  We propose a new class of high-order time-marching schemes with dissipation
user-control and unconditional stability for parabolic equations. High-order
time integrators can deliver the optimal performance of highly-accurate and
robust spatial discretizations such as isogeometric analysis. The
generalized-$\alpha$ method delivers unconditional stability and second-order
accuracy in time and controls the numerical dissipation in the discrete
spectrum's high-frequency region. Our goal is to extend the generalized-$alpha$
methodology to obtain a high-order time marching methods with high accuracy and
dissipation in the discrete high-frequency range. Furthermore, we maintain the
stability region of the original, second-order generalized-$alpha$ method foe
the new higher-order methods. That is, we increase the accuracy of the
generalized-$\alpha$ method while keeping the unconditional stability and
user-control features on the high-frequency numerical dissipation. The
methodology solve $k>1, k\in \mathbb{N}$ matrix problems and updates the system
unknowns, which correspond to higher-order terms in Taylor expansions to obtain
$(3/2k)^{th}$-order method for even $k$ and $(3/2k+1/2)^{th}$-order for odd
$k$. A single parameter $\rho^\infty$ controls the dissipation, and the update
procedure follows the formulation of the original second-order method.
Additionally, we show that our method is A-stable and setting $\rho^\infty=0$
allows us to obtain an L-stable method. Lastly, we extend this strategy to
analyze the accuracy order of a generic method.
