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Theoretical and numerical studies of the P N P M DG schemes in one space dimension

Abdulatif Badenjki, Gerald G. Warnecke (2019)

Applications of Mathematics

We give a proof of the existence of a solution of reconstruction operators used in the P N P M DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the P N P M DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several P N P M DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency...

Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping

R. Belaouar, T. Colin, G. Gallice, C. Galusinski (2006)

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

In this paper, we study a Zakharov system coupled to an electron diffusion equation in order to describe laser-plasma interactions. Starting from the Vlasov-Maxwell system, we derive a nonlinear Schrödinger like system which takes into account the energy exchanged between the plasma waves and the electrons via Landau damping. Two existence theorems are established in a subsonic regime. Using a time-splitting, spectral discretizations for the Zakharov system and a finite difference scheme for the...

Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping

R. Belaouar, T. Colin, G. Gallice, C. Galusinski (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we study a Zakharov system coupled to an electron diffusion equation in order to describe laser-plasma interactions. Starting from the Vlasov-Maxwell system, we derive a nonlinear Schrödinger like system which takes into account the energy exchanged between the plasma waves and the electrons via Landau damping. Two existence theorems are established in a subsonic regime. Using a time-splitting, spectral discretizations for the Zakharov system and a finite difference scheme for...

Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Nikolaos Bournaveas, Georgios E. Zouraris (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Nikolaos Bournaveas, Georgios E. Zouraris (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion

Sören Bartels, Tomáš Roubíček (2011)

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

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful successive...

Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion*

Sören Bartels, Tomáš Roubíček (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful...

Time domain computational modelling of 1D arterial networks in monochorionic placentas

Victoria E. Franke, Kim H. Parker, Ling Y. Wee, Nicholas M. Fisk, Spencer J. Sherwin (2003)

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

In this paper we outline the hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/ h p element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance....

Time domain computational modelling of 1D arterial networks in monochorionic placentas

Victoria E. Franke, Kim H. Parker, Ling Y. Wee, Nicholas M. Fisk, Spencer J. Sherwin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we outline the hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance....

Time domain decomposition in final value optimal control of the Maxwell system

John E. Lagnese, G. Leugering (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a boundary optimal control problem for the Maxwell system with a final value cost criterion. We introduce a time domain decomposition procedure for the corresponding optimality system which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. In the limit full recovery of the coupling conditions is achieved, and, hence, the local solutions and controls converge to the global ones. The process is inherently parallel and is suitable for real-time...

Time Domain Decomposition in Final Value Optimal Control of the Maxwell System

John E. Lagnese, G. Leugering (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a boundary optimal control problem for the Maxwell system with a final value cost criterion. We introduce a time domain decomposition procedure for the corresponding optimality system which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. In the limit full recovery of the coupling conditions is achieved, and, hence, the local solutions and controls converge to the global ones. The process is inherently parallel and is suitable for real-time...

Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes

D. J. Mavriplis, Z. Yang (2011)

Mathematical Modelling of Natural Phenomena

For flows with strong periodic content, time-spectral methods can be used to obtain time-accurate solutions at substantially reduced cost compared to traditional time-implicit methods which operate directly in the time domain. However, these methods are only applicable in the presence of fully periodic flows, which represents a severe restriction for many aerospace engineering problems. This paper presents an extension of the time-spectral approach...

Time splitting for wave equations in random media

Guillaume Bal, Lenya Ryzhik (2004)

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

Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified...

Time splitting for wave equations in random media

Guillaume Bal, Lenya Ryzhik (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the...

Time-delay regularization of anisotropic diffusion and image processing

Abdelmounim Belahmidi, Antonin Chambolle (2005)

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

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell 12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell 14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

Time-delay regularization of anisotropic diffusion and image processing

Abdelmounim Belahmidi, Antonin Chambolle (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

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