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A numerical study of non-cavitating and cavitating liquid flow around a hydrofoil

François Beux, Maria-Vittoria Salvetti, Alexey Ignatyev, Ding Li, Charles Merkle, Edoardo Sinibaldi (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The results of a workshop concerning the numerical simulation of the liquid flow around a hydrofoil in non-cavitating and cavitating conditions are presented. This workshop was part of the conference “Mathematical and Numerical aspects of Low Mach Number Flows” (2004) and was aimed to investigate the capabilities of different compressible flow solvers for the low Mach number regime and for flows in which incompressible and supersonic regions are simultaneously present. Different physical models...

A numerical study of non-cavitating and cavitating liquid flow around a hydrofoil

François Beux, Maria-Vittoria Salvetti, Alexey Ignatyev, Ding Li, Charles Merkle, Edoardo Sinibaldi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The results of a workshop concerning the numerical simulation of the liquid flow around a hydrofoil in non-cavitating and cavitating conditions are presented. This workshop was part of the conference “Mathematical and Numerical aspects of Low Mach Number Flows” (2004) and was aimed to investigate the capabilities of different compressible flow solvers for the low Mach number regime and for flows in which incompressible and supersonic regions are simultaneously present. Different physical models...

A parabolic system involving a quadratic gradient term related to the Boussinesq approximation.

Jesús Ildefonso Díaz, Jean-Michel Rakotoson, Paul G. Schmidt (2007)

RACSAM

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. Based on adequate notions of weak and strong...

A parallel algorithm for two phase multicomponent contaminant transport

Todd Arbogast, Clint N. Dawson, Mary F. Wheeler (1995)

Applications of Mathematics

We discuss the formulation of a simulator in three spatial dimensions for a multicomponent, two phase (air, water) system of groundwater flow and transport with biodegradation kinetics and wells with multiple screens. The simulator has been developed for parallel, distributed memory, message passing machines. The numerical procedures employed are a fully implicit expanded mixed finite element method for flow and either a characteristics-mixed method or a Godunov method for transport and reactions...

A penalty algorithm for the spectral element discretization of the Stokes problem

Christine Bernardi, Adel Blouza, Nejmeddine Chorfi, Nizar Kharrat (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

A penalty algorithm for the spectral element discretization of the Stokes problem*

Christine Bernardi, Adel Blouza, Nejmeddine Chorfi, Nizar Kharrat (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems

Josef Dalík (1991)

Applications of Mathematics

A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation - ϵ u n + p u ' + q u = f are presented and analyzed theoretically. The positive number ϵ is supposed to be much less than the discretization step and the values of p , q . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

A phase-field method applied to interface tracking for blood clot formation

Marek Čapek (2020)

Applications of Mathematics

The high shear rate thrombus formation was only recently recognized as another way of thrombosis. Models proposed in Weller (2008), (2010) take into account this type of thrombosis. This work uses the phase-field method to model these evolving interface problems. A loosely coupled iterative procedure is introduced to solve the coupled system of equations. Convergence behavior on two levels of refinement of perfusion chamber geometry and cylinder geometry is then studied. The perfusion chamber simulations...

A piecewise P2-nonconforming quadrilateral finite element

Imbunm Kim, Zhongxuan Luo, Zhaoliang Meng, Hyun NAM, Chunjae Park, Dongwoo Sheen (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...

A plane problem of incompressible magnetohydro-dynamics with viscosity and resistivity depending on the temperature

Giovanni Cimatti (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The plane flow of a fluid obeying the equations of magnetohydrodynamics is studied under the assumption that both the viscosity and the resistivity depend on the temperature. Some results of existence, non-existence, and uniqueness of solution are proved.

A positivity preserving central scheme for shallow water flows in channels with wet-dry states

Jorge Balbás, Gerardo Hernandez-Duenas (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from...

A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

Alexandre Ern, Sébastien Meunier (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u , is governed by an elliptic equation and the other, say p , by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u - and p -components to obtain optimally convergent a priori bounds for all the terms in the error energy norm....

A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

Alexandre Ern, Sébastien Meunier (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u- and p-components to obtain optimally convergent a priori bounds for all the terms in the error energy...

A posteriori error analysis of the fully discretized time-dependent Stokes equations

Christine Bernardi, Rüdiger Verfürth (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

A posteriori error analysis of the fully discretized time-dependent Stokes equations

Christine Bernardi, Rüdiger Verfürth (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Mario Ohlberger (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation c t + ∇ · ( 𝐮 f ( c ) ) - ∇ · ( D ∇ c ) + λ c = 0 . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1 -norm, independent of the diffusion parameter D . The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...

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