2,540 research outputs found
Global Strong Solutions for a Class of Heterogeneous Catalysis Models
We consider a mathematical model for heterogeneous catalysis in a finite
three-dimensional pore of cylinder-like geometry, with the lateral walls acting
as a catalytic surface. The system under consideration consists of a
diffusion-advection system inside the bulk phase and a
reaction-diffusion-sorption system modeling the processes on the catalytic wall
and the exchange between bulk and surface. We assume Fickian diffusion with
constant coefficients, sorption kinetics with linear growth bound and a network
of chemical reactions which possesses a certain triangular structure. Our main
result gives sufficient conditions for the existence of a unique global strong
-solution to this model, thereby extending by now classical results on
reaction-diffusion systems to the more complicated case of heterogeneous
catalysis.Comment: 30 page
Preventing blow up by convective terms in dissipative PDEs
We study the impact of the convective terms on the global solvability or
finite time blow up of solutions of dissipative PDEs. We consider the model
examples of 1D Burger's type equations, convective Cahn-Hilliard equation,
generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish
the following common scenario: adding sufficiently strong (in comparison with
the destabilizing nonlinearity) convective terms to equation prevents the
solutions from blowing up in finite time and makes the considered system
globally well-posed and dissipative and for weak enough convective terms the
finite time blow up may occur similarly to the case when the equation does not
involve convective term.
This kind of result has been previously known for the case of Burger's type
equations and has been strongly based on maximum principle. In contrast to
this, our results are based on the weighted energy estimates which do not
require the maximum principle for the considered problem
Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation
In this article we consider the numerical approximation of the
convection-diffusion-reaction equation. One of the main challenges of designing
a numerical method for this problem is that boundary layers occurring in the
convection-dominated case can lead to non-physical oscillations in the
numerical approximation, often referred to as Gibbs phenomena. The idea of this
article is to consider the approximation problem as a residual minimization in
dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a
non-standard, non-linear PetrovGalerkin discretization, that is applicable to
reflexive Banach spaces such that the space itself and its dual are strictly
convex. Similar to discontinuous Petrov-Galerkin methods, this method is based
on minimizing the residual in a dual norm. Replacing the intractable dual norm
by a suitable discrete dual norm gives rise to a non-linear inexact mixed
method. This generalizes the Petrov-Galerkin framework developed in the context
of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the
convection-diffusion-reaction equation, this yields a generalization of a
similar approach from the L2-setting to the Lq-setting. A key advantage of
considering a more general Banach space setting is that, in certain cases, the
oscillations in the numerical approximation vanish as q tends to 1, as we will
demonstrate using a few simple numerical examples
Operation of a perfusive catalytic membrane with nonlinear kinetics
Operation of a perfusive catalytic curved membrane is systematized into different transport-reaction regimes. The internal viscous permeation improves the catalyst performance, measured here by the effectiveness factor and by its enhancement relative to purely diffusive conditions. A theoretical analysis is presented for nonlinear kinetic expressions, which are suitable to describe the consumption of a reactant in many (bio)catalytic systems. The kinetic and transport parameters required to attain maximum enhancement are related by simple design rules, which depend on the form of the reaction rate law (namely on the order of reaction and dimensionless inhibition constant). For zero-order reactions, these optimum conditions correspond to attaining negligible concentration at a position inside the membrane, while may be interpreted in general as separating situations of severe mass transfer resistance from cases of high effectiveness. It is important to incorporate the correct form of the kinetic expression in the analysis, so that the predictions can be used in a quantitative manner. The results for the different regimes are compiled in enhancement plots and in Peclet–Thiele diagrams. Moreover, the study also yielded new results for the nonlinear reaction–diffusion problem in a curved membrane with its two surfaces exposed to different concentrations, a case of relevance in membrane reactors
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
Unified convergence analysis of numerical schemes for a miscible displacement problem
This article performs a unified convergence analysis of a variety of
numerical methods for a model of the miscible displacement of one
incompressible fluid by another through a porous medium. The unified analysis
is enabled through the framework of the gradient discretisation method for
diffusion operators on generic grids. We use it to establish a novel
convergence result in of the approximate
concentration using minimal regularity assumptions on the solution to the
continuous problem. The convection term in the concentration equation is
discretised using a centred scheme. We present a variety of numerical tests
from the literature, as well as a novel analytical test case. The performance
of two schemes are compared on these tests; both are poor in the case of
variable viscosity, small diffusion and medium to small time steps. We show
that upstreaming is not a good option to recover stable and accurate solutions,
and we propose a correction to recover stable and accurate schemes for all time
steps and all ranges of diffusion
Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure
We present a numerical method for approximating the solutions of degenerate
parabolic equations with a formal gradient flow structure. The numerical method
we propose preserves at the discrete level the formal gradient flow structure,
allowing the use of some nonlinear test functions in the analysis. The
existence of a solution to and the convergence of the scheme are proved under
very general assumptions on the continuous problem (nonlinearities, anisotropy,
heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the
efficiency and of the robustness of our approach
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