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Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations

William Layton, Nathaniel Mays, Monika Neda, Catalin Trenchea (2014)

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

We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing...

Numerical analysis of nonlinear elliptic-parabolic equations

Emmanuel Maitre (2002)

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

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern’s iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

Numerical analysis of nonlinear elliptic-parabolic equations

Emmanuel Maitre (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

Numerical analysis of the meshless element-free Galerkin method for hyperbolic initial-boundary value problems

Yaozong Tang, Xiaolin Li (2017)

Applications of Mathematics

The meshless element-free Galerkin method is developed for numerical analysis of hyperbolic initial-boundary value problems. In this method, only scattered nodes are required in the domain. Computational formulae of the method are analyzed in detail. Error estimates and convergence are also derived theoretically and verified numerically. Numerical examples validate the performance and efficiency of the method.

Numerical analysis of the Navier-Stokes equations

Rolf Rannacher (1993)

Applications of Mathematics

This paper discusses some conceptional questions of the numerical simulation of viscous incompressible flow which are related to the presence of boundaries.

Numerical approach to a rate-independent model of decohesion in laminated composites

Zeman, Jan, Gruber, Pavel (2010)

Programs and Algorithms of Numerical Mathematics

In this paper, we present a numerical approach to evolution of decohesion in laminated composites based on incremental variational problems. An energy-based framework is adopted, in which we characterize the system by the stored energy and dissipation functionals quantifying reversible and irreversible processes, respectively. The time-discrete evolution then follows from a solution of incremental minimization problems, which are converted to a fully discrete form by employing the conforming finite...

Numerical approaches to rate-independent processes and applications in inelasticity

Alexander Mielke, Tomáš Roubíček (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence of subsequences of space-time discretizations even in case where the limit problem does not have a unique solution and we need no additional assumptions on higher regularity of the limit solution. The variety of general perspectives thus obtained is illustrated on several...

Numerical approaches to the modelling of quasi-brittle crack propagation

Jiří Vala (2023)

Archivum Mathematicum

Computational analysis of quasi-brittle fracture in cement-based and similar composites, supplied by various types of rod, fibre, etc. reinforcement, is crucial for the prediction of their load bearing ability and durability, but rather difficult because of the risk of initiation of zones of microscopic defects, followed by formation and propagation of a large number of macroscopic cracks. A reasonable and complete deterministic description of relevant physical processes is rarely available. Thus,...

Numerical Approximation of a Fractional-In-Space Diffusion Equation (II) – with Nonhomogeneous Boundary Conditions

Ilic, M., Liu, F., Turner, I., Anh, V. (2006)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 26A33 (primary), 35S15In this paper, a space fractional diffusion equation (SFDE) with nonhomogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the...

Numerical Approximation of a Fractional-In-Space Diffusion Equation, I

Ilic, M., Liu, F., Turner, I., Anh, V. (2005)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 26A33 (primary), 35S15 (secondary)This paper provides a new method and corresponding numerical schemes to approximate a fractional-in-space diffusion equation on a bounded domain under boundary conditions of the Dirichlet, Neumann or Robin type. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix...

Numerical approximation of flow in a symmetric channel with vibrating walls

Sváček, Petr, Horáček, Jaromír (2010)

Programs and Algorithms of Numerical Mathematics

In this paper the numerical solution of two dimensional fluid-structure interaction problem is addressed. The fluid motion is modelled by the incompressible unsteady Navier-Stokes equations. The spatial discretization by stabilized finite element method is used. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian (ALE) method. The time-space problem is solved with the aid of multigrid method. The method is applied onto a problem of interaction of channel...

Numerical approximation of Knudsen layer for the Euler-Poisson system

Fréderique Charles, Nicolas Vauchelet, Christophe Besse, Thierry Goudon, Ingrid Lacroix–Violet, Jean-Paul Dudon, Laurent Navoret (2011)

ESAIM: Proceedings

In this work, we consider the computation of the boundary conditions for the linearized Euler–Poisson derived from the BGK kinetic model in the small mean free path regime. Boundary layers are generated from the fact that the incoming kinetic flux might be far from the thermodynamical equilibrium. In [2], the authors propose a method to compute numerically the boundary conditions in the hydrodynamic limit relying on an analysis of the boundary layers....

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Noel J. Walkington (2011)

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

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations*

Noel J. Walkington (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

Numerical approximation of the inviscid 3D primitive equations in a limited domain

Qingshan Chen, Ming-Cheng Shiue, Roger Temam, Joseph Tribbia (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

Numerical approximation of the inviscid 3D primitive equations in a limited domain

Qingshan Chen, Ming-Cheng Shiue, Roger Temam, Joseph Tribbia (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

Currently displaying 21 – 40 of 105