Convection Diffusion Reaction Equations Research Articles

Standard-BEM solutions to two types of anisotropic-diffusion convection reaction equations with variable coefficients

Two versions of diffusion–convection–reaction equations for anisotropic functionally graded materials (FGMs) are discussed again in this paper to find their numerical solutions by using the boundary element method (BEM). The numerical results obtained show the consistency and accuracy of the BEM solutions. Also, the solutions exhibit the impact of the anisotropy and inhomogeneity (spatial variability) of the media.

A robust numerical scheme for the simulation of nonlinear convection–diffusion–reaction equation

In this article, the polynomial based differential quadrature method is used to develop a numerical scheme for solving convection–diffusion–reaction model with a general reaction term. The numerical scheme is found applicable to a wide set of problems including diffusion, convection–diffusion and diffusion–reaction. The validation of the method is performed through simulations of Newell-Whitehead, Burger-Fisher, and Burger-Huxley equations. The proposed numerical method is found better than many numerical methods presented in the literature. Vector form of numerical scheme is presented to implement algorithm efficiently. The compatibility, error and computational complexity of the numerical scheme are studied in detail.

A goal-oriented anisotropic [formula omitted]-mesh adaptation method for linear convection–diffusion–reaction problems

We deal with the numerical solution of linear convection–diffusion–reaction equations using the hp-variant of the discontinuous Galerkin method on triangular grids. We develop a mesh adaptive algorithm which modifies the size and shape of mesh elements and the corresponding polynomial approximation degrees in order to decrease the error in a target functional under the given tolerance. We recall some theoretical results and describe in detail several technical aspects of this approach. Numerical experiments demonstrating the accuracy, efficiency and robustness of the algorithm are presented.

Radial integration boundary element method for two-dimensional non-homogeneous convection–diffusion–reaction problems with variable source term

This paper describes a new formulation of the radial integration boundary element method (RIBEM) for two-dimensional non-homogeneous convection–diffusion–reaction problems with variable source terms. The radial integration method (RIM) is implemented to transform the resulting domain integral into equivalent boundary integrals, and thus a boundary-only integral equation formulation can be achieved. The fundamental solution of the steady-state convection–diffusion–reaction equation with constant coefficients is employed. The integral representation formula for the convection–diffusion–reaction problem with source term is obtained from Green’s second identity. Numerical applications are included for five different cases, for which analytical solutions are available, to establish the validity of the proposed approach and to demonstrate convergence and efficiency of the proposed technique. Results obtained show that the RIBEM produced an excellent agreement with the analytical solutions and the results do not present oscillations or damping of the wave front, as may appear in other numerical techniques.

High-order compact difference scheme of 1D nonlinear degenerate convection–reaction–diffusion equation with adaptive algorithm

This research describes a high accuracy adaptive difference strategy on 1D reaction–diffusion equation with convection item, which explains quenching phenomena of nonlinear singular degenerate problem. It originates from the first and the second central difference scheme and the forward difference operator. The former is used to discrete the first-order and the second-order spatial derivative and the latter is utilized to approximate temporal derivative. After its accuracy and stability discussed, the typical quenching cases are illustrated singularity of the nonlinear problem by using the proposed strategy with adaptation mechanism and comparing with the low-order approach.

Variational multiscale finite element methods for a nonlinear convection-diffusion-reaction equation

This paper is devoted to developing finite-element methods for solving a two-dimensional boundary value problem for a singularly perturbed time-dependent convection–diffusion–reaction equation. The solution to this problem may vary rapidly in thin layers. As a result, spurious oscillations occur in the solution if the standard Galerkin method is used. In the multiscale finite-element methods, the original problem is split into the grid-scale and subgrid-scale problems, which allows capturing the problem features at a scale smaller than the element mesh size. In this study two methods are considered: the variational multiscale method with algebraic sub-scale approximation (VMM-ASA) and the residual-free bubbles (RFB) method. In the first method the subgrid-scale problem is simulated by the residual of the grid-scale equation and intrinsic time scales. In the second method the subgrid-scale problem is approximated by special functions. The grid-scale and subgrid-scale problems are formulated via the linearization procedure on the subgrid component applied to the original problem. The computer implementation of the methods was carried out using a commercial finite-element package. The efficiency of the developed methods is evaluated by solving a test boundary value problem for the nonlinear equation. Cases with different values of the diffusion coefficient have been analyzed. Based on the numerical investigation, it is shown that the multiscale methods enable improving the stability of the numerical solution and decreasing the quantity and the amplitude of oscillations compared to the standard Galerkin method. In the case of a small diffusion coefficient, the developed methods can yield a satisfactory numerical solution on a sufficiently coarse mesh.

