 Time-symmetric numerical method

A time-symmetric numerical method is a one-step numerical method with a locally invertible step map ${\Phi }_h$ satisfyingFor 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 $y^{\prime }=\lambda y$, 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.