  In this study we consider unconditionally non-oscillatory, high order
implicit time marching based on time-limiters. The first aspect of our work is
to propose the high resolution Limited-DIRK3 (L-DIRK3) scheme for conservation
laws and convection-diffusion equations in the method-of-lines framework. The
scheme can be used in conjunction with an arbitrary high order spatial
discretization scheme such as 5th order WENO scheme. It can be shown that the
strongly S-stable DIRK3 scheme is not SSP and may introduce strong oscillations
under large time step. To overcome the oscillatory nature of DIRK3, the key
idea of L-DIRK3 scheme is to apply local time-limiters (K.Duraisamy,
J.D.Baeder, J-G Liu), with which the order of accuracy in time is locally
dropped to first order in the regions where the evolution of solution is not
smooth. In this way, the monotonicity condition is locally satisfied, while a
high order of accuracy is still maintained in most of the solution domain. For
convenience of applications to systems of equations, we propose a new and
simple construction of time-limiters which allows flexible choice of reference
quantity with minimal computation cost. Another key aspect of our work is to
extend the application of time-limiter schemes to multidimensional problems and
convection-diffusion equations. Numerical experiments for scalar/systems of
equations in one- and two-dimensions confirm the high resolution and the
improved stability of L-DIRK3 under large time steps. Moreover, the results
indicate the potential of time-limiter schemes to serve as a generic and
convenient methodology to improve the stability of arbitrary DIRK methods.
