A Conservative a-Posteriori Time-Limiting Procedure in Quinpi Schemes

The superior stability properties of implicit time schemes allow to avoid small time steps required to satisfy restrictive stability conditions for stiff hyperbolic systems. In Puppo et al. (Commun Appl Math Comput 2022) an implicit third order finite volume scheme based on a third order DIRK combined with a third order CWENO reconstruction for the space-limiting was proposed. The originality of the proposed method, named Quinpi, lies in the computation of a first order implicit predictor which is used to fix the nonlinear weights of the space reconstruction, thus simplifying considerably the non-linearity of the scheme. However, the time-limiting in the above mentioned paper, which is necessary to control spurious oscillations in the implicit time integration, requires a conservative correction. In this work, we address this problem and we propose a conservative a-posteriori time-limiting procedure inspired by the MOOD method. The numerical experiments show the reliability of the proposed scheme and include both linear and nonlinear scalar conservation laws.