A sliding mode observer for a linear reaction–convection–diffusion equation with disturbances

A sliding mode observer is designed and analyzed for a linear reaction–convection–diffusion equation subject to external disturbances. Based on a discontinuous input, the proposed observer ensures both state estimation and disturbance rejection using only one boundary measurement. An appropriate Lyapunov function is used to prove the asymptotic convergence of the observation error. The performance of the proposed observer is illustrated by some numerical simulation results.

A compact integrated RBF method for time fractional convection–diffusion–reaction equations

In this paper, a local compact integrated radial basis function (CIRBF) method is proposed to solve the time fractional convection–diffusion–reaction equations. The proposed CIRBF scheme derives high accuracy and fast convergence rate with grid refinement. The spatial derivatives are discretized by combining with the integrated RBF approximation and compact approximation in a stencil. The nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the RBF weight space. The time discretization is replaced by the second order shifted Grünwald scheme. Moreover, the unconditional stability and convergence of the proposed method are shown. Finally, some numerical examples on 2D and 3D domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations.

Boundary Control Problem for a Nonlinear Convection–Diffusion–Reaction Equation

The solvability of boundary-value and extremum problems for a nonlinear convection–diffusion–reaction equation with mixed boundary conditions is proved in the case where the coefficient in the boundary condition is a fairly arbitrary function of the solution to the boundary value problem. For the mass transfer coefficient equal to the modulus of the substance concentration, local stability estimates of the solution to the extremum problem with respect to relatively small perturbations in the cost functional and the given functions of the boundary value problem are obtained.

Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

This paper has focused on unknown functions identification in nonlinear boundary conditions of an inverse problem of a time‐fractional reaction–diffusion–convection equation. This inverse problem is generally ill‐posed in the sense of stability, that is, the solution of problem does not depend continuously on the input data. Thus, a combination of the mollification regularization method with Gauss kernel and a finite difference marching scheme will be introduced to solve this problem. The generalized cross‐validation choice rule is applied to find a suitable regularization parameter. The stability and convergence of the numerical method are investigated. Finally, two numerical examples are provided to test the effectiveness and validity of the proposed approach.

A multiquadric RBF–FD scheme for simulating the financial HHW equation utilizing exponential integrator

We propose a computational scheme to solve the financial time-dependent 3D Heston–Hull–White PDE. In fact, a novel radial basis function (RBF) generated finite difference (FD) scheme associated with multiquadric RBF is introduced for solving this convection–diffusion–reaction equation. Non-uniform grids alongside the multiquadric RBF–FD technique are applied to obtain results of high accuracy in significant areas, at which the PDE problem is degenerate and discontinuous. The efficacy of the new scheme is shown through a series of numerical experiments.

RBF-based meshless local Petrov Galerkin method for the multi-dimensional convection–diffusion-reaction equation

In this paper, the meshless local Petrov Galerkin (MLPG) method is employed to analyze convection–diffusion-reaction equation based on radial basis function (RBF) collocation method. Compared with traditional Petrov Galerkin method, MLPG method is not limited to the particular interpolation domain and integration domain. In the MLPG method, interpolation domain is chosen through a special method which ensures that the number of neighboring nodes around the interpolation point (or collocation point) is constant. And all integrations are carried out locally over small quadrature domains of regular shapes such as two-dimensional (2D) squares and three-dimensional (3D) cubes. Crank–Nicolson scheme is applied to the time discretization to guarantee the unconditional stability of present method. Thanks to the Kronecker delta function property of the shape functions for RBF interpolation, we can easily obtain the error estimates that MLPG method has second order convergent rate in time and space simultaneously. Finally, numerical examples are presented to show the accuracy and efficiency of the MLPG method.

