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Volume 36, Issue 1
Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes

Gabriella Puppo, Matteo Semplice & Giuseppe Visconti

Commun. Comput. Phys., 36 (2024), pp. 30-70.

Published online: 2024-07

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  • Abstract

Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g. because the discrete system is nonlinear also on 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 of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler’s system of gas-dynamics.

  • AMS Subject Headings

65M08, 65M20, 35L65, 65L04

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COPYRIGHT: © Global Science Press

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@Article{CiCP-36-30, author = {Puppo , GabriellaSemplice , Matteo and Visconti , Giuseppe}, title = {Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {1}, pages = {30--70}, abstract = {

Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g. because the discrete system is nonlinear also on 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 of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler’s system of gas-dynamics.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0199}, url = {http://global-sci.org/intro/article_detail/cicp/23296.html} }
TY - JOUR T1 - Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes AU - Puppo , Gabriella AU - Semplice , Matteo AU - Visconti , Giuseppe JO - Communications in Computational Physics VL - 1 SP - 30 EP - 70 PY - 2024 DA - 2024/07 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2023-0199 UR - https://global-sci.org/intro/article_detail/cicp/23296.html KW - Implicit methods, essentially non-oscillatory schemes, finite volumes, hyperbolic systems, entropy indicators. AB -

Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g. because the discrete system is nonlinear also on 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 of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler’s system of gas-dynamics.

Gabriella Puppo, Matteo Semplice & Giuseppe Visconti. (2024). Quinpi: Integrating Stiff Hyperbolic Systems with Implicit High Order Finite Volume Schemes. Communications in Computational Physics. 36 (1). 30-70. doi:10.4208/cicp.OA-2023-0199
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