Semi–implicit Euler

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In mathematics, the semi–implicit Euler method, also called symplectic Euler, semi–explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), 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.

Contents 1 Setting 2 The method 3 Example 4 References

[edit] Setting

The semi–implicit Euler method can be applied to a pair of differential equations of the form

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

The differential equations are to be solved with the initial condition

[edit] The method

The semi–implicit Euler method produces an approximate discrete solution by iterating

where Δt is the time step and is the time after n steps.

The difference with the standard Euler method is that the semi–implicit Euler method uses v_{n + 1} in the equation for x_{n + 1}, while the Euler method uses v_n.

The semi–implicit Euler 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 semi–implicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semi–implicit Euler 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

The semi–implicit Euler for this equation is

[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. 

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