A thermodynamically compatible semi-implicit finite volume scheme for the compressible Euler equations
Abstract
In this work we construct a new semi-implicit (SI) hyperbolic and thermodynamically compatible (HTC) scheme for the compressible Euler equations of gasdynamics. The challenge therein lies in solving the entropy inequality as the primary evolution equation instead of the total energy conservation law, which in our framework is an additional conservation law that needs to be fulfilled by the numerical scheme as the consequence of a compatible discretization. The proposed method has a predictor-corrector character, where we first solve the isentropic problem without considering the entropy production. Therein, the entropy and the nonlinear convective terms in the momentum flux are discretized explicitly, while the mass flux and the pressure terms are taken implicitly. The discrete momentum equation is then inserted into the discrete mass conservation equation, leading to a mildly nonlinear system for the pressure, which can be efficiently solved with a nested Newton-type algorithm. Based on this prediction, a non-negative production term in the entropy is added, ensuring a cell entropy inequality for the numerical scheme. To guarantee global conservation of total energy at the discrete level, a new modified global Abgrall-type flux correction is introduced. The presented numerical method allows time steps that are restricted only by the material wave speed and not by the sound speed. This makes the scheme particularly useful for flows in the low-Mach-number regime. Numerical results are shown in one and two space dimensions to assess the theoretical properties of the new scheme.
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BibTeX
@misc{dumbser2026thermodynamically,
title = {{A thermodynamically compatible semi-implicit finite volume scheme for the compressible Euler equations}},
author = {Michael Dumbser and Gabriella Puppo and Sara Rinaldi and Andrea Thomann},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31065},
eprint = {2609.31065},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05835871v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A thermodynamically compatible semi-implicit finite volume scheme for the compressible Euler equations
AU - Michael Dumbser
AU - Gabriella Puppo
AU - Sara Rinaldi
AU - Andrea Thomann
PY - 2026
DA - 2026/09/25
PB - HAL
DO - 10.48550/arxiv.2609.31065
UR - https://hal.science/hal-05835871v1
AB - In this work we construct a new semi-implicit (SI) hyperbolic and thermodynamically compatible (HTC) scheme for the compressible Euler equations of gasdynamics. The challenge therein lies in solving the entropy inequality as the primary evolution equation instead of the total energy conservation law, which in our framework is an additional conservation law that needs to be fulfilled by the numerical scheme as the consequence of a compatible discretization. The proposed method has a predictor-corrector character, where we first solve the isentropic problem without considering the entropy production. Therein, the entropy and the nonlinear convective terms in the momentum flux are discretized explicitly, while the mass flux and the pressure terms are taken implicitly. The discrete momentum equation is then inserted into the discrete mass conservation equation, leading to a mildly nonlinear system for the pressure, which can be efficiently solved with a nested Newton-type algorithm. Based on this prediction, a non-negative production term in the entropy is added, ensuring a cell entropy inequality for the numerical scheme. To guarantee global conservation of total energy at the discrete level, a new modified global Abgrall-type flux correction is introduced. The presented numerical method allows time steps that are restricted only by the material wave speed and not by the sound speed. This makes the scheme particularly useful for flows in the low-Mach-number regime. Numerical results are shown in one and two space dimensions to assess the theoretical properties of the new scheme.
ER -