In part I of the paper (see Zlámal [13]) finite element solutions of the nonstationary semiconductor equations were constructed. Two fully discrete schemes were proposed. One was nonlinear, the other partly linear. In this part of the paper we justify the nonlinear scheme. We consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable. Further, we show that the approximate solution, extended to the whole time interval as a piecewise...

We consider the use of finite volume methods for the approximation of a parabolic variational inequality arising in financial mathematics. We show, under some regularity conditions, the convergence of the upwind implicit finite volume scheme to a weak solution of the variational inequality in a bounded domain. Some results, obtained in comparison with other methods on two dimensional cases, show that finite volume schemes can be accurate and efficient.

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step...

In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation $u_t(x,t)+\text{div}F(x,t,u(x,t))=0$ with the initial condition $u_0\in L^{\infty }\left({ℝ}^N\right)$. Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in L^{i}nfty({ℝ}^N\times {ℝ}_{+})$. Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L_{loc}^p({ℝ}^N\times {ℝ}_{+})$, 1 ≤ p ≤ +∞. Furthermore, if $u_0\in L^{\infty }\cap {\text{BV}}_{loc}\left({ℝ}^N\right)$, we show that $u\in {\text{BV}}_{loc}({ℝ}^N\times {ℝ}_{+})$, which leads to an “$h^{\frac{1}{4}}$” error estimate between the approximate and the entropy solutions (where h defines the size of the...

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics...

In this paper we deal with a problem of segmentation (including missing boundary completion) and subjective contour creation. For the corresponding models we apply the semi-implicit finite volume numerical schemes leading to methods which are robust, efficient and stable without any restriction to a time step. The finite volume discretization enables to use the spatial adaptivity and thus improve significantly the computational time. The computational results related to image segmentation with partly...

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. Non-Newtonian...

This paper aims at the development of numerical schemes for nonlinear reaction diffusion problems with a convection that blows up in a finite time. A full discretization of this problem that preserves the blow - up property is presented as well as a numerical simulation. Efficiency of the method is derived via a numerical comparison with a classical scheme based on the Runge Kutta scheme.

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order...