We propose a new section-averaged one-dimensional model for blood flows in deformable arteries.The model is derived from the three-dimensional Navier-Stokes equations,written in cylindrical coordinates,under the“thin...We propose a new section-averaged one-dimensional model for blood flows in deformable arteries.The model is derived from the three-dimensional Navier-Stokes equations,written in cylindrical coordinates,under the“thin-artery”assumption(similar to the“shallow-water”assumption for free surface models).The blood flow/artery interaction is taken into account through suitable boundary conditions.The obtained equations enter the scope of the nonlinear convection-diffusion problems.We show that the resulting model is energetically consistent.The proposed model extends most extant models by adding more scope,depending on an additional viscous term.We compare both models computationally based on an Incomplete Interior Penalty Galerkin(IIPG)method for the parabolic part,and on a Runge Kutta Discontinuous Galerkin(RKDG)method for the hyperbolic part.The time discretization explicit/implicit is based on the well-known Additive Runge-Kutta(ARK)method.Moreover,through a suitable change of variables,by construction,we show that the numerical scheme is well-balanced,i.e.,it preserves exactly still-steady state solutions.To end,we numerically investigate its efficiency through several test cases with a confrontation to an exact solution.展开更多
In this article,we study the convergence of an IIPG(Incomplete Interior Penalty Galerkin)Discontinuous Galerkin numerical method for the Richards equation.The Richards equation is a degenerate parabolic nonlinear equa...In this article,we study the convergence of an IIPG(Incomplete Interior Penalty Galerkin)Discontinuous Galerkin numerical method for the Richards equation.The Richards equation is a degenerate parabolic nonlinear equation for modeling flows in porous media with variable saturation.The numerical solution of this equation is known to be difficult to calculate numerically,due to the abrupt displacement of the wetting front,mainly as a result of highly nonlinear hydraulic properties.As time scales are slow,implicit numerical methods are required,and the convergence of nonlinear solvers is very sensitive.We propose an original method to ensure convergence of the numerical solution to the exact Richards solution,using a technique of auto-calibration of the penalty parameters derived from the Galerkin Discontinuous method.The method is constructed using nonlinear 1D and 2D general elliptic problems.We show that the numerical solution converges toward the unique solution of the continuous problem under certain conditions on the penalty parameters.Then,we numerically demonstrate the efficiency and robustness of the method through test cases with analytical solutions,laboratory test cases,and large-scale simulations.展开更多
基金supported by the ADEN-MED project(Adaptability to Extreme Events and Natural Risks-Application to the Mediterranean and Djibouti)funded by the Région Sud Provence-Alpes-Côte d’Azur under the AAP MEDCLIMAT“Natural Risks and Food Sovereignty”.
文摘We propose a new section-averaged one-dimensional model for blood flows in deformable arteries.The model is derived from the three-dimensional Navier-Stokes equations,written in cylindrical coordinates,under the“thin-artery”assumption(similar to the“shallow-water”assumption for free surface models).The blood flow/artery interaction is taken into account through suitable boundary conditions.The obtained equations enter the scope of the nonlinear convection-diffusion problems.We show that the resulting model is energetically consistent.The proposed model extends most extant models by adding more scope,depending on an additional viscous term.We compare both models computationally based on an Incomplete Interior Penalty Galerkin(IIPG)method for the parabolic part,and on a Runge Kutta Discontinuous Galerkin(RKDG)method for the hyperbolic part.The time discretization explicit/implicit is based on the well-known Additive Runge-Kutta(ARK)method.Moreover,through a suitable change of variables,by construction,we show that the numerical scheme is well-balanced,i.e.,it preserves exactly still-steady state solutions.To end,we numerically investigate its efficiency through several test cases with a confrontation to an exact solution.
基金supported by the ADEN-MED project(Adaptability to Extreme events and Natural risks-application to the Mediterranean and Djibouti),funded by the Region Sud Provence-Alpes-Cote d’Azur under the AAP MEDCLIMAT“Natural risks and food sovereignty”by France 2030 through the Priority Research Program and Equipment(PEPR)“Maths-Vives-Mathematics in Interactions”,targeted project HYDRAUMATH(ANR-23-EXMA-007),operated by ANR.
文摘In this article,we study the convergence of an IIPG(Incomplete Interior Penalty Galerkin)Discontinuous Galerkin numerical method for the Richards equation.The Richards equation is a degenerate parabolic nonlinear equation for modeling flows in porous media with variable saturation.The numerical solution of this equation is known to be difficult to calculate numerically,due to the abrupt displacement of the wetting front,mainly as a result of highly nonlinear hydraulic properties.As time scales are slow,implicit numerical methods are required,and the convergence of nonlinear solvers is very sensitive.We propose an original method to ensure convergence of the numerical solution to the exact Richards solution,using a technique of auto-calibration of the penalty parameters derived from the Galerkin Discontinuous method.The method is constructed using nonlinear 1D and 2D general elliptic problems.We show that the numerical solution converges toward the unique solution of the continuous problem under certain conditions on the penalty parameters.Then,we numerically demonstrate the efficiency and robustness of the method through test cases with analytical solutions,laboratory test cases,and large-scale simulations.
