Non-Oscillatory Limited-Time Integration for Conservation Laws and Convection-Diffusion Equations

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.