Convergence analysis of finite difference schemes for semi-linear initial-value problems
Convergence and Order Reduction of Runge-Kutta Schemes Applied to Evolutionary Problems in Partial Differential Equations.
Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical...
Convergence of a lattice numerical method for a boundary-value problem with free boundary and nonlinear Neumann boundary conditions.
Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem
We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...
Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem
We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...
Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation
We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation’s gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We...
Convergence of an entropic semi-discretization for nonlinear Fokker-Planck equations in Rd
Convergence of approximate solutions to scalar conservation laws by degenerate diffusion
Convergence of discretized attractors for parabolic equations on the line.
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 finite element approximation to quasilinear initial boundary value problems
Convergence of functionals and its applications to parabolic equations.
Convergence of iterative methods applied to generalized Fisher equation.
Convergence of Rothe's method in Hölder spaces
The convergence of Rothe’s method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe’s approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.
Convergence of the 'relativistic' heat equation to the heat equation as c → ∞.
We prove that the entropy solutions of the so-called relativistic heat equation converge to solutions of the heat equation as the speed of light c tends to ∞ for any initial condition u0 ≥ 0 in L1(RN) ∩ L∞(RN).
Convergence Properties of the Runge-Kutta-Chebyshev Method.
Convergence to a positive equilibrium for some nonlinear evolution equations in a ball.
Convergence to stationary solutions in a model of self-gravitating systems
We study convergence of solutions to stationary states in an astrophysical model of evolution of clouds of self-gravitating particles.