Nonlinear Convection Diffusion Equation Research Articles

An adaptive multigrid conjugate gradient method for the inversion of a nonlinear convection-diffusion equation

Abstract This paper considers the problem of estimating the permeability in a nonlinear convection-diffusion equation. To overcome the large calculation burden of conventional methods, we apply an adaptive multigrid conjugate gradient method to solve this inverse problem. This new method combines the multigrid multiscale idea with the conjugate gradient method, and adopts the necessary condition that the optimum solution should be the fixed point of the multigrid inversion method. Some numerical results verify that the proposed method both dramatically reduces the required computations and improves the inversion quality.

Perturbed invariant subspaces and approximate generalized functional variable separation solution for nonlinear diffusion–convection equations with weak source

In this paper, we propose the concept of the perturbed invariant subspaces (PISs), and study the approximate generalized functional variable separation solution for the nonlinear diffusion–convection equation with weak source by the approximate generalized conditional symmetries (AGCSs) related to the PISs. Complete classification of the perturbed equations which admit the approximate generalized functional separable solutions (AGFSSs) is obtained. As a consequence, some AGFSSs to the resulting equations are explicitly constructed by way of examples.

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

AbstractWe propose an integrable discrete model of one‐dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time‐dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self‐adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.

Convergence analysis of mixed volume element-characteristic mixed volume element for three-dimensional chemical oil-recovery seepage coupled problem

The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in l2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.

A heuristic optimization method of fractional convection reaction: An application to diffusion process

The convection differential models play an essential role in studying different chemical process and effects of the diffusion process. This paper intends to provide optimized numerical results of such equations based on the conformable fractional derivative. Subsequently, a well-known heuristic optimization technique, differential evolution algorithm, is worked out together with the Taylor?s series expansion, to attain the optimized results. In the scheme of the Taylor optimization method (TOM), after expanding the functions with the Taylor?s series, the unknown terms of the series are then globally optimized using differential evolution. Moreover, to illustrate the applicability of TOM, some examples of linear and non-linear fractional convection diffusion equations are exemplified graphically. The obtained assessments and comparative demonstrations divulged the rapid convergence of the estimated solutions towards the exact solutions. Comprising with an effective expander and efficient optimizer, TOM reveals to be an appropriate approach to solve different fractional differential equations modeling various problems of engineering.

An efficient solving method for nonlinear convection diffusion equation

PurposeThis paper aims to provide a well-behaved nonlinear scheme and accelerating iteration for the nonlinear convection diffusion equation with fundamental properties illustrated.Design/methodology/approachA nonlinear finite difference scheme is studied with fully implicit (FI) discretization used to acquire accurate simulation. A Picard–Newton (PN) iteration with a quadratic convergent ratio is designed to realize fast solution. Theoretical analysis is performed using the discrete function analysis technique. By adopting a novel induction hypothesis reasoning technique, the L∞ (H1) convergence of the scheme is proved despite the difficulty because of the combination of conservative diffusion and convection operator. Other properties are established consequently. Furthermore, the algorithm is extended from first-order temporal accuracy to second-order temporal accuracy.FindingsTheoretical analysis shows that each of the two FI schemes is stable, its solution exists uniquely and has second-order spatial and first/second-order temporal accuracy. The corresponding PN iteration has the same order of accuracy and quadratic convergent speed. Numerical tests verify the conclusions and demonstrate the high accuracy and efficiency of the algorithms. Remarkable acceleration is gained.Practical implicationsThe numerical method provides theoretical and technical support to accelerate resolving convection diffusion, non-equilibrium radiation diffusion and radiation transport problems.Originality/valueThe FI schemes and iterations for the convection diffusion problem are proposed with their properties rigorously analyzed. The induction hypothesis reasoning method here differs with those for linearization schemes and is applicable to other nonlinear problems.

Analytical Solutions for Nonlinear Convection–Diffusion Equations with Nonlinear Sources

Nonlinear equations of the convection–diffusion type with nonlinear sources are used for the description of many processes and phenomena in physics, mechanics, and biology. In this work, we consider the family of nonlinear differential equations which are the traveling wave reductions of the nonlinear convection–diffusion equation with polynomial sources. The question of the construction of the general analytical solutions to these equations is studied. The steady-state and nonstationary cases with and without account of convection are considered. The approach based on the application of nonlocal transformations generalizing the Sundman transformation is applied for construction of the analytical solutions. It is demonstrated that in the steady-state case without account of convection, the general analytical solution can be found without any constraints on the equation parameters and it is expressed in terms of the Weierstrass elliptic function. In the general case, this solution has a cumbersome form; the constraints on the parameters for which it has a simple form are found, and the corresponding analytical solutions are obtained. It is shown that in the nonstationary case, both with and without account of convection, the general solution to the studied equations can be constructed under certain constraints on the parameters. The integrability criteria for the Lienard-type equations are used for this purpose. The corresponding general analytical solutions to the studied equations are explicitly constructed in terms of exponential or elliptic functions.

