  We examine whether the $O(a)$ improved quark action on anisotropic lattices
can be used as a framework for the heavy quark, which enables precision
computation of matrix elements of heavy-light mesons. To this end, it is
crucial to verify that a mass independent and nonperturbative tuning of the
parameters is possible. As a first step, we observe the dispersion relation of
heavy-light mesons on a quenched lattice using the action which is
nonperturbatively tuned only for the leading terms. On a lattice with the
spatial cutoff $a_\sigma^{-1} \simeq$ 1.6 GeV and the anisotropy $\xi=4$, the
relativity relation holds within 2% accuracy in the quark mass region $a_\sigma
m_Q \leq 1.2$ with the bare anisotropy parameter tuned for the massless quark.
We also apply the action to a calculation of heavy-light decay constants in the
charm quark mass region.
