# Which numerical methods preserve time reversal symmetry?

If I have a physical system which contains a time reversal symmetry (for example a Hamiltonian $H(x,p)=p^2/2m + V(x)$ with $V(x)$ real) and I want to solve the differential equations which describe this system, which solver for ODEs should I use in order to keep the time reversal symmetry (for example in mathematica)? Which solvers break this symmetry?

EDIT: I want to extend this question. Let us consider a system of coupled first order differential equations $$\dot{a}_1 (t) = f_1(a_1,a_2,a_3,\ldots,a_n;t) \\ \dot{a}_2(t) = f_2(a_1,a_2,a_3,\ldots,a_n;t) \\ \dot{a}_3(t) = f_3(a_1,a_2,a_3,\ldots,a_n;t) \\ \vdots$$ What integration method is best used if the underlying system contains a time reversal symmetry?

• I think that a Symplectic integrator might do the trick. For example, Verlet integrator. – nicoguaro Dec 13 '15 at 18:29
• @nicoguaro As I intended to use mathematica: is there a Verlet method included? – Merlin1896 Dec 14 '15 at 10:36
• I have barely used Mathematica myself. You can check this post – nicoguaro Dec 14 '15 at 13:48
• @nicoguaro 's answer (in the comments) is correct. Select it. – Inon Dec 15 '15 at 8:17
• @R.M. is your previous comment missing a "not"? – Federico Poloni Jan 15 '16 at 14:11

$$\Delta t \to -\Delta t$$ $$a^{n+j} \to a^{n-j}$$
(here $a^n$ is the numerical approximation of the solution $a(t^n)$, so the second substitution is implied by the first).
I will give two examples to illustrate. The explicit Euler method $$a^{n+1} = a^n + \Delta t f(a^n)$$ does not preserve time symmetry; applied backward in time it becomes the implicit Euler method: $$a^n = a^{n-1} + \Delta t f(a^n).$$ On the other hand the midpoint (or leapfrog) method $$a^{n+1} = a^{n-1} + 2\Delta t f(a^n)$$ does preserve time-reversal symmetry. Other well-known methods that preserve time-reversal symmetry include the trapezoidal method and (as mentioned in the comments) the Verlet method.