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Parabolic Shot

Parabolic shot simulation with different types of integrators. Velocity is calculated by the numeric integration of the following differential equation:

\[\frac{dv}{dt}=f(t,v),\] \[v_{i+1}=v_n+\phi \Delta t\]

\(\phi\) value depends on each numeric integrator (press their own key to change):

-(E) Explicit Euler: \[v_{n+1}=v_n+f(t_n,v_n)\Delta t\] \[x_{n+1}=x_n+v_n\Delta t\] -(S) Symplectic (Semi-implicit) Euler: \[v_{n+1}=v_n+f(t_n,v_n)\Delta t\] \[x_{n+1}=x_n+v_{n+1}\Delta t\] -(H) Heun: \[v'_{n+1}=v_n+f(t_n,v_n)\Delta t\] \[v_{n+1}=v_n+[f(t_n,v_n)+ f(t_{n+1},v'_{n+1})]\frac {\Delta t} 2\] \[x_{n+1}=x_n+v_{n+1}\Delta t\] -(2) RK2: \[v_{n+1}=v_n+ \frac {\Delta t} 2 (k_1 + k_2)\] -(4) RK4: \[v_{n+1}=v_n + \frac {\Delta t} 6 (k_1+2k_2+2k_3+k_4)\]

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\[k_1=f(t_n,v_n)\] \[k_2= f(t_n+\frac {\Delta t} 2, v_n + k_1 \frac {\Delta t} 2 )\] \[k_3= f(t_n+\frac {\Delta t} 2, v_n + k_2 \frac {\Delta t} 2 )\] \[k_4= f(t_n+\Delta t, v_n + k_3 \Delta t )\]