Nonlinear Advection Diffusion Reaction Research Articles

A sixth-order finite difference method for the two-dimensional nonlinear advection diffusion reaction equation

A novel higher-order finite difference method is developed to solve the two-dimensional nonlinear advection diffusion reaction equation. Firstly, a new five-point sixth-order finite difference scheme is proposed to discretize spatial derivatives and the sixth-order backward difference method is applied to teat temporal derivative. As a result, a new fully implicit scheme with sixth-order accuracy in both time and space is proposed. Subsequently, for the calculation of the starting step of the sixth-order backward difference method, the Crank-Nicolson method combined with the Richardson extrapolation technique is employed to obtain the equivalent temporal accuracy as the main scheme. Finally, the accuracy and reliability of the presented method are validated through various numerical examples.

An efficient hybrid method based on cubic B-spline and fourth-order compact finite difference for solving nonlinear advection–diffusion–reaction equations

An efficient hybrid method based on cubic B-spline and fourth-order compact finite difference for solving nonlinear advection–diffusion–reaction equations

Enhanced Unconditionally Positive Finite Difference Method for Advection–Diffusion–Reaction Equations

In this study, we develop the enhanced unconditionally positive finite difference method (EUPFD), and use it to solve linear and nonlinear advection–diffusion–reaction (ADR) equations. This method incorporates the proper orthogonal decomposition technique to the unconditionally positive finite difference method (UPFD) to reduce the degree of freedom of the ADR equations. We investigate the efficiency and effectiveness of the proposed method by checking the error, convergence rate, and computational time that the method takes to converge to the exact solution. Solutions obtained by the EUPFD were compared with the exact solutions for validation purposes. The agreement between the solutions means the proposed method effectively solved the ADR equations. The numerical results show that the proposed method greatly improves computational efficiency without a significant loss in accuracy for solving linear and nonlinear ADR equations.

On the convergence of NSFD schemes for a new class of advection–diffusion–reaction equations

This paper deals with the two classes of non-linear advection–diffusion–reaction (ADR) equations subject to certain initial and boundary conditions. We derive exact finite difference (FD) schemes with the help of solitary wave solutions of ADR equations. Furthermore, we construct non-standard FD schemes for both the equations. The positivity and boundedness properties of the equations are satisfied by the proposed non-standard FD schemes. We also prove that these schemes are stable under certain conditions. Moreover, they are shown to be first order consistent in both time and space. We compute solutions of the ADR equations by applying proposed non-standard FD schemes under given initial and boundary conditions. The obtained results demonstrate that the proposed methods are simple and powerful tool in finding the numerical solutions of ADR equations. The schemes give good accuracy even for few spatial divisions.

Some Finite Difference Methods to Model Biofilm Growth and Decay: Classical and Non-Standard

The study of biofilm formation is undoubtedly important due to micro-organisms forming a protected mode from the host defense mechanism, which may result in alteration in the host gene transcription and growth rate. A mathematical model of the nonlinear advection–diffusion–reaction equation has been studied for biofilm formation. In this paper, we present two novel non-standard finite difference schemes to obtain an approximate solution to the mathematical model of biofilm formation. One explicit non-standard finite difference scheme is proposed for biomass density equation and one property-conserving scheme for a coupled substrate–biomass system of equations. The nonlinear term in the mathematical model has been handled efficiently. The proposed schemes maintain dynamical consistency (positivity, boundedness, merging of colonies, biofilm annihilation), which is revealed through experimental observation. In order to verify the accuracy and effectiveness of our proposed schemes, we compare our results with those obtained from standard finite difference schemes and earlier known results in the literature. The proposed schemes (NSFD1 and NSFD2) show good performance. The NSFD2 scheme reveals that the processes of biofilm formation and nutritive substrate growth are intricately linked.

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

PurposeThe purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.Design/methodology/approachThe strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.FindingsSeveral test problems are provided to confirm the validity and efficiently of the proposed method.Originality/valueFor the first time, some famous examples are solved by using the proposed high-order technique.

