= Time-symmetric numerical method
= Symmetric numerical method
{synonym}
A <time-symmetric numerical method> is a one-step <numerical method> with a locally invertible step map $\Phi_h$ satisfying
$$
\Phi_{-h}=\Phi_h^{-1},\qquad \Phi_{-h}\circ\Phi_h=I.
$$
For a nonautonomous <ordinary differential equation>, include time in the state or explicitly reverse the step's time endpoints. The definition means that applying the same rule backward undoes its forward step; it does not by itself claim small discretization error.
The <implicit midpoint rule> illustrates this directly: swapping $y_n,y_{n+1}$ and replacing $h$ by $-h$ leaves its midpoint equation unchanged, wherever the local solve is unique. By contrast, <forward Euler method> has factor $1+h\lambda$ on $y'=\lambda y$, and the backward factor $1-h\lambda$ gives product $1-h^2\lambda^2$, not one.
Assume a consistent method has a smooth integer-power local-error expansion of finite order $p$, with nonzero leading defect $h^{p+1}E(x)$. Composing its forward and backward steps leaves leading defect $[1+(-1)^{p+1}]h^{p+1}E(x)$. Symmetry forces this coefficient to vanish. Thus $p+1$ is odd and $p$ is even. The finite leading-error and regular expansion assumptions are needed for this parity argument.
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