, Franz J. (2001). Computational Physics: An Introduction, 2nd edition, Springer, page 117. ISBN 978-0-306-46631-1.