Mathematical Models and Methods in Applied Sciences

The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully nonlinearly coupled formulation for the study of fluid structure interactions.

Many numerical methods have been introduced to reduce the difficulties related to the nonlinear coupling between the structure and the fluid evolution. However numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates of the variational formulation of the immersed boundary method.

A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly.

We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.

In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve some known results in Sobolev space. The local existence of classical solutions to the Cauchy problem is obtained by the regularized means and compactness argument. Using the high- and low-frequency decomposition method, we prove the global exponential stability of classical solutions (close to equilibrium). Furthermore, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The main analytic tools are the Littlewood–Paley decomposition and Bony's para-product formula.

The aim of this paper is to develop on the asymptotics of some one-dimensional nonlinear Schrödinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proved rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed. Several shortcomings of spectral time-splitting schemes are investigated.

We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by , where γ(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary, γ is a bounded function and we show the convergence of the solutions in H¹ and C^α norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.

This paper focuses on the optimization of traffic flow on a road network, modeled by a fluid-dynamic approach. Three cost functionals that measure average velocity, average traveling time, and total flux of cars, are considered. First, such functionals are optimized for two simple networks that consist of a single junction: one with two incoming and one outgoing roads (junctions of 2 × 1 type), and the other with one incoming and two outgoing roads (junctions of 1 × 2 type). The optimization is made with respect to right of way parameters and traffic distribution coefficients, obtaining an explicit solution. Then, through simulations, the traffic behavior for complex networks is studied. The main result is that the local optimization ensures a very good result also for the complete network. This is shown by the case study of Re di Roma Square, a big traffic circle of the urban network of Rome.

A convergence theory is established for a truncation method in solving polyconvex elasticity problems involving the Lavrentiev phenomenon. Numerical results on a recent example by Foss et al., which has a polyconvex integrand and admits continuous singular minimizers, not only verify our convergence theorems but also provide a sharper estimate on the upper bound of a perturbation parameter for the existence of the Lavrentiev phenomenon in the example.