  We explore the Lagrangian perturbation theory (LPT) at 1-loop order with
Gaussian initial conditions. We present an expansion method to approximately
compute the power spectrum in LPT. Our approximate solution has good
convergence in the series expansion and enables us to compute the power
spectrum in LPT accurately and quickly. Non-linear corrections in the
Lagrangian perturbation theory naturally satisfy the law of conservation of
mass because the relation between matter density and the displacement vector of
dark matter corresponds to the conservation of mass. By matching the 1-loop
solution in LPT to the 2-loop solution in standard perturbation theory, we
present an approximate solution of the power spectrum which has higher order
corrections than the 2-loop order in standard perturbation theory with the
conservation of mass satisfied. With this approximation, we can use LPT to
compute a non-linear power spectrum without any free parameters, and this
solution agrees with numerical simulations at $k=0.2$ $h{\rm Mpc}^{-1}$ and
$z=0.35$ to better than 2%.
