  We propose a short-memory operator splitting scheme for solving the
constant-Q wave equation, where the fractional stress-strain relation contains
multiple Caputo fractional derivatives with order much smaller than 1. The key
is to exploit its extension problem by converting the flat singular kernels
into strongly localized ones, so that the major contribution of weakly singular
integrals over a semi-infinite interval can be captured by a few Laguerre
functions with proper asymptotic behavior. Despite its success in reducing both
memory requirement and arithmetic complexity, we show that numerical accuracy
under prescribed memory variables may deteriorate in time due to the dynamical
increments of projection errors. Fortunately, it can be considerably alleviated
by introducing a suitable scaling factor $\beta > 1$ and pushing the
collocation points closer to origin. An operator splitting scheme is introduced
to solve the resulting set of equations, where the auxiliary dynamics can be
solved exactly, so that it gets rid of the numerical stiffness and
discretization errors. Numerical experiments on both 1-D diffusive wave
equation and 2-D constant-Q $P$- and $S$-wave equations are presented to
validate the accuracy and efficiency of the proposed scheme.
