  We present a variationally separable splitting technique for the
generalized-$\alpha$ method for solving parabolic partial differential
equations. We develop a technique for a tensor-product mesh which results in a
solver with a linear cost with respect to the total number of degrees of
freedom in the system for multi-dimensional problems. We consider finite
elements and isogeometric analysis for the spatial discretization. The overall
method maintains user-controlled high-frequency dissipation while minimizing
unwanted low-frequency dissipation. The method has second-order accuracy in
time and optimal rates ($h^{p+1}$ in $L^2$ norm and $h^p$ in $L^2$ norm of
$\nabla u$) in space. We present the spectrum analysis on the amplification
matrix to establish that the method is unconditionally stable. Various
numerical examples illustrate the performance of the overall methodology and
show the optimal approximation accuracy.
