High Resolution Time-Limiter Schemes for Conservation Laws

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This study investigates the high resolution time-limiter schemes for conservation laws. These schemes are proposed (K.Duraisamy, J.D.Baeder, J-G Liu, 2003) to enhance the stability of high order implicit time marching when the time step is beyond the original TVD limit. The improved stability is realized by taking local convex combination of a higher order oscillatory method (accurate mode) with a first order unconditionally TVD method (stable mode). The application of time-limiters, which detects the local smoothness, enables the self-adjusting switch between different modes. One of the main aspects of this work is employing time-limiters to improve the stability of the strongly S-stable DIRK3 scheme, which is shown to be non-SSP and thus may generate strong oscillations in non-smooth problems. The new Limited-DIRK3 scheme (L-DIRK3) is proposed. For convenience of applications to systems of equations, we also propose a new and convenient construction of time-limiter, which allows an arbitrary choice of reference quantity with minimal computation cost. Another innovation of our work is the extension of time-limiter schemes to multi-dimensional problems and convection-diffusion problems. The numerical results for one- and two-dimensional problems confirm that the L-DIRK3 scheme generate high resolution and less oscillatory solutions under large time step. Particularly, the L-DIRK3 scheme shows a clear improvement against the original DIRK3 in convection-diffusion problems when a large CFL number is taken.