Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes (Communications in Computational Physics, 2024)
Many interesting physical problems described by systems of hyperbolicconservation laws are stiff, and thus impose a very small time-step because of therestrictive CFL stability condition. In this case, one can exploit the superior stabilityproperties of implicit time integration which allows to choose the time-step only fromaccuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systemswithout tailoring it to a specific problem. The nonlinearity of high order schemes, dueto space- and time-limiting procedures which control nonphysical oscillations, makesthe implicit time integration difficult, e.g. because the discrete system is nonlinear alsoon linear problems. This nonlinearity of the scheme is circumvented as proposed in(Puppo et al., Comm. Appl. Math. & Comput., 2023) for scalar conservation laws,where a first order implicit predictor is computed to freeze the nonlinear coefficients ofthe essentially non-oscillatory space reconstruction, and also to assist limiting in time.In addition, we propose a novel conservative flux-centered a-posteriori time-limitingprocedure using numerical entropy indicators to detect troubled cells. The numericaltests involve classical and artificially devised stiff problems using the Euler’s systemof gas-dynamics.