78 research outputs found
Numerical Methods for Solving Convection-Diffusion Problems
Convection-diffusion equations provide the basis for describing heat and mass
transfer phenomena as well as processes of continuum mechanics. To handle flows
in porous media, the fundamental issue is to model correctly the convective
transport of individual phases. Moreover, for compressible media, the pressure
equation itself is just a time-dependent convection-diffusion equation.
For different problems, a convection-diffusion equation may be be written in
various forms. The most popular formulation of convective transport employs the
divergent (conservative) form. In some cases, the nondivergent (characteristic)
form seems to be preferable. The so-called skew-symmetric form of convective
transport operators that is the half-sum of the operators in the divergent and
nondivergent forms is of great interest in some applications.
Here we discuss the basic classes of discretization in space: finite
difference schemes on rectangular grids, approximations on general polyhedra
(the finite volume method), and finite element procedures. The key properties
of discrete operators are studied for convective and diffusive transport. We
emphasize the problems of constructing approximations for convection and
diffusion operators that satisfy the maximum principle at the discrete level
--- they are called monotone approximations.
Two- and three-level schemes are investigated for transient problems.
Unconditionally stable explicit-implicit schemes are developed for
convection-diffusion problems. Stability conditions are obtained both in
finite-dimensional Hilbert spaces and in Banach spaces depending on the form in
which the convection-diffusion equation is written
Flux-splitting schemes for parabolic problems
To solve numerically boundary value problems for parabolic equations with
mixed derivatives, the construction of difference schemes with prescribed
quality faces essential difficulties. In parabolic problems, some possibilities
are associated with the transition to a new formulation of the problem, where
the fluxes (derivatives with respect to a spatial direction) are treated as
unknown quantities. In this case, the original problem is rewritten in the form
of a boundary value problem for the system of equations in the fluxes. This
work deals with studying schemes with weights for parabolic equations written
in the flux coordinates. Unconditionally stable flux locally one-dimensional
schemes of the first and second order of approximation in time are constructed
for parabolic equations without mixed derivatives. A peculiarity of the system
of equations written in flux variables for equations with mixed derivatives is
that there do exist coupled terms with time derivatives
Intensity-dependent reflectance modulation of femtosecond laser pulses in GaAs nanocylinders with magnetic resonances
Abstract We experimentally demonstrate modulation of reflectance in periodic arrays of subwavelength gallium arsenide nanocylinders with Mie-type resonances due to absorption saturation and changes in the refractive index of the semiconductor material of metasurface. The intensity-dependent reflectance modulation of up to 30% in the vicinity of the magnetic dipole resonance at a low laser fluence below 200 μ J/cm 2 is shown by I-scan measurements
On numerical solution of 1-D poroelasticity equations in a multilayered domain
In soil mechanics assumption of only vertical subsidence is often invoked and this leads to the one-dimensional model of poroelasticity. The classical model of linear poroelasticity is obtained by Biot [1], detailed derivation can be found e.g., in [2]. This model is applicable also to modelling certain processes in geomechanics, hydrogeology, petroleum engineering (see, e.g., [3, 8], in biomechanics (e.g., [9, 10]), in filtration (e.g., filter cake formation, see [15, 16, 17]), in paper manufacturing (e.g., [11, 12]), in printing (e.g., [13]), etc. Finite element and finite difference methods were applied by many authors for numerical solution of the Biot system of PDEs, see e.g. [3, 4, 5] and references therein. However, as it is wellknown, the standard FEM and FDM methods are subject to numerical instabilities at the first time steps. To avoid this, discretization on staggered grid was suggested in [4, 5]. A single layer deformable porous medium was considered there. This paper can be viewed as extension of [4, 5] to the case of multilayered deformable porous media. A finite volume discretization to the interface problem for the classical one-dimensional Biot model of consolidation process is applied here. Following assumptions are supposed to be valid: each of the porous layers is composed of incompressible solid matrix, it is homogeneous and isotropic. Furthermore, one of two following assumptions is valid: porous medium is not completely saturated and fluid is incompressible or porous medium is completely saturated and fluid is slightly compressible. The reminder of the paper is organised as follows. Next section presents the mathematical model. Third section is devoted to the dicsretization of the continuous problem. Fourth section contains the results from the numerical experiments
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