A nonlinear multigrid method for inverse problem in the multiphase porous media flow

In this paper, we consider a parameter identification problem for the nonlinear convection–diffusion equation in the multiphase porous media flow. A nonlinear multigrid method is proposed for the recovery of permeability. This method works by dynamically adjusting the objective functionals at different grids so that they are consistent with each other, and ultimately reduce, the finest grid objective functional. In this manner, the nonlinear multigrid method can efficiently compute the solution to a desired fine grid inverse problem. Numerical results illustrate that the proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality.

A Model for Subsurface Oil Pollutant Migration

The subsurface spill of hydrocarbon fluids (from an underground leaking tank or from pipelines) is a widespread problem in the oil industry and hydrology. Under the assumption of low oil saturation, the migration of oil pollutant may be modeled by a nonlinear convection–diffusion equation with power law nonlinearities. The saturation-dependent diffusion coefficient is proportional to a power of saturation for which the exponent may be negative or positive depending on the porous medium and the pollutant characteristics. This paper presents general analytical solutions for the one-dimensional transient saturation distribution in immiscible zone. The solutions describe the longtime behavior of a contaminant oil pollutant caused by a continuous source of contaminant (e.g., leakage from pipelines, disposal dumps or storage tanks). Two general forms corresponding to positive and negative power exponents of the diffusion coefficient are presented. The general forms express implicitly the position as function of oil saturation and time. An alternative explicit closed-form solution near the front position is also proposed. An exact explicit solution is obtained for the particular case of constant diffusion coefficient. The proposed solutions are fully analytical in the sense that they do not require any numerical implementation. The solutions may be used to study the porous medium and pollutant parameters. They can be applied to aquifer contamination problems, and they are also important for the verification of numerical solutions.

Superconvergence of the local discontinuous Galerkin method for nonlinear convection-diffusion problems

In this paper, we discuss the superconvergence of the local discontinuous Galerkin methods for nonlinear convection-diffusion equations. We prove that the numerical solution is (k+3/2)th-order superconvergent to a particular projection of the exact solution, when the upwind flux and the alternating fluxes are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k (kgeq1). The numerical experiments reveal that the property of superconvergence actually holds true for general fluxes.

A family of analytical solutions of a nonlinear diffusion–convection equation

Despite its popularity in many engineering fields, the nonlinear diffusion–convection equation has no general analytical solutions. This work presents a family of closed-form analytical traveling wave solutions for the nonlinear diffusion–convection equation with power law nonlinearities. This kind of equations typically appears in nonlinear problems of flow and transport in porous media. The solutions that are addressed are simple and fully analytical. Three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient (increasing, decreasing or constant). It has shown that the structure of the traveling wave solution is strongly related to the diffusion term. The main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters. The proposed closed-form solutions are simple to use, do not require any numerical implementation, and may be implemented in a simple spreadsheet. The analytical expressions are also useful to mathematically analyze the structure and properties of the solutions.

A wavelet multiscale method for the inverse problem of a nonlinear convection–diffusion equation

This paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized using finite-difference methods and the inverse problem is formulated as a minimization problem with regularization terms. In order to overcome disturbance of local minimum, a wavelet multiscale method is applied to solve this parameter identification inverse problem. This method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub-inverse problems relying on scale variables, and successively solving these sub-inverse problems according to the size of scale from the smallest to the largest. The stable and fast regularized Gauss–Newton method is applied to each scale. Numerical simulations show that the proposed algorithm is widely convergent, computationally efficient, and has the anti-noise and de-noising abilities. • Permeability identification of a nonlinear convection–diffusion equation is studied. • The finite difference scheme for nonlinear convection–diffusion equation is given. • A wavelet multiscale method is proposed for parameter identification. • This method is one of multiscale inversion method. • Numerical simulations testify the effectiveness of the wavelet multiscale method.

Lie Group Analysis and Invariant Solutions for Nonlinear Time-Fractional Diffusion-Convection Equations**Supported by the Natural Science Foundation of China under Grant Nos. 11371287 and 61663043

On the basis of Lie group theory, (1 + N)-dimensional time-fractional partial differential equations are studied and the expression of is given. As applications, two special forms of nonlinear time-fractional diffusion-convection equations are investigated by Lie group analysis method. Then the equations are reduced into fractional ordinary differential equations under group transformations. Therefore, the invariant solutions and some exact solutions are obtained.

Metastability for nonlinear convection–diffusion equations

We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity $$\varepsilon $$ . In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple $$(\xi ,v)$$ , where $$\xi $$ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in $$\varepsilon $$ . As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion.

Numerical solution of a multi-class model for batch settling in water resource recovery facilities

• The numerical solution for several batch settling unit processes in WRRFs is computed. • Multi-class concentration-driven model based on settling velocity distributions used. • Non-linear convection–diffusion equations appear due to the multi-class extension. • IMEX-RK schemes combined with high order shock-capturing technology are applied. In Torfs et al. (2017) a new unified framework to model settling tanks in water resource recovery facilities was proposed providing a set of partial differential equations (PDEs) modelling different settling unit processes in wastewater treatment such as primary and secondary settling tanks (PSTs and SSTs). The extension to a multi-class framework to deal with the distributed properties of the settling particles leads to a system of non-linear hyperbolic-parabolic PDEs whose solutions may contain very sharp transitions. This necessitates the use of a consistent and robust numerical method to obtain well-resolved and reliable approximations to the PDE solutions. The use of implicit–explicit Runge–Kutta (IMEX-RK) schemes, along with the weighted essentially non-oscillatory (WENO) shock-capturing technology for the discretization of the set of equations, is advocated in this work. The versatility of the proposed unified framework is demonstrated through a set of numerical examples for batch settling occurring in both PSTs and SSTs, along with the efficiency and reliability of the numerical scheme.

Identification of diffusion parameters in a non-linear convection–diffusion equation using adaptive homotopy perturbation method

This paper is concerned with the numerical identification of diffusion parameters in a non-linear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. In order to overcome the defect of the local convergence of traditional methods, an adaptive homotopy perturbation method is applied to solve this parameter identification inverse problem. The adaptive homotopy perturbation method provides a simple way to adapt computational refinement to the choice of the homotopy parameter. Numerical simulations illustrate that the proposed algorithm is globally convergent and computationallyefficient.

An improved homogenization result for immiscible compressible two-phase flow in porous media

The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneous porous media. We consider liquid and gas phases (water and hydrogen) flow in a porous reservoir, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. The gas phase is supposed compressible and obeying the ideal gas law. The flow is then described by the conservation of the mass for each phase. The model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. The aim of this paper is to extend our previous results to the case of an ideal gas. In this case a new degeneracy appears in the pressure equation. With the help of an appropriate regularization we show the existence of a weak solution to the studied system. We also consider the corresponding nonlinear homogenization problem and provide a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.

Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank-Nicolson and Time Efficient ADI

To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of L2, L∞ norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

A New Augmented Singular Transform and its Partial Newton-Correction Method for Finding More Solutions

With the information provided by previously found solutions, an augmented singular transform is introduced in Xie et al. (JCAM 286:145---157, 2015) to change the local basin structure of the original problem for finding new solutions. However the old formulation in Xie et al. (JCAM 286:145---157, 2015) involves the kernel of an unknown solution to be found or the kernels of all previously found solutions, thus left several theoretical issues unsolved and prevents from further development. In this paper, we derive a new augmented singular transform which changes only the local basin/barrier structure around $$u=0$$u=0 for finding more solutions. Comparing to the old formulation, the new one is much easier to apply and resolves all unsolved theoretical issues left in Xie et al. (JCAM 286:145---157, 2015). A corresponding partial Newton-correction method is then designed to solve the augmented problem on the solution set. Mathematical justification of the new formulation, method and its local convergence are established. The new method is first tested on two very different variational problems and then applied to solve a nonvariational nonlinear convection-diffusion equation for multiple solutions, which are, for the first time, numerically computed and visualized with their profile and contour plots. Several interesting phenomena are observed for the first time and open for mathematical verification. Since the new formulation is general and simple, it can also be modified to treat other problems, e.g., quasilinear PDEs, a large system of PDEs with equality constraints, for finding multiple solutions.

A posteriorierror analysis for a viscous flow-transport problem

In this paper we develop an a posteriori error analysis for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem. The governing system corresponds to a scalar, nonlinear convection-diffusion equation coupled with a Stokes problem with variable viscosity, and it serves as a prototype model for sedimentation-consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest. The solvability of the continuous mixed-primal formulation along with a priori error estimates for a finite element scheme using Raviart−Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤ k + 1 for both velocity and concentration, have been recently established in [M. Alvarez et al., ESAIM: M2AN 49 (2015) 1399–1427]. Here we derive two efficient and reliable residual-based a posteriori error estimators for that scheme: for the first estimator, and under suitable assumptions on the domain, we apply a Helmholtz decomposition and exploit local approximation properties of the Clement interpolant and Raviart−Thomas operator to show its reliability. On the other hand, its efficiency follows from inverse inequalities and the localization arguments based on triangle-bubble and edge-bubble functions. Secondly, an alternative error estimator is proposed, whose reliability can be proved without resorting to Helmholtz decompositions. Our theoretical results are then illustrated via some numerical examples, highlighting also the performance of the scheme and properties of the proposed error indicators.