Local projection stabilization for convection–diffusion–reaction equations on surfaces

The numerical solution of convection–diffusion–reaction equations in two and three dimensional domains Ω is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle boundary or interior layers and to localize and suppress unphysical oscillations. Much less is known on convection–diffusion–reaction equations on surfaces Γ=∂Ω. We propose a fitted finite element local projection stabilization (LPS) for convection–diffusion–reaction equations on surfaces based on a linear surface approximation and continuous, piecewise linear finite elements. Unique solvability of the continuous and discrete problem is established. A-priori error estimates show the O(h3∕2) convergence in mesh-dependent norms for smooth solutions in the convection-dominated case. The associated bilinear form satisfies an inf–sup condition in a norm stronger than the standard LPS-norm leading to improved stability properties. Numerical test examples demonstrate the properties and the potential of the proposed method.

Two types of spurious oscillations at layers diminishing methods for convection–diffusion–reaction equations on surface

In this article, we consider the steady convection–diffusion–reaction equation posed on general surface in convection-dominated regimes, which is an important problem in many models for simulating substances transport on ultrathin material and solid surfaces. Two types of spurious oscillations at layers diminishing (SOLD) methods are considered when the streamline diffusion method cannot completely eliminates the spurious oscillations at layers for convection-dominated problem with nonsmooth solution. One is the edge stabilization method that uses least square stabilization of the gradient jumps a across element boundaries, the other, a Petrov–Galerkin method, is called the Mizukami–Hughes method. An error analysis of the edge stabilization method on surface is provided. Finally, numerical experiments including the comparisons among these methods are presented to illustrate the accuracy and efficiency of the SOLD methods.

A Study of the von Neumann Stability Analysis of a Semi-implicit Numerical Method Applied Over the Radial Impurity Transport Equation in Tokamak Plasma

The radial impurity transport equation for tokamak plasma is a form of diffusion–convection–reaction equation. The impurity transport equation is solved to determine the distribution of impurity (non-fuel) ion species with different ionization states perpendicular to magnetic surfaces of tokamak plasma. The equation for each charge (ionization) state Z is a non-linear, second-order in space, first-order in time, parabolic partial differential equation coupled to the previous Z − 1 and the next Z + 1 charge states of the impurity species through its reaction term. The number of differential equations to be solved simultaneously is hence determined by the number of ionization states of the impurity species studied. The solution to the set of these coupled equations can be obtained using a semi-implicit numerical method applied on it. The present study describes the application of von Neumann stability analysis over the semi-implicit numerical method applied over the radial impurity transport equation and determines a generic stability criterion for the method. The stability analysis is further illustrated using the geometry of Aditya tokamak installed at the Institute for Plasma Research Gandhinagar, India as an example. The impurity species considered is oxygen (Atomic number = 8). This leads to a set of eight coupled equations for charge states Z = 1 to 8 over which von Neumann analysis is illustrated in present study.

A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains

A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.

A Meshfree Stochastic Algorithm for Solving Diffusion—Convection—Reaction Equations in Domains with Complex Geometry

A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.

A posteriori error estimators for stabilized finite element approximations of an optimal control problem

We derive a posteriori error estimators for an optimal control problem governed by a convection–reaction–diffusion equation; control constraints are also considered. We consider a family of low-order stabilized finite element methods to approximate the solutions to the state and adjoint equations. We obtain a fully computable a posteriori error estimator for the optimal control problem. All the constants that appear in the upper bound for the error are fully specified. Therefore, the proposed estimator can be used as a stopping criterion in adaptive algorithms. We also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative. Numerical examples, in two and three dimensions, are presented to illustrate the theory.

Convergence of a positive nonlinear Control Volume Finite Element scheme for solving an anisotropic degenerate breast cancer development model

In this paper, a nonlinear control volume finite element (CVFE) scheme for solving an anisotropic degenerate breast cancer development model is introduced and analyzed. This model includes both ordinary differential equations and convection–diffusion–reaction equations modeling the stepwise mutations from a normal breast stem cell to a tumor cell. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized on a dual mesh by means of the piecewise linear conforming finite element method and using the Godunov scheme to approximate the diffusion fluxes provided by the conforming finite element reconstruction. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh, where the dual volumes are constructed around the vertices of the original mesh. This technique ensures the positivity and boundedness of discrete solutions without any restriction on the diffusion tensor nor the transmissibility coefficients. The convergence of the scheme is proved, only supposing the shape regularity condition for the original mesh and using a priori estimates as well as the Kolmogorov relative compactness theorem. The proposed scheme is robust, locally conservative, efficient, and stable, which is confirmed by numerical experiments over a general mesh.

A stabilised finite element method for the convection–diffusion–reaction equation in mixed form

This paper is devoted to the approximation of the convection–diffusion–reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems.