Convergence of a continuous BGK model for initial boundary-value problems for conservation laws.
Convergence of a difference method for a system of first order partial differential equations of hyperbolic type
Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions
Convergence of a two-grid algorithm for the control of the wave equation
We analyze the problem of boundary observability of the finite-difference space semidiscretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the meshsize) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm...
Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh.
Convergence of approximate solutions of a differential equation
Convergence of approximate solutions to scalar conservation laws by degenerate diffusion
Convergence of discretization procedures for problems whose entropy solutions are uniquely characterized by additional relations
Weak solutions of given problems are sometimes not necessarily unique. Relevant solutions are then picked out of the set of weak solutions by so-called entropy conditions. Connections between the original and the numerical entropy condition were often discussed in the particular case of scalar conservation laws, and also a general theory was presented in the literature for general scalar problems. The entropy conditions were realized by certain inequalities not generalizable to systems of equations...
Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients
We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough” coefficient function . We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, is in , thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations...
Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients
We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a "rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion...
Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction
In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.
Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function
This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial...
Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary
This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.
Convergence of numerical algorithms for semilinear hyperbolic system
Convergence of solutions of generalized Korteweg-de Vries-Burgers equations to those of first order equations
Convergence of the boundary control for the wave equation in domains with holes of critical size.
Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition
The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...
Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition
The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...
Convergence to a positive equilibrium for some nonlinear evolution equations in a ball.
Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system
We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.