Nonlinear Advection-diffusion Equation Research Articles

The Splitting Characteristic Finite Difference Domain Decomposition Scheme for Solving Time-Fractional MIM Nonlinear Advection–Diffusion Equations

The Splitting Characteristic Finite Difference Domain Decomposition Scheme for Solving Time-Fractional MIM Nonlinear Advection–Diffusion Equations

A nonlinear system to model communication between yeast cells during their mating process

In this work, we develop a model to describe some aspects of communication between yeast cells. It consists in a coupled system of two one-dimensional non-linear advection-diffusion equations in which the advective field is given by the Hilbert transform. We give some sufficient condition for the blow-up in finite time of the coupled system (formation of a singularity). We provide a biological interpretation of these mathematical results.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

AbstractOn the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.

Effect of electromigration dispersion and non-Newtonian rheology of a charged solute in a microcapillary

The present work is concerned with the electromigration interaction of non-Newtonian fluid in a rectangular micro-capillary under the influence of an external electric field to predict the spatiotemporal dynamics of the solute concentration due to an effective dispersion and migration velocity. The solute concentration is optimized by dispersion and a driving force exploiting the interplay between the sequential ionic distribution and the local electrical conductivity coupled with the characteristics of the fluid. The incompressible Navier–Stokes equation combined with the Poisson equation for the electric field is considered for the flow transport incorporated with the Nernst–Planck equation for the ion transport. The numerical computations are performed for the coupled electro-osmosis/electrophoresis migrated nonlinear equations by a control volume approach for effective dispersion. The analytical observation of electrical conductivity in the case of a planar uniformly charged substrate is found to be varied locally near the sample peak and majorly concentration dependent. The asymptotic analysis for the velocity is made by using the lubrication approximation. The solutal species calculation is made from an area averaged nonlinear advection diffusion equation incorporating the coupled momentum equation. It is observed that the Taylor–Aries dispersion effect is dependent on the flow behavior index of the power law fluid, the flow strength, and the local sample concentration. The study of the time regime and the flow strength dependent instantaneous dispersion has also been conducted.

Characterizing water-on-deck propagation with a nonlinear advection-diffusion equation

Characterizing water-on-deck propagation with a nonlinear advection-diffusion equation

Approximation rates of DeepONets for learning operators arising from advection–diffusion equations

We present the analysis of approximation rates of operator learning in Chen and Chen (1995) and Lu et al. (2021), where continuous operators are approximated by a sum of products of branch and trunk networks. In this work, we consider the rates of learning solution operators from both linear and nonlinear advection–diffusion equations with or without reaction. We find that the approximation rates depend on the architecture of branch networks as well as the smoothness of inputs and outputs of solution operators.

A brief review on the solutions of advection-diffusion equation

In this work both linear and nonlinear advection-diffusion equations are considered and discussed their analytical solutions with different initial and boundary conditions. The work of Ogata and Banks, Harleman and Rumer, Cleary and Adrian, Atul Kumar et al., Mojtabi and Deville are reviewed for linear advection-diffusion equations and for nonlinear, we have chosen the work of Sakai and Kimura. Some enthusiastic functions used in the articles, drawbacks and applications of the results are discussed. Reduction of the advection-diffusion equations into diffusion equations make the governing equation solvable by using integral transform method for analytical solution. For nonlinear advection-diffusion equations, the Cole-Hopf transformation is used to reduce into the diffusion equation. Different dispersion phenomena in atmosphere, surface and subsurface area are outlined.

Strongly Nonlinear Diffusion in Turbulent Environment: A Problem with Infinitely Many Couplings

The field theoretic renormalization group is applied to the strongly nonlinear stochastic advection-diffusion equation. The turbulent advection is modelled by the Kazantsev–Kraichnan “rapid-change” ensemble. As a requirement of the renormalizability, the model necessarily involves infinite number of coupling constants (“charges”). The one-loop counterterm is calculated explicitly. The corresponding renormalization group equation demonstrates existence of a pair of two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain infrared attractive regions, the problem allows for the large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant), the critical dimensions of the scalar field Δθ, the response field Δθ′ and the frequency Δω are nonuniversal (through the dependence on the effective couplings) but satisfy certain exact identities. For the second surface (advection is relevant), the dimensions are universal and they are found exactly.

Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order

In this article, neural network method (NNM) is presented to solve the spatiotemporal variable-order fractional advection-diffusion equation with a nonlinear source term. The network is established by using shifted Legendre orthogonal polynomials with adjustable coefficients. According to the properties of variable fractional derivative, the loss function of neural network is deduced theoretically. Assume that the source function satisfies the Lipschitz hypothesis, the reasonable range for learning rate is discussed in details. Then neural networks are trained repeatedly on the training set to reduce the loss functions for two numerical examples. Numerical results show that the neural network method is better than the finite difference method in solving some nonlinear variable fractional order problems. Finally, several graphs and some numerical analysis are given to confirm our conclusions.

Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model

Abstract In this study, we give the numerical scheme to the stochastic nonlinear advection diffusion equation. This models is considered with white noise (or random process) having same intensity by changing frequencies. Furthermore, the stability and consistency of proposed scheme are also discussed. Moreover, it is concerned about the analytical solutions, the Riccati equation mapping method is adopted. The different families of single (shock and singular) and mixed (complex solitary-shock, shock-singular, and double-singular) form solutions are obtained with the different choices of free parameters. The graphical behavior of solutions is also depicted in 3D and corresponding contours.

A novel three-level time-split approach for solving two-dimensional nonlinear unsteady convection-diffusion-reaction equation

This paper considers a deep analysis of a three-level explicit time-split MacCormack method, namely the locally one-dimensional explicit MacCormack for the numerical solution of the two-dimensional nonlinear evolutionary advection-diffusion equation subjects to suitable initial and boundary conditions. The splitting reduces the computational cost of the algorithm. Under a suitable time-step restriction, both theoretical and numerical results on the stability and error estimates of the scheme are deeply analyzed in \(L^{m}(0,T;L^{2})\)-norm (\(m=1,2,\infty\)). The numerical experiments suggest that the proposed algorithm is easy to implement, temporal second-order convergent and fourth-order accurate in space. This shows the utility and efficiency of the considered method.

A multi level linearized Crank–Nicolson scheme for Richards equation under variable flux boundary conditions

The Richards equation is a nonlinear degenerate advection diffusion equation that models flow in saturated/unsaturated porous media, it's crucially important for prediction of disasters when heavy rain attacks. Efficient and precise linearized numerical schemes are necessary, but there is few study related it, and the numerical theory is incomplete because of the degeneracy and strong nonlinearity. In this paper, we establish a linearized Crank–Nicolson finite difference scheme which is a three-level scheme with almost second-order accuracy. In stability analysis, we develop a creative technique to overcome the degeneracy by adding a small positive perturbation ϵ. We also propose the error estimates by applying Young's inequality and prove the convergence order is approximate to second-order. Numerical examples are also provided to verify our main results and show the relationship between the computational error and ϵ is linear.

A Nitsche stabilized finite element method: Application for heat and mass transfer and fluid–structure interaction

A Nitsche stabilized finite element method is proposed for heat & mass transfer and fluid–structure interaction. The GLS/PSPG stabilization is employed to stabilize the finite element formulation. The Nitsche’s methods are employed to weakly impose the Dirichlet condition for heat & mass transfer. An upwind term is included in Nitsche’s methods to enhance the stability for fast moving interfaces. The Ghost penalty method is employed to control the jumps across the cut cells. The projection-based adaptive Gauss quadrature (PAGQ) scheme is used for the numerical integration of the discontinuous function. The nonlinear advection–diffusion equations are linearized by Newton procedure. The second-order accurate unconditionally stable generalized-α time integration is implemented to march the solution in time. The fluid and the structure equations are weakly coupled by a second-order accurate staggered-partitioned scheme. Numerical examples include the cases of fixed/vibrating/rotating cylinder(s) for heat & mass transfer and fluid–structure interaction in enclosure and external flow. The obtained numerical results match well with the experiments, the empirical correlation and the numerical simulations in literature. This methodology efficiently and accurately simulates the complex physics of heat & mass transfer.

Numerical solution of highly non-linear fractional order reaction advection diffusion equation using the cubic B-spline collocation method

Abstract In this article, the approximate solution of the fractional-order reaction advection-diffusion equation with the prescribed initial and boundary conditions is found with the help of a cubic B-spline collocation method, which is unconditionally stable and convergent. The accuracy of the scheme is validated by applying the method on four existing problems having analytical solutions and through the evaluation of the absolute errors between numerical results and the exact solutions for different particular cases. Applying the proposed method on the last two numerical problems, it is shown that the method performs better than the existing methods even for very less number of spatial and temporal discretizations. The main contribution of the article is to develop an efficient method to solve the proposed fractional order nonlinear problem and to find the effect on solute concentration graphically due to increase in the non-linearity in the diffusion term for different particular values of parameters.

A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences

In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium. Firstly, we classify all (point) symmetries of the equation, which prompt three cases of n. Next, for each case, we construct an optimal system of one-dimensional subalgebras and use them to perform symmetry reductions and symmetry invariant solutions. In a bid to explain the physical significance of some invariant solutions secured, we present a graphic display of some solutions in 3D, 2D as well as density plots via the exploitation of numerical simulations. Besides, we categorically state here that the results obtained in this study are new when compared with the outcomes previously achieved by Loubens et al., 2011 Quart. Appl. Math. 69 389–401. Interestingly, kink shape soliton, dark soliton, singular soliton together with exponential function solution wave profiles are displayed to make this work more valuable. Furthermore, we determine the conserved vectors in two different ways: engaging the general multiplier approach and Ibragimov’s conservation law theorem. Finally, we provide the physical meaning of these conservation laws.

A meshless technique based on the moving least squares shape functions for nonlinear fractal-fractional advection-diffusion equation

This paper introduces a fractal-fractional version of the nonlinear 2D advection-diffusion equation and proposes a meshless method based on the moving least squares shape functions for its numerical solution. The fractal-fractional derivative in the Atangana-Riemann-Liouville is considered to define this equation. The proposed method includes the following steps: We first approximate the fractal-fractional derivative using the finite differences method and derive a recursive algorithm by applying the θ-weighted method. Next, using the moving least squares shape functions, we expand the solution of the problem and its corresponding partial derivatives and substitute them into the recurrence formula. Finally, in accordance with the previous step, we obtain a linear system of algebraic equations which must be solved at each time step. The validity and accuracy of the method are investigated by solving some numerical examples.

A coarse-grid projection method for accelerating incompressible MHD flow simulations

Coarse grid projection (CGP) is a multiresolution technique for accelerating numerical calculations associated with a set of nonlinear evolutionary equations along with stiff Poisson’s equations. In this article, we use CGP for the first time to speed up incompressible magnetohydrodynamics (MHD) flow simulations. Accordingly, we solve the nonlinear advection–diffusion equation on a fine mesh, while we execute the electric potential Poisson equation on the corresponding coarsened mesh. Mapping operators connect two grids together. A pressure correction scheme is used to enforce the incompressibility constrain. The study of incompressible flow past a circular cylinder in the presence of Lorentz force is selected as a benchmark problem with a fixed Reynolds number but various Stuart numbers. We consider two different situations. First, we only apply CGP to the electric potential Poisson equation. Second, we apply CGP to the pressure Poisson equation as well. The maximum speed-up factors achieved here are approximately 3 and 23, respectively, for the first and second situations. For the both situations, we examine the accuracy of velocity and vorticity fields as well as the lift and drag coefficients. In general, the results obtained by CGP are in an excellent to reasonable range of accuracy. The CGP results are significantly more accurate compared to the numerical simulations of the advection–diffusion and electric potential Poisson equations on pure coarse scale grids.

A perturbative approach to a nonlinear advection-diffusion equation of particle transport

We explore analytical techniques for modeling the nonlinear cosmic ray transport in various astrophysical environments which is of significant current research interest. While nonlinearity is most often described by coupled equations for the dynamics of the thermal plasma and the cosmic ray transport or for the transport of the plasma waves and the cosmic rays, we study the case of a single but nonlinear advection-diffusion equation. The latter can be approximately solved analytically or semi-analytically, with the advantage that these solutions are easy to use and, thus, can facilitate a quantitative comparison to data. In the present study, we extend our previous work in a twofold manner. First, instead of employing an integral method to the case of pure nonlinear diffusion, we apply an expansion technique to the advection-diffusion equation. We use the technique systematically to analyze the effect of nonlinear diffusion for the cases of constant and spatially varying advection combined with time-varying source functions. Second, we extend the study from the one-dimensional, Cartesian geometry to the radially symmetric case, which allows us to treat more accurately the nonlinear diffusion problems on larger scales away from the source.