A conservative a-posteriori time-limiting procedure in Quinpi schemes (Q6613482)

The study addresses a significant issue in computational mathematics concerning the numerical solution of stiff hyperbolic systems, a class of partial differential equations characterised by waves of vastly differing speeds. These equations pose challenges for explicit numerical schemes due to restrictive stability conditions imposed by the Courant-Friedrichs-Lewy (CFL) criterion.\N\NThe scientific problem examined in this work involves advancing the implicit Quinpi scheme, initially introduced by Puppo et al., to ensure numerical stability while achieving high accuracy for hyperbolic systems. The original Quinpi scheme uses a third-order diagonally implicit Runge-Kutta (DIRK) method combined with a third-order central weighted essentially non-oscillatory (CWENO) spatial reconstruction. Although this method exhibits promising stability and non-oscillatory behaviour, the implicit time integration still suffers from spurious oscillations, particularly for large Courant numbers. To address these oscillations, the authors propose a novel conservative a posteriori time-limiting procedure inspired by the multi-dimensional optimal order detection (MOOD) method.\N\NThe methodology is innovative in combining implicit DIRK schemes with an a posteriori limiting approach to correct for oscillations. The proposed modification involves blending the solutions obtained from a low-order implicit predictor and the high-order solution derived from the Quinpi scheme. Two predictors are investigated: one based on the composite backward Euler method and another using a continuous extension of backward Euler. The study rigorously compares these methods in terms of numerical accuracy and computational efficiency.\N\NThe results demonstrate the effectiveness of the conservative time-limiting strategy. Numerical experiments are conducted on both linear and nonlinear scalar conservation laws, including the Burgers equation, and show that the proposed schemes maintain third-order accuracy while suppressing spurious oscillations. The MOOD-inspired time-limiting ensures mass conservation, a significant improvement over previous Quinpi implementations. Additionally, the computational performance of the new schemes is evaluated, revealing reduced computational costs compared to the original Quinpi method, especially for the scheme employing continuous extension predictors.\N\NThe findings underline the significance of blending predictor-corrector methods with implicit schemes for tackling stiff hyperbolic systems. The novel conservative a posteriori time-limiting procedure offers a robust solution to spurious oscillations, making the Quinpi schemes more reliable and efficient. This advancement has potential implications for broader applications in computational fluid dynamics, where implicit methods are often required to handle stiff systems efficiently. The authors conclude with prospects for extending this approach to systems of conservation laws and exploring alternative time integration schemes, which could further enhance the applicability and versatility of the Quinpi framework.\N\NFor the entire collection see [Zbl 1530.65011].