The paper deals with numerical simulation of a compressible flow in time-dependent 2D domains with a special interest in medical applications to airflow in the human vocal tract. The mathematical model of this process is described by the compressible Navier-Stokes equations. For the treatment of the time-dependent domain, the arbitrary Lagrangian-Eulerian (ALE) method is used. The discontinuous Galerkin finite element method (DGFEM) is used for the space semidiscretization of the governing equations...

In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$u_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta u(t,x)+\delta |u_t{(t,x)|}^{p-1}{u}_t(t,x)={\mu \left|u(t,x)\right|}^{q-1}u(t,x),x\in \Omega ,t\ge 0,v_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta v(t,x)+\delta |v_t{(t,x)|}^{p-1}{v}_t(t,x)={\mu \left|v(t,x)\right|}^{q-1}v(t,x),x\in \Omega ,t\ge 0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \Omega ,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),x\in \Omega ,u{{|}_{{}_{\partial \Omega }}=v|}_{{}_{\partial \Omega }}=0$$ where $q>1$, $p\ge 1$, $\delta >0$, $\alpha >0$, $\beta \ge 0$, $\mu \in ℝ$ and $\Delta$ is the Laplacian in ${ℝ}^N$.

A three-parameter family of Boussinesq type systems in two space dimensions is considered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized...

Numerical simulations of time-dependent behaviour of advances structures need the analysis of systems of partial differential equations of hyperbolic type, whose semi-discretization, using the Fourier multiplicative decomposition together with the finite element or similar techniques, leads to large sparse systems of ordinary differential equations. Effective and robust methods for numerical evaluation of their solutions in particular time steps are required; thus still new computational schemes...

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem...

This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann...

In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last...

In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This...

We investigate the approximation of the evolution of compact hypersurfaces of ${ℝ}^N$ depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.

We investigate the approximation of the evolution of compact hypersurfaces of ${ℝ}^N$ depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface.