Nonlinear Advection-diffusion Equation Research Articles

Solution of Nonlinear Advection-Diffusion Equations via Linear Fractional Map Type Nonlinear QCA

Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schrodinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that some nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from LFMT NLQCA.

Lattice-free models of directed cell motility

Directed cell migration often occurs when individual cells move in response to an external chemical stimulus. Cells can respond by moving in either the direction of increasing (chemoattraction) or decreasing (chemorepulsion) concentration. Many previous models of directed cell migration use a lattice-based framework where agents undergo a lattice-based random walk and the direction of nearest-neighbour motility events is biased in a preferred direction. Such lattice-based models can lead to unrealistic configurations of agents, since the agents always move on an artificial lattice structure which is never observed experimentally. We present a lattice-free model of directed cell migration that incorporates two key features. First, agents move on a continuous domain, with the possibility that there is some preferred direction of motion. Second, to be consistent with experimental observations, we enforce a crowding mechanism so that motility events that would lead to agent overlap are not permitted. We compare simulation data from the new lattice-free model with a more traditional lattice-based model. To provide additional insight into the lattice-free model, we construct an approximate conservation statement which corresponds to a nonlinear advection–diffusion equation in the continuum limit. The solution of this mean-field model compares well with averaged data from the individual-based model.

Using generating functions to convert an implicit (3,3) finite difference method to an explicit form on diffusion equation with different boundary conditions

In this article, our main goal is to develop an idea to convert an implicit (3,3) ??-scheme finite difference method to an explicit form for both linear and nonlinear diffusion equations and also for nonlinear advection-diffusion equation with different boundary conditions. Accordingly, we assist power series generating functions which are a routine method in discrete mathematics. Also, the stability analysis of ??---scheme to implement in nonlinear advection---diffusion equation has been investigated. Finally, the new approach has been implemented for Fisher, reaction---diffusion, Burgers and coupled Burgers equations as test problems to verify the ability and efficiency of the method proposed in this paper.

Viscous fluid injection into a confined channel

We analyze the injection of a viscous fluid into a two-dimensional horizontal confined channel initially filled with another viscous fluid of different density and viscosity. We study the flow using the lubrication approximation and assume that the mixing between the fluids and their interfacial tension are negligible. When the injection rate is maintained constant, the evolution of the fluid-fluid interface can be described by a nonlinear advection-diffusion equation dependent only on the viscosity ratio between the two fluids. In the early time period, the advection-diffusion equation reduces to a well-known nonlinear diffusion equation, and a self-similar solution is obtained. In the late time period, the advection-diffusion equation is approximated by a nonlinear hyperbolic equation, and a compound wave solution is constructed to describe the time evolution of the fluid-fluid interface. Numerical solutions of the full equation show good agreement with the analytical solutions in both the early and late time periods. Finally, a regime diagram is obtained to summarize the flow behaviours with regard to two dimensionless groups: the viscosity ratio of the two fluids and the dimensionless time; three different dynamical behaviours are identified in the regime diagram: a nonlinear diffusion regime, a hyperbolic regime, and a transition regime. This problem is analogous to the corresponding injection flow problem into a confined porous medium.

Self-similar evolution of self-gravitating viscous accretion discs

A new one-dimensional, dynamical model is proposed for geometrically thin, self-gravitating viscous accretion discs. The vertically integrated equations are simplified using the slow accretion limit and the monopole approximation with a time-dependent central point mass to account for self-gravity and accretion. It is shown that the system of partial differential equations can be reduced to a single non-linear advection diffusion equation which describes the time evolution of angular velocity. In order to solve the equation three different turbulent viscosity prescriptions are considered. It is shown that for these parametrizations the differential equation allows for similarity transformations depending only on a single non-dimensional parameter. A detailed analysis of the similarity solutions reveals that this parameter is the initial power law exponent of the angular velocity distribution at large radii. The radial dependence of the self-similar solutions is in most cases given by broken power laws. At small radii the rotation law always becomes Keplerian with respect to the current central point mass. In the outer regions the power law exponent of the rotation law deviates from the Keplerian value and approaches asymptotically the value determined by the initial condition. It is shown that accretion discs with flatter rotation laws at large radii yield higher accretion rates. The methods are applied to self-gravitating accretion discs in active galactic nuclei. Fully self-gravitating discs are found to evolve faster than nearly Keplerian discs. The implications on supermassive black hole formation and Quasar evolution are discussed.

Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection–diffusion equations

In this study, we consider free boundary problems for nonlinear advection–diffusion equations of the form ut−uxx+βux=f(u) for t>0, g(t)<x<h(t), where x=g(t) and x=h(t) are free boundaries. This problem may be used to describe the spreading of a biological or chemical species where the free boundaries represent the expanding fronts. When f is a logistic nonlinearity, it has been shown that the asymptotic spreading speeds of the two fronts h(t) and g(t) are different due to the advection term. In this study, for monostable, bistable, and combustion types nonlinearities, we give much sharper estimates of the different spreading speeds of the fronts, and we also prove that the solution converges to a semi-wave in C2-norm as t→∞ when spreading occurs. We develop new approaches and extend a previous result to the problem with the advection term.

Nonlinear Tempering of Subdiffusion with Chemotaxis, Volume Filling, and Adhesion

The purpose of this paper is to implement nonlinear particle interactions into subdiffusive transport, involving chemotaxis and nonlinear effects such as volume filling and adhesion. We systematically derive nonlinear subdiffusive fractional equations with chemoattractant dependent forcing. We consider the diffusion limit of the master equation and analyse the role of nonlinear tempering in the stationary case. We study the interaction between attractive forces of anomalous aggregation and chemotaxis, with repulsive forces induced by nonlinear reactions. We show that this nonlinear interaction can prevent the phenomenon of anomalous aggregation when the local particle concentration grows too high. We also show that the effect of nonlinear tempering is to suppress the intermediate subdiffusive behaviour which results in a nonlinear advection diffusion equation in the long time limit.

Existence of solutions to nonlinear advection-diffusion equation applied to Burgers’ equation using Sinc methods

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers’ equation. The approximate solution is shown to converge to the exact solution at an exponential rate. A numerical example is given to illustrate the accuracy of the method.

Late-time asymptotic solution of nonlinear advection-diffusion equations with equal exponents

We derive the late-time asymptotic solution of a nonlinear advection-diffusion equation, u t = [ α / ( q − 1 ) ] ( u q ) x + ( 1 / q ) ( u q ) x x u_t = [\alpha /(q-1)](u^q)_x + (1/q) (u^q)_{xx} , where α ≠ 0 \alpha \ne 0 and q &gt; 2 q &gt; 2 . The equation is a more general form of the purely quadratic nonlinearity for advection and diffusion considered previously. For initial conditions with compact support, the solution has left and right moving boundaries, the distance between which is the width of the “plume”. We show the width to grow as t 1 / q t^{1/q} , with a constant correction term. The outer solution is dominated by the nonlinear advective term, the leading-order solution of which is shown to satisfy the partial differential equation and the right boundary condition exactly, but with a t t -dependent shifted argument. To satisfy the left boundary of vanishing plume thickness, a boundary layer is introduced, for which the inner solution may be obtained up to second order, again by using a shifted coordinate with respect to the wetting front. A leading-order composite solution for u u , uniformly correct to O ( 1 / t 1 / q ) O(1/t^{1/q}) , is obtained. The first and second-order terms are correct to O ( ( 1 / t 2 / q ) ln ⁡ t ) O((1/t^{2/q})\ln t) and O ( 1 / t 2 / q ) O(1/t^{2/q}) respectively. The composite second-order correction involves an arbitrary constant, implying its dependence on an unknown initial condition. Numerical results that agree with the analytical solutions are given along with an expression for the unknown constant computed with an impulse initial data.

Electromigration dispersion in a capillary in the presence of electro-osmotic flow

The differential migration of ions in an applied electric field is the basis for separation of chemical species by capillary electrophoresis. Axial diffusion of the concentration peak limits the separation efficiency. Electromigration dispersion is observed when the concentration of sample ions is comparable to that of the background ions. Under such conditions, the local electrical conductivity is significantly altered in the sample zone making the electric field, and therefore, the ion migration velocity concentration dependent. The resulting nonlinear wave exhibits shock like features, and, under certain simplifying assumptions, is described by Burgers' equation (S. Ghosal and Z. Chen Bull. Math. Biol. 201072, pg. 2047). In this paper, we consider the more general situation where the walls of the separation channel may have a non-zero zeta potential and are therefore able to sustain an electro-osmotic bulk flow. The main result is a one dimensional nonlinear advection diffusion equation for the area averaged concentration. This homogenized equation accounts for the Taylor-Aris dispersion resulting from the variation in the electro-osmotic slip velocity along the wall. It is shown that in a certain range of parameters, the electro-osmotic flow can actually reduce the total dispersion by delaying the formation of a concentration shock. However, if the electro-osmotic flow is sufficiently high, the total dispersion is increased because of the Taylor-Aris contribution.

New Analytical Solution to Water Content Simulation in Porous Media

AbstractThe nonlinear advection-diffusion equation, also known as Richard’s equation, is one of the most famous equations expressing water content in unsaturated porous media with broad applications in hydrology, engineering, and soil sciences. Because of the inherent nonlinearity in the equation, its closed-form analytical solution is rare. The traveling wave solution (TWS) is one of the recent exact approaches used in solving nonlinear partial differential equations (PDEs) and seems to be a suitable and efficient approach to obtain analytical solutions for Richard's equation. This paper presents three major cases of Richard’s equation that are tackled with the TWS method using general and modified forms of tanh⁡ function scheme to obtain the exact solution for each equation. The typical forms of diffusivity and conductivity functions proposed by Brooks and Corey are considered, and the exact solution for Richard’s equation is presented. Solutions have a broad applicability in pertinent engineering and s...

Asymptotic solution of a nonlinear advection-diffusion equation

We carry out an asymptotic analysis as t → ∞ t\rightarrow \infty for the nonlinear advection-diffusion equation, ∂ t u = 2 α u ∂ x u + ∂ x ( u ∂ x u ) \partial _t u = 2\alpha u \partial _x u + \partial _x(u \partial _x u) , where α \alpha is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with α \alpha having a specific physical significance related to the bed inclination. For compactly supported initial data, the solution is characterized by two moving boundaries propagating with finite speed and spanning a distance of O ( t ) \mathcal {O}(\sqrt {t}) . We construct an exact outer solution to the PDE that satisfies the right boundary condition. The vanishing condition at the left boundary is enforced by introducing a moving boundary layer, for which we obtain a closed-form expression. The leading-order composite solution is uniformly correct to O ( 1 / t ) \mathcal {O}(1/\sqrt {t}) . A higher-order correction to the inner and the composite solutions is also derived analytically. As a result, we obtain late-time asymptotic expansions for the two moving boundaries, correct to O ( 1 ) \mathcal {O}(1) , as well as a composite solution correct to O ( 1 / t ) \mathcal {O}(1/t) . The findings of this paper are illustrated and verified by numerical computations.

A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations

According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.

Experimental investigation into segregating granular flows down chutes

We experimentally investigated how a binary granular mixture made up of spherical glass beads (size ratio of 2) behaved when flowing down a chute. Initially, the mixture was normally graded, with all the small particles on top of the coarse grains. Segregation led to a grading inversion, in which the smallest particles percolated to the bottom of the flow, while the largest rose toward the top. Because of diffusive remixing, there was no sharp separation between the small-particle and large-particle layers, but a continuous transition. Processing images taken at the sidewall, we were able to measure the evolution of the concentration and velocity profiles. These experimental profiles were used to test a recent theory developed by Gray and Chugunov [J. Fluid Mech. 569, 365 (2006)], who derived a nonlinear advection diffusion equation that describes segregation and remixing in dense granular flows of binary mixtures. We found that this theory was able to provide a consistent description of the segregation/remixing process under steady uniform flow conditions. To obtain the correct depth-averaged concentration far downstream, it was very important to use an accurate approximation to the downstream velocity profile through the avalanche depth. The S-shaped concentration profile in the far field provided a useful way of determining what we refer to as the Péclet number for segregation, a dimensionless number that quantifies how large the segregation is compared to diffusive remixing. While the theory was able to closely match the final fully developed concentration profile far downstream, there were some discrepancies in the inversion region (i.e., the region in which the mixing occurs). The reasons for this are not clear. The difficulty to set up the experiment with both well controlled initial conditions and a steady uniform bulk flow field is one of the most plausible explanations. Another interesting lead is that the flux of segregating particles, which was assumed to be a quadratic function of the concentration in small beads, takes a more complicated form.

Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advection-diffusion equations, there are very few analytical results on their stability, even for the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably, the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters. Such optimal stability is reached owing to the free (“kinetic”) relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative equilibrium functions (the distribution divided by the local mass), often labeled as “unphysical”. We prove that the non-negativity condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT sub-classes, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT sub-class more efficient than the BGK one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes.

An efficient solver of the saturation equation in liquid composite molding processes

A major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy’s law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of saturation. The modified saturation equation is obtained as a result, which is a non-linear advection-diffusion equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified saturation equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds.

The Footprint of the CO2 Plume during Carbon Dioxide Storage in Saline Aquifers: Storage Efficiency for Capillary Trapping at the Basin Scale

We study a sharp-interface mathematical model of CO2 migration in deep saline aquifers, which accounts for gravity override, capillary trapping, natural groundwater flow, and the shape of the plume during the injection period. The model leads to a nonlinear advection–diffusion equation, where the diffusive term is due to buoyancy forces, not physical diffusion. For the case of interest in geological CO2 storage, in which the mobility ratio is very unfavorable, the mathematical model can be simplified to a hyperbolic equation. We present a complete analytical solution to the hyperbolic model. The main outcome is a closed-form expression that predicts the ultimate footprint on the CO2 plume, and the time scale required for complete trapping. The capillary trapping coefficient and the mobility ratio between CO2 and brine emerge as the key parameters in the assessment of CO2 storage in saline aquifers. Despite the many approximations, the model captures the essence of the flow dynamics and therefore reflects proper dependencies on the mobility ratio and the capillary trapping coefficient, which are basin-specific. The expressions derived here have applicability to capacity estimates by capillary trapping at the basin scale.

Exclusion processes on a growing domain

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker–Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection–diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.

A mathematical model of the footprint of the CO2 plume during and after injection in deep saline aquifer systems

Abstract We present a sharp-interface mathematical model of CO 2 migration in saline aquifers, which accounts for gravity override, capillary trapping, natural groundwater flow, and the shape of the plume during the injection period. The model leads to a nonlinear advection–diffusion equation, where the diffusive term is due to buoyancy forces, not physical diffusion. For the case of interest in geological CO 2 storage, in which the mobility ratio is very unfavorable, the mathematical model can be simplified to a hyperbolic equation. We present a complete analytical solution to the hyperbolic model. The main outcome is a closed-form expression that predicts the ultimate footprint on the CO 2 plume, and the time scale required for complete trapping. The capillary trapping coefficient emerges as the key parameter in the assessment of CO 2 storage in saline aquifers. The expressions derived here have immediate applicability to the risk assessment and capacity estimates of CO 2 sequestration at the basin scale. In a companion paper [Szulczewski and Juanes, GHGT-9, Paper 463 (2008)] we apply the model to specific geologic basins.

Multi-species simple exclusion processes

A motility mechanism based on a simple exclusion process, where the movement of discrete agents on a lattice is either unbiased (symmetric) or biased (asymmetric) is considered. Estimates of diffusivities from tracking data do not describe the population-level response of the system. This mismatch between the individual-level and population-level behaviour can be resolved by averaging the individual-level mechanism in terms of an expected site occupancy. New insight into simple exclusion processes is obtained by representing the system as a series of interacting subpopulations. This formalism leads to a system of nonlinear advection–diffusion equations which can be interpreted in terms of the agent fluxes. These interactions have consequences for both agent-based modelling and continuum modelling in cell biology, such as tracking subpopulations of cells within a total cell population.