  We present a lattice QCD calculation of the $x$-dependent pion and kaon
distribution amplitudes (DA) in the framework of large momentum effective
theory. This calculation is performed on a fine lattice of $a=0.076$~fm at
physical pion mass, with the pion boosted to $1.8$~GeV and kaon boosted to
$2.3$~GeV. We renormalize the matrix elements in the hybrid scheme and match to
$\overline{\rm MS}$ with a subtraction of the leading renormalon in the
Wilson-line mass. The perturbative matching is improved by resumming the large
logarithms related to the small quark and gluon momenta in the soft-gluon
limit. After resummation, we demonstrate that we are able to calculate a range
of $x\in[x_0,1-x_0]$ with $x_0=0.25$ for pion and $x_0=0.2$ for kaon with
systematics under control. The kaon DA is shown to be slighted skewed, and
narrower than pion DA. Although the $x$-dependence cannot be direct calculated
beyond these ranges, we estimate higher moments of the pion and kaon DAs {by
complementing} our calculation with short-distance factorization.
