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The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems

Edwige Godlewski, Kim-Claire Le Thanh, Pierre-Arnaud Raviart (2005)

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

We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We...

The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems

Edwige Godlewski, Kim-Claire Le Thanh, Pierre-Arnaud Raviart (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem....

The problem of data assimilation for soil water movement

François-Xavier Le Dimet, Victor Petrovich Shutyaev, Jiafeng Wang, Mu Mu (2004)

ESAIM: Control, Optimisation and Calculus of Variations

The soil water movement model governed by the initial-boundary value problem for a quasilinear 1-D parabolic equation with nonlinear coefficients is considered. The generalized statement of the problem is formulated. The solvability of the problem is proved in a certain class of functional spaces. The data assimilation problem for this model is analysed. The numerical results are presented.

The problem of data assimilation for soil water movement

François-Xavier Le Dimet, Victor Petrovich Shutyaev, Jiafeng Wang, Mu Mu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The soil water movement model governed by the initial-boundary value problem for a quasilinear 1-D parabolic equation with nonlinear coefficients is considered. The generalized statement of the problem is formulated. The solvability of the problem is proved in a certain class of functional spaces. The data assimilation problem for this model is analysed. The numerical results are presented.

The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics

Dana Lauerová (1990)

Aplikace matematiky

The existence of a periodic solution of a nonlinear equation z ' + A 0 z + B 0 z = F is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.

Theoretical analysis of the upwind finite volume scheme on the counter-example of Peterson

Daniel Bouche, Jean-Michel Ghidaglia, Frédéric P. Pascal (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

When applied to the linear advection problem in dimension two, the upwind finite volume method is a non consistent scheme in the finite differences sense but a convergent scheme. According to our previous paper [Bouche et al., SIAM J. Numer. Anal.43 (2005) 578–603], a sufficient condition in order to complete the mathematical analysis of the finite volume scheme consists in obtaining an estimation of order p, less or equal to one, of a quantity that depends only on the mesh and on the advection ...

Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations

Anne de Bouard, Arnaud Debussche, Laurent Di Menza (2001)

Journées équations aux dérivées partielles

We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...

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...

Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

Astha Chauhan, Rajan Arora (2019)

Communications in Mathematics

In this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are...

Transonic flow calculation via finite elements

Petr Klouček, Josef Málek (1988)

Aplikace matematiky

Using new results based on a convenient entropy condition, two types of algorithms for computing transonic flows are constructed. A sequence of solutions of the linearised problem with a posteriori control is constructed and its convergence to the physical solution of transonic flow in some special situations is proved. This paper contains also numerical results and their analysis for the case of flow past NACA 230012 airfoil. Some numerical improvements of the general algorithms, based on our...

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