Convection-diffusion Systems Research Articles

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

AbstractEntropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.

High Order Approximations for LQR Control and LQG Estimation of Convection Diffusion Systems

In this paper we discuss higher order methods for approximating linear control systems governed by convection diffusion equations. In particular, we employ a hp - finite element method to construct low order approximations of the Riccati equations that arise in linear quadratic control and optimal state estimation. The method is based on using high order piecewise polynomial approximations that achieves accuracy by combining mesh refinement (h → 0) with high order polynomials (p → +∞). By employing high order methods one can obtain accuracy with fewer degrees of freedom than can be achieved with mesh refinement alone. These methods are well known in the simulation community and can be used to achieve high order simulations, but hp – methods have not been investigated as potential methods for approximating control systems governed by partial differential equations. We employ the hp – finite element method to develop approximations of the infinite dimensional Riccati operators that define optimal LQR and LQG controllers. Convergence and rates of convergence are presented. A numerical example is provided to illustrate the convergence rates for the hp – method and we close with a discussion of how such methods might be used to enhance the development of reduced order models.

An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.

Mesh Optimization for Discontinuous Galerkin Methods Using a Continuous Mesh Model

A method for anisotropic mesh adaptation and optimization for high-order discontinuous Galerkin schemes is presented. Given the total number of degrees of freedom, a metric-based method is proposed, which aims to globally optimize the mesh with respect to the norm of the error. This is done by minimizing a suitable error model associated with the approximation space. Advantages of using a metric-based method in this context are several. First, it facilitates changing and manipulating the mesh in a general anisotropic way. Second, defining a suitable continuous interpolation operator allows the use of an analytic optimization framework that operates on the metric field, rather than the discrete mesh. The formulation of the method is presented as well as numerical experiments in the context of convection–diffusion systems.

A cholera epidemic model in a spatiotemporally heterogeneous environment

The transmission of cholera, a water- and food-borne intestinal infection, involves complex interactions among human hosts, pathogens, and the environment, and is impacted by the spatial variation and seasonal fluctuation. In this paper, we propose a new deterministic model to investigate the spatiotemporal dynamics of cholera transmission. The model employs a reaction–convection–diffusion system to represent the spatial movement of the hosts and pathogens, and incorporates time-periodic parameters to describe the seasonality of the disease transmission and bacterial growth rates. Using the theory of principle eigenvalues, we define and analyze the basic reproduction number of this model, based on which we establish the threshold type results for cholera transmission in a spatiotemporally heterogeneous environment.

Some classes of inverse problems of determining the source function in convection–diffusion systems

We consider the generalized solvability of convection–diffusion systems and study the inverse problem of determining the right-hand side (the source function) of such a system from integral overdetermination data. The solution of a parabolic system is understood in the generalized sense, and distributions of certain classes are allowed as right-hand sides. Under certain conditions on the problem data, we show that the inverse problem is well-posed in the Sobolev classes.

Spacetime discontinuous Galerkin methods for solving convection-diffusion systems

In this paper, we present two new approaches for solving systems of hyperbolic conservation laws with correct physical viscosity and heat conduction terms such as the compressible Navier–Stokes equations. Our methods are extensions of the spacetime discontinuous Galerkin method for hyperbolic conservation laws developed by Hiltebrand and Mishra [26]. Following this work, we use entropy variables as degrees of freedom and entropy stable fluxes. For the discretization of the diffusion term, we consider two different approaches: the interior penalty approach, resulting in the ST-SIPG and the ST-NIPG method, and a variant of the local discontinuous Galerkin method, resulting in the ST-LDG method. We show entropy stability of the ST-NIPG and the ST-LDG method when applied to the compressible Navier–Stokes equations. For the ST-SIPG method, this result holds under an assumption on the computed solution. All schemes incorporate shock capturing terms. Therefore, the schemes can handle both regimes of underresolved and fully resolved physical diffusion. We present a numerical comparison of the three methods in one space dimension.

Mathematical modeling and optimal control problems in brain tumor targeted drug delivery strategies

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions–diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection–diffusion system (which includes the developed model for brain tumor targeted drug delivery system).

Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models

We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive, and accurately inferring parameters can be expensive, too. We address both problems by proposing an offline-online strategy for controlling systems with time-varying parameters. During the offline phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reflects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider two numerical test problems in the form of partial differential equations: a convection-diffusion system, and a model for flow through a porous medium. We demonstrate on those models that the proposed method successfully stabilizes the system model in the presence of process noise.

Control of chemical chaos through medium viscosity in a batch ferroin-catalysed Belousov-Zhabotinsky reaction.

In this paper we show that the active interplay of nonlinear kinetics and transport phenomena in a chemical oscillator can be exploited to induce and control chaos. To this aim we use as a model system the ferroin-catalysed Belousov-Zhabotinsky (BZ) oscillating reaction, which is known to evolve to characteristic chaotic transient dynamics when carried out under batch and unstirred conditions. In particular, chemical chaos was found to appear and disappear by following a Ruelle-Takens-Newhouse (RTN) scenario. Here we use medium viscosity as a bifurcation parameter to tune the reaction-diffusion-convection (RDC) interplay and force the reaction in a specific sequence of dynamical regimes: either (i) periodic → quasi-periodic → chaotic or (ii) periodic → quasi-periodic or (iii) only periodic. The medium viscosity can be set by adding different amounts of surfactant (sodium dodecyl sulphate), known to have a little impact on the reaction mechanism, above its critical micelle concentration. Experimental results are supported by means of numerical simulations of a RDC model, which combines self-sustained oscillations to the related chemically-induced buoyancy convection.

Entropy Stable Schemes for Degenerate Convection-Diffusion Equations

In this work, we construct entropy stable schemes for nonlinear degenerate convection-diffusion systems by extending the theoretical framework introduced by Tadmor for conservation laws. Entropy stable finite difference schemes are developed for systems with viscous terms in both conservative and nonconservative formulations. Extended versions that include extra viscosity are proposed for both formulations in order to prevent the appearance of numerical spurious oscillations at discontinuities of the physical solution. Finally, some numerical simulations are shown in order to illustrate the behavior and accuracy of the entropy stable methods for degenerate parabolic problems.

Parallel algorithm for convection-diffusion system based on least-squares procedure.

Combining subspace correction method with least-squares finite element procedure, we construct a new overlapping domain decomposition parallel algorithm for solving the first-order time-dependent convection–diffusion system. This algorithm is fully parallel. We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to given accuracy at each time step.

On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations

AbstractImplicit time integration schemes are popular because their relaxed stability constraints can result in better computational efficiency. For time-accurate unsteady simulations, it has been well recognized that the inherent dispersion and dissipation errors of implicit Runge-Kutta schemes will reduce the computational accuracy for large time steps. Yet for steady state simulations using the time-dependent governing equations, these errors are often overlooked because the intermediate solutions are of less interest. Based on the model equationdy/dt=(μ+iλ)yof scalar convection diffusion systems, this study examines the stability limits, dispersion and dissipation errors of four diagonally implicit Runge-Kutta-type schemes on the complex (μ+iλ)Δtplane. Through numerical experiments, it is shown that, as the time steps increase, the A-stable implicit schemes may not always have reduced CPU time and the computations may not always remain stable, due to the inherent dispersion and dissipation errors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate the convergence rate, and the dispersion errors may cause large oscillations of the numerical solutions. These errors, especially those of high wavenumber components, grow at large time steps. They lead to difficulty in the convergence of the numerical computations, and result in increasing CPU time or even unstable computations as the time step increases. It is concluded that an optimal implicit time integration scheme for steady state simulations should have high dissipation and low dispersion.

Spacetime discontinuous Galerkin methods for convection-diffusion equations

We have developed two new methods for solving convection-diffusion systems, with particular focus on the compressible Navier-Stokes equations. Our methods are extensions of a spacetime discontinuous Galerkin method for solving systems of hyperbolic conservation laws [3]. Following the original scheme, we use entropy variables as degrees of freedom and entropy stable numerical fluxes for the nonlinear convection term. We examine two different approaches for incorporating the diffusion term: the interior penalty method and the local discontinuous Galerkin approach. For both extensions, we can show an entropy stability result for convection-diffusion systems. Although our schemes are designed for systems, we focus on scalar convectiondiffusion equations in this contribution. This allows us to highlight our main ideas behind the stability proofs, which are the same for scalar equations and systems, in a simplified setting.

