  Unconditionally stable implicit time-marching methods are powerful in solving
stiff differential equations efficiently. In this work, a novel framework to
handle stiff physical terms implicitly is proposed. Both physical and numerical
stiffness originating from convection, diffusion and source terms (typically
related to reaction) can be handled by a set of predefined Time-Accurate and
highly-Stable Explicit (TASE) operators in a unified framework. The proposed
TASE operators act as preconditioners on the stiff terms and can be deployed to
any existing explicit time-marching methods straightforwardly. The resulting
time integration methods remain the original explicit time-marching schemes,
yet with nearly unconditional stability. The TASE operators can be designed to
be arbitrarily high-order accurate with Richardson extrapolation such that the
accuracy order of original explicit time-marching method is preserved.
Theoretical analyses and stability diagrams show that the $s$-stages
$s$th-order explicit Runge-Kutta (RK) methods are unconditionally stable when
preconditioned by the TASE operators with order $p \leq s$ and $p \leq 2$. On
the other hand, the $s$th-order RK methods preconditioned by the TASE operators
with order of $p \leq s$ and $p > 2$ are nearly unconditionally stable. The
only free parameter in TASE operators can be determined a priori based on
stability arguments. Unlike classical implicit methods, the TASE methodology
allows for solving non-linear problems with arbitrary order without requiring
solving a nonlinear system of equations. A set of benchmark problems with
strong stiffness is simulated to assess the performance of the TASE method.
Numerical results suggest that the proposed framework preserves the high-order
accuracy of the explicit time-marching methods with very-large time steps for
all the considered cases.
