In mathematics, the semiimplicit Euler method, also called symplectic Euler, semiexplicit 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.
Setting
The semiimplicit Euler method can be applied to a pair of differential equations of the form
 $\{dx\; \backslash over\; dt\}\; =\; f(t,v)$
 $\{dv\; \backslash over\; dt\}\; =\; g(t,x),$
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
 $H\; =\; T(t,v)\; +\; V(t,x).\; \backslash ,$
The differential equations are to be solved with the initial condition
 $x(t\_0)\; =\; x\_0,\; \backslash qquad\; v(t\_0)\; =\; v\_0.$
The method
The semiimplicit Euler method produces an approximate discrete solution by iterating
 $\backslash begin\{align\}$
v_{n+1} &= v_n + g(t_n, x_n) \, \Delta t\\[0.3em]
x_{n+1} &= x_n + f(t_n, v_{n+1}) \, \Delta t
\end{align}
where Δt is the time step and t_{n} = t_{0} + nΔt is the time after n steps.
The difference with the standard Euler method is that the semiimplicit Euler method uses v_{n+1} in the equation for x_{n+1}, while the Euler method uses v_{n}.
Applying the method with negative time step to the computation of $(x\_n,v\_n)$ from $(x\_\{n+1\},v\_\{n+1\})$ and rearranging leads to the second variant of the semiimplicit Euler method
 $\backslash begin\{align\}$
x_{n+1} &= x_n + f(t_n, v_n) \, \Delta t\\[0.3em]
v_{n+1} &= v_n + g(t_n, x_{n+1}) \, \Delta t
\end{align}
which has similar properties.
The semiimplicit Euler is a firstorder integrator, just as the standard Euler method. This means that it commits a global error of the order of Δt. However, the semiimplicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semiimplicit Euler method almost conserves the energy (when the Hamiltonian is timeindependent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.
Alternating between the two variants of the semiimplicit Euler method leads in one simplification to the StörmerVerlet integration and in a slightly different simplification to the leapfrog integration, increasing both the order of the error and the order of preservation of energy.^{[1]}
The stability region of the semiimplicit method was presented in ^{[2]} although the method was misleadingly called symmetric Euler. The semiimplicit method models the simulated system correctly if the complex roots of the characteristic equation are within the circle shown below. For real roots the stability region extends outside the circle for which the criteria is $s\; >\; \; 2/\backslash Delta\; t$
As can be seen, the semiimplicit method can simulate correctly both stable systems that have their roots in the left half plane and unstable systems that have their roots in the right half plane. This is clear advantage over forward (standard) Euler and backward Euler. Forward Euler tends to have less damping than the real system when the negative real parts of the roots get near the imaginary axis and backward Euler may show the system be stable even when the roots are in the right half plane.
Example
The motion of a spring satisfying Hooke's law is given by
 $\backslash begin\{align\}$
\frac{dx}{dt} &= v(t)\\[0.2em]
\frac{dv}{dt} &= \frac{k}{m}\,x=\omega^2\,x.
\end{align}
The semiimplicit Euler for this equation is
 $\backslash begin\{align\}$
v_{n+1} &= v_n  \omega^2\,x_n\,\Delta t \\[0.2em]
x_{n+1} &= x_n + v_{n+1} \,\Delta t.
\end{align}
The iteration preserves the modified energy functional $E\_h(x,v)=\backslash tfrac12\backslash left(v^2+\backslash omega^2\backslash ,x^2\backslash omega^2\backslash Delta\; t\backslash ,vx\backslash right)$ exactly, leading to stable periodic orbits that deviate by $O(\backslash Delta\; t)$ from the exact orbits. The exact circular frequency $\backslash omega$ increases in the numerical approximation by a factor of $1+\backslash tfrac1\{24\}\backslash omega^2\backslash Delta\; t^2+O(\backslash Delta\; t^4)$.
References
Numerical integration 

 Firstorder methods  

 Secondorder methods  

 Higherorder methods  


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