  In this paper, we consider numerical approximation to periodic measure of a
time periodic stochastic differential equations (SDEs) under weakly dissipative
condition. For this we first study the existence of the periodic measure
$\rho_t$ and the large time behaviour of $\mathcal{U}(t+s,s,x) :=
\mathbb{E}\phi(X_{t}^{s,x})-\int\phi d\rho_t,$ where $X_t^{s,x}$ is the
solution of the SDEs and $\phi$ is a test function being smooth and of
polynomial growth at infinity. We prove $\mathcal{U}$ and all its spatial
derivatives decay to 0 with exponential rate on time $t$ in the sense of
average on initial time $s$. We also prove the existence and the geometric
ergodicity of the periodic measure of the discretized semi-flow from the
Euler-Maruyama scheme and moment estimate of any order when the time step is
sufficiently small (uniform for all orders). We thereafter obtain that the weak
error for the numerical scheme of infinite horizon is of the order $1$ in terms
of the time step. We prove that the choice of step size can be uniform for all
test functions $\phi$. Subsequently we are able to estimate the average
periodic measure with ergodic numerical schemes.