Numerical solution of nonlinear advection diffusion reaction equation using high-order compact difference method

In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order Padé formula of the first-order derivative, and the time derivative term is discretized by using the fourth-order backward differencing formula. An unconditionally stable five-step fourth-order fully implicit compact difference scheme is developed. This scheme has fourth-order accuracy in both time and space. Secondly, for the calculations of the start-up time steps, the time derivative term is discretized by the Crank-Nicolson method, and Richardson extrapolation formula is used to improve the accuracy in time direction from the second-order to the fourth-order. Thirdly, convergence and stability of the difference scheme in H1 seminorm, L∞ and L2 norms, existence and uniqueness of the numerical solutions are proved, respectively. Fourthly, the Thomas algorithm is used to solve the nonlinear algebraic equations at each time step, and a time advancement algorithm with linearized iteration strategy is established. Finally, the accuracy, stability and efficiency of the present approach are verified by some numerical experiments.

A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable coefficients

The radial basis function method for 3D advection diffusion reaction equations with variable coefficients is presented. The proposed method implements the linear combination of radial basis functions which impose boundary conditions in advance, and thus such a combination with weighted parameters can be used to construct the final approximation. Furthermore, the weighted parameters are solved by substituting the approximation into governing equations. This method leads to crucial improvements in the feasibility and accuracy which can now be easily applied to general 3D nonlinear problems through linearized techniques. Finally, accuracy and efficiency of the proposed method are verified by several examples.

Learning and meta-learning of stochastic advection–diffusion–reaction systems from sparse measurements

Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. We obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.

Simulation of linear and nonlinear advection–diffusion–reaction problems by a novel localized scheme

This paper presents a new meshless scheme based on the compactly supported radial basis functions for solving linear and nonlinear advection–diffusion–reaction equations. The simulation of proposed problems is divided into the rough approximation obtained from the boundary conditions and corrected approximation using compactly supported radial basis functions where a sparse system matrix will be obtained. The behaviors of the method are verified by several examples which indicate its accuracy and efficiency.

An accurate meshless formulation for the simulation of linear and fully nonlinear advection diffusion reaction problems

In this paper, a new meshless semi-analytical collocation technique is presented for the simulation of time-dependent linear and fully nonlinear advection diffusion reaction equations (ADREs). The time-dependent nonlinear problem is reduced to the system of linear problems in each time layer by using the quasilinearization technique in combination with the Crank–Nicolson scheme for the temporal approximation. Then it may be solved by the meshless backward substitution method. In the backward substitution method the approximation is given to satisfy the boundary conditions as a prior. Then the approximation is substituted back to the governing equations where the unknown weighted parameters are obtained using the collocation approach. The developed method has been tested on seven concrete problems for a range of free parameters. The comparison of the obtained numerical results with those available in literatures reveals that the proposed method can accurately approximate linear and nonlinear advection diffusion reaction problems. This method can also be easily extended onto other linear and nonlinear transient problems in 1D, 2D, and 3D due to its general idea and easy implementation.

Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations

Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs.

Computing the Allee Effect and Population Dynamics with Density Dependent Migeration Using Homotopy Perturbation Method

Solutions of nonlinear models are of great importance and their significance has increased a lot. In given paper, the homotopy perturbation method (HPM) is implemented to show the numerical assumption of the population dynamics model with density-dependent migeration and the Allee effect. The resemblance of the numerical solutions attained by HPM with exact solution allows the order of this method. The results show applicability, accuracy and efficiency of HPM in solving the parabolic equation. HPM is effective for solving the transitary non-linear advection diffusion reaction equation.

Remediation of contaminated groundwater by meshless local weak forms

In this paper, four meshless local weak form methods such as direct meshless local Petrov–Galerkin method (DMLPG), meshless local Petrov–Galerkin method (MLPG), local weak radial point interpolation method (LWRPIM) and local weak moving kriging method (LWMK) are applied to find the numerical solution of coupled non-linear advection–diffusion–reaction system arising in the prevention of groundwater contamination problem. A comparison between these methods is done from the perspective of accuracy and computational efficiency. An efficient fourth-order exponential time differencing Runge–Kutta method is utilized for the time discretization. The computational efficiency is the most significant advantage of the DMLPG method in comparison with the other meshless local weak form methods, because DMLPG reduces the computational costs, significantly. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions. The main aim of this paper is to show that the meshless local weak form methods can be used for solving the system of non-linear partial differential equations especially coupled non-linear advection–diffusion–reaction system. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Nonlinear Advection–Diffusion–Reaction Phenomena Involved in the Evolution and Pumping of Oil in Open Sea: Modeling, Numerical Simulation and Validation Considering the Prestige and Oleg Naydenov Oil Spill Cases

The main goal of this article is to improve upon a previous model used to simulate the evolution of oil spots in the open sea and the effect of a skimmer ship pumping oil out from the spots. The concentration of the pollutant is subject to the effects of wind and sea currents, diffusion, and the pumping action of a skimmer (i.e., cleaning) ship that follows a pre-assigned trajectory. This implies that the mathematical model is of the advection---diffusion---reaction type. A drawback of our previous model was that diffusion was propagating with infinite velocity; in this article, we use an improved model relying on a nonlinear diffusion term, implying that diffusion propagates with finite velocity. To reduce numerical diffusion when approximating the advection part of the model, we consider second order discretization schemes with nonlinear flux limiters. We consider also absorbing boundary conditions to insure accurate results near the boundary. To reduce CPU time we use an operator-splitting scheme for the time discretization. Finally, we also introduce the modeling of coastlines and dynamic sources of pollutant. The novel approach we advocate in this article is validated by comparing our numerical results with real life measurements from the Oleg Naydenov and the Prestige oil spills, which took place in Spain in 2015 and 2002, respectively.

Krylov single-step implicit integration factor WENO methods for advection–diffusion–reaction equations

Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods were combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (2013) [19] to efficiently solve stiff nonlinear advection–diffusion–reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection–diffusion–reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.

Coupled nonlinear advection–diffusion–reaction system for prevention of groundwater contamination by modified upwind finite volume element method

The coupled advection-dominated diffusion reaction equations arising in the prevention of groundwater contamination problem are approximated by the modified upwind finite volume element method. In order to solve the resulting nonlinear system efficiently, we use a two-grid algorithm to decompose the nonlinear system into a small nonlinear system on a coarse grid with mesh size H and a linear system on a fine grid with mesh size h. It is shown that the approximation still achieves asymptotically optimal as long as the mesh sizes satisfy H=O(h1/3). Numerical examples are presented to illustrate efficiency and accuracy of the proposed method.

Numerical solution of a multidimensional sedimentation problem using finite volume-element methods

We are interested in the reliable simulation of the sedimentation of monodisperse suspensions under the influence of body forces. At the macroscopic level, the complex interaction between the immiscible fluid and the sedimentation of a compressible phase may be governed by the Navier–Stokes equations coupled to a nonlinear advection–diffusion–reaction equation for the local solids concentration. A versatile and effective finite volume element (FVE) scheme is proposed, whose formulation relies on a stabilized finite element (FE) method with continuous piecewise linear approximation for velocity, pressure and concentration. Some numerical simulations in two and three spatial dimensions illustrate the features of the present FVE method, suggesting their applicability in a wide range of problems.

Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection–diffusion–reaction equations

Implicit integration factor (IIF) methods are originally a class of efficient “exactly linear part” time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection–diffusion–reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection–diffusion–reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.

Point source identification in nonlinear advection–diffusion–reaction systems

We consider a problem of identification of point sources in time-dependent advection–diffusion systems with a nonlinear reaction term. The linear counterpart of the problem in question can be reduced to solving a system of nonlinear algebraic equations via the use of adjoint equations. We extend this approach by constructing an algorithm that solves the problem iteratively to account for the nonlinearity of the reaction term. We study the question of improving the quality of source identification by adding more measurements adaptively using the solution obtained previously with a smaller number of measurements.