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Volume 38, Issue 4
Relaxation Schemes for Entropy Dissipative Systems of Viscous Conservation Laws

Tuowei Chen & Jiequan Li

DOI: 10.4208/cicp.OA-2024-0299

Commun. Comput. Phys., 38 (2025), pp. 953-986.

Published online: 2025-09

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  • Abstract

In this paper, a hyperbolic relaxation model is designed for a class of entropy dissipative systems of viscous conservation laws, such as the 1-D viscous Burgers and 2-D Navier-Stokes equations. An artificial variable is introduced to relax both the convective and viscous fluxes together. Based on the entropy dissipative property of the original system, a dissipation condition is proposed for the resulting relaxation model, and used to prove the entropy inequality of the relaxation model for linear convection-diffusion equations. Lax-Wendroff type second-order finite-volume schemes are developed to discretize the relaxation model. A number of numerical experiments, including viscous compressible flow problems from subsonic to supersonic speeds, are used to validate the relaxation model and evaluate the performance of the current schemes.

  • Keywords

Relaxation method, viscous conservation laws, entropy dissipation, Lax-Wendroff type solver, Navier-Stokes equations, generalized Riemann problem.

  • AMS Subject Headings

65M08, 76M12, 35Q30, 76N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-38-953, author = {Chen , Tuowei and Li , Jiequan}, title = {Relaxation Schemes for Entropy Dissipative Systems of Viscous Conservation Laws}, journal = {Communications in Computational Physics}, year = {2025}, volume = {38}, number = {4}, pages = {953--986}, abstract = {

In this paper, a hyperbolic relaxation model is designed for a class of entropy dissipative systems of viscous conservation laws, such as the 1-D viscous Burgers and 2-D Navier-Stokes equations. An artificial variable is introduced to relax both the convective and viscous fluxes together. Based on the entropy dissipative property of the original system, a dissipation condition is proposed for the resulting relaxation model, and used to prove the entropy inequality of the relaxation model for linear convection-diffusion equations. Lax-Wendroff type second-order finite-volume schemes are developed to discretize the relaxation model. A number of numerical experiments, including viscous compressible flow problems from subsonic to supersonic speeds, are used to validate the relaxation model and evaluate the performance of the current schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0299}, url = {http://global-sci.org/intro/article_detail/cicp/24350.html} }
TY - JOUR T1 - Relaxation Schemes for Entropy Dissipative Systems of Viscous Conservation Laws AU - Chen , Tuowei AU - Li , Jiequan JO - Communications in Computational Physics VL - 4 SP - 953 EP - 986 PY - 2025 DA - 2025/09 SN - 38 DO - http://doi.org/10.4208/cicp.OA-2024-0299 UR - https://global-sci.org/intro/article_detail/cicp/24350.html KW - Relaxation method, viscous conservation laws, entropy dissipation, Lax-Wendroff type solver, Navier-Stokes equations, generalized Riemann problem. AB -

In this paper, a hyperbolic relaxation model is designed for a class of entropy dissipative systems of viscous conservation laws, such as the 1-D viscous Burgers and 2-D Navier-Stokes equations. An artificial variable is introduced to relax both the convective and viscous fluxes together. Based on the entropy dissipative property of the original system, a dissipation condition is proposed for the resulting relaxation model, and used to prove the entropy inequality of the relaxation model for linear convection-diffusion equations. Lax-Wendroff type second-order finite-volume schemes are developed to discretize the relaxation model. A number of numerical experiments, including viscous compressible flow problems from subsonic to supersonic speeds, are used to validate the relaxation model and evaluate the performance of the current schemes.

Chen , Tuowei and Li , Jiequan. (2025). Relaxation Schemes for Entropy Dissipative Systems of Viscous Conservation Laws. Communications in Computational Physics. 38 (4). 953-986. doi:10.4208/cicp.OA-2024-0299
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