Mathematical Models and Methods in Applied Sciences

This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier–Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier–Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

This paper is concerned with the direct and inverse time-harmonic electromagnetic scattering problems for a finite number of isotropic point-like obstacles in three dimensions. In the first part, we show that the representation of scattered fields obtained using the Foldy physical assumption "on the proportionality of the strength of the scattered wave on a given scatterer to the external field on it" is the same as the one derived from the model corresponding to the scattering by Dirac-like refraction indices. Using the regularization approach known in quantum mechanics, we rigorously deduce the solution operator (Green's tensor) of the last model in appropriate weighted spaces. Intermediate levels of the scattering between the Born and Foldy models are also described. In the second part, we apply the MUSIC algorithm to the inverse problem of detecting both the position of point-like scatterers and the scattering coefficients attached to them from the far-field measurements of finitely many incident plane waves, with an emphasis on discussing the effect of multiple scattering.

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary differential equations (ODEs) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini's type. The coupling of the partial differential equation (PDE) and the ODE's is given by nonlinear Robin boundary conditions. This paper provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit–explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved efficiently by monotone multigrid. Numerical experiments are provided using the DUNE numerics framework.

We present a new approximate Riemann solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general nonlinear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated entropic shocks and we prove that it is positive, Lipschitz-continuous and satisfies an entropy inequality. Finally, the ARS is used to develop either a classical entropy conservative Godunov-type method, or a Glimm-type (random sampling-based Godunov-type) method able to generate infinitely sharp discrete shock profiles. Numerical experiments are proposed to prove the validity of these approaches.

We investigate the unique solvability of radiative transfer problems without strictly positive lower bounds on the absorption and scattering parameters. The analysis is based on a reformulation of the transfer equation as a mixed variational problem with penalty term for which we establish the well-posedness. We also prove stability of the solution with respect to perturbations in the parameters. This allows to approximate stationary radiative transfer problems by even-parity formulations even in the case of vanishing absorption. The mixed variational framework used for the analysis also enables a systematic investigation of discretization obtained by Galerkin methods. We show that, in contrast to the full problem, the widely used P_N-approximations, and discretizations based on these, are not stable in the case of vanishing absorption. Some consequences and possible remedies yielding stable approximations are discussed.

We recall a recently introduced mixed formulation of thin film magnetization problems for type-II superconductors written in terms of two variables, the electric field and the magnetization function, see [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. A finite element approximation, , based on this mixed formulation, involving the lowest-order Raviart–Thomas element for approximating the electric field, was also introduced in [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. Here h, τ are the spatial and temporal discretization parameters, and with p-1 the value of power in the current–voltage relation characterizing the superconducting material. In this paper, we establish well-posedness of , and prove convergence of the unique solution of to a solution of the power law model (Q_r), for a fixed r > 1, as h, τ → 0. In addition, we prove convergence of a solution of (Q_r) to a solution of the critical state model (Q), as r → 1. Hence, we prove existence of solutions to (Q_r), for a fixed r > 1, and (Q). Finally, numerical experiments are presented.

We introduce the concept of amplitude–phase decompositions and apply it to the study of growth and decay of solutions of the incompressible Navier–Stokes and Euler equations. Amplitudes coincide with functionals of physical and mathematical interest. We are able to explicitly solve for the amplitudes in terms of the phases. The results obtained provide insights into the growth and decay of enstrophy and viscous dissipation, and identify new criteria for solutions to remain smooth for all time or blow-up in finite time.