Near-Wall Hindered Diffusion in Convective Systems: Transport Limitations in Colloidal and Nanoparticulate Systems

Redox flow cells have a significant potential as efficient, scalable energy storage, and use of nanomaterials is likely to increase the energy density even further. Efficient cell design requires understanding of mass transport effects, and for colloidal systems the theoretical assumptions commonly used for molecular species require re-evaluation. In the present work the effect of near-wall hindered diffusion is investigated in the convective–diffusive system of a colloidal suspension of nanoparticles. The rotating disk electrode system is used as a model due to the wide applicability of the technique for battery testing. A major influence of near-wall hindered diffusion is observed in the resulting concentration profiles of the nanoparticles (aqueous concentration as a function of distance) and the current responses in the case of the colloidal suspensions, and the finding is likely to have a significant impact on the understanding of physical processes underlying the practical cell design and modeling.

High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

The main purpose of the paper is to show how to use implicit---explicit Runge---Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a large class of problems. This approach gives a great flexibility, and allows, in many cases the construction of simple linearly implicit schemes without any Newton's iteration. This is obtained by identifying the (possibly linear) dependence on the unknown of the system which generates the stiffness. Only the stiff dependence is treated implicitly, then making the whole method much simpler than fully implicit ones. The resulting schemes are denoted as semi-implicit R---K. We adopt several semi-implicit R---K methods up to order three. We illustrate the effectiveness of the new approach with many applications to reaction---diffusion, convection diffusion and nonlinear diffusion system of equations.

Counter-extrapolation method for conjugate interfaces in computational heat and mass transfer

In this paper a conjugate interface method is developed by performing extrapolations along the normal direction. Compared to other existing conjugate models, our method has several technical advantages, including the simple and straightforward algorithm, accurate representation of the interface geometry, applicability to any interface-lattice relative orientation, and availability of the normal gradient. The model is validated by simulating the steady and unsteady convection-diffusion system with a flat interface and the steady diffusion system with a circular interface, and good agreement is observed when comparing the lattice Boltzmann results with respective analytical solutions. A more general system with unsteady convection-diffusion process and a curved interface, i.e., the cooling process of a hot cylinder in a cold flow, is also simulated as an example to illustrate the practical usefulness of our model, and the effects of the cylinder heat capacity and thermal diffusivity on the cooling process are examined. Results show that the cylinder with a larger heat capacity can release more heat energy into the fluid and the cylinder temperature cools down slower, while the enhanced heat conduction inside the cylinder can facilitate the cooling process of the system. Although these findings appear obvious from physical principles, the confirming results demonstrates the application potential of our method in more complex systems. In addition, the basic idea and algorithm of the counter-extrapolation procedure presented here can be readily extended to other lattice Boltzmann models and even other computational technologies for heat and mass transfer systems.

Accurate boundary treatments for lattice Boltzmann simulations of electric fields and electro-kinetic applications

In this paper a novel boundary method is proposed for lattice Boltzmann simulations of electric potential fields with complex boundary shapes and conditions. A shifted boundary from the physical surface location is employed in simulations to achieve a better finite-difference approximation of the potential gradient at the physical surface. Simulations are presented to demonstrate the accuracy and capability of this method in dealing with complex surface situations. An example simulation of the electrical double layer and electro-osmotic flow around a three-dimensional spherical particle is also presented. These simulated results are compared with analytical predictions and are found to be in excellent agreement. This method could be useful for electro-kinetic and colloidal simulations with complex boundaries, and can also be readily extended to other phenomena and processes, such as heat transfer and convection-diffusion systems.

Global existence and blowing-up of solutions to a class of coupled reaction–convection–diffusion systems

Abstract This paper concerns the global existence and blowing-up of solutions to the homogeneous Neumann problem of a coupled reaction–convection–diffusion system. The critical Fujita curve is determined and blowing-up theorem of Fujita type is established. An interesting phenomenon is that the critical Fujita curve even could be the infinite due to the convection.

Analysis of a Singular Convection Diffusion System Arising in Turbulence Modelling

We shall study some singular stationary convection diffusion system governing the steady state of a turbulence model closely related to the one. We shall establish existence, positivity, and regularity results in a very general framework.