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128
We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when and the integral kernel in the nonlocal boundary condition is symmetric.
We consider space semi-discretizations of the 1-d wave equation in a bounded
interval with homogeneous Dirichlet boundary conditions. We analyze the problem
of boundary observability, i.e., the problem of whether the total energy of
solutions can be estimated uniformly in terms of the energy concentrated on the
boundary as the net-spacing h → 0. We prove that, due to the spurious modes
that the numerical scheme introduces at high frequencies, there is no such a
uniform bound. We prove however a...
In this article, we present a numerical scheme based on a finite element method in order to solve a time-dependent convection-diffusion equation problem and satisfy some conservation properties. In particular, our scheme is able to conserve the total energy for a heat equation or the total mass of a solute in a fluid for a concentration equation, even if the approximation of the velocity field is not completely divergence-free. We establish a priori errror estimates for this scheme and we give some...
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.
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.
A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
This paper is aimed at the description of the multi-dimensional finite volume solver EULER, which has been developed for the numerical solution of the compressible Euler equations during several last years. The present overview of numerical schemes and the explanation of numerical techniques and tricks which have been used for EULER could be of certain interest not only for registered users but also for numerical mathematicians who have decided to implement a finite volume solver themselves. This...
This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves , . The curves are driven by the normal velocity which is the function of curvature and the position. The evolution law reads as: . The motion law is treated using direct approach numerically solved by two schemes, i. e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved...
The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation...
We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the...
The Rothe-Galerkin method is used for discretization. The rate of convergence in for the approximate solution of a quasilinear parabolic equation with a Volterra operator on the right-hand side is established.
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