Convection Diffusion Reaction Equations Research Articles

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

Convergence analysis of a class of iterative methods for propagation of reaction fronts in porous media

We present an iterative scheme for the numerical analysis of propagating reaction front problems in porous media satisfying an Arrhenius-type law. The governing equations consist of the Darcy equations for the pressure and flow field coupled to two convection–diffusion–reaction equations for the temperature and depth of conversion. Well-posedness, existence and uniqueness of the weak solution are first studied using a fixed-point approach and then, analysis of the proposed iterative scheme is investigated. Numerical results are also presented in order to validate the theoretical estimates and to illustrate the performance of the proposed scheme. The obtained results are in line with our expectations for a good numerical resolution with high accuracy and stability behaviors.

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

A high resolution Physics-informed neural networks for high-dimensional convection–diffusion–reaction equations

In practical problems, some partial differential equations defined in high-dimensional domains or complex surfaces are difficult to calculate by traditional methods. In this paper, a novel data-driven deep learning algorithm is proposed to solve high-dimensional convection–diffusion–reaction equations. The main idea of the method is to use the neural network which combines the physical characteristics of the equation to get high accuracy numerical solution. The proposed method not only avoids the high cost of mesh generation, but also effectively reduces the numerical oscillation caused by the domination of the convection. In addition, two types of loss functions are designed to force physical properties, such as the positivity or maximum principle of the solution. Various numerical examples are performed to demonstrate the validity and accuracy of the proposed method.

A streamline-derivative-based local projection stabilization virtual element method for nonlinear convection–diffusion–reaction equation

A streamline-derivative-based local projection stabilization virtual element method for nonlinear convection–diffusion–reaction equation

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.

VEM discretization allowing small edges for the reaction–convection–diffusion equation: source and spectral problems

In this paper we analyze a lowest order virtual element method for the classic load reaction–convection–diffusion problem and the convection–diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration.

A numerical investigation of unsteady space–time dependent coefficients anisotropic-diffusion convection reaction equation

Two dimensional transient problems for anisotropic FGMs governed by a diffusion convection reaction (DCR) equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation which can be written in forms of boundary-only integral equations and solved using a standard BEM for the solutions. These BEM solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials with compressible or incompressible flow are solved. The numerical solutions obtained are quite accurate.

A numerical solution of fractional reaction–convection–diffusion for modeling PEM fuel cells based on a meshless approach

The purpose of this contribution is to present or implement generalized finite difference method (GFDM) for the first time in order to solve the reaction convection Diffusion equation (FRCDE) model. The FRCDE equations are nonlinear time fractional differential equations that can be used to describe mathematical models in physics and engineering, such as fuel cells and transport in inhomogeneous media. This model is a generalization of the classical RCDE, with time terms considered by Caputo derivative sense for 0<α(x,t)<1. The meshless GFDM, in conjunction with the ϑ-weighted finite difference method, is developed to approximate processes in the spatial direction. GFDM is based on the Taylor series expansion and the moving least squares (MLS) method to evaluate the derivatives of unknown variables. The proposed method, as a truly meshless approach, is very promising in the numerical approximation of engineering problems within convex, non-convex, irregular, and regular domains (complex domains). The reliability and accuracy of the proposed method are shown by considering a variety of computational domains. The sensitivity of the selection of local collocation points and ϑ is also reported.

The HCVEFG method for 3D steady convection–diffusion–reaction problems

In this paper, a hybrid complex variable element-free Galerkin (HCVEFG) method is proposed for solving three-dimensional (3D) steady convection–diffusion–reaction equations. By introducing the dimension splitting method (DSM), a 3D convection–diffusion–reaction problem is changed into a series of two-dimensional (2D) ones. For each 2D form, the improved complex variable moving least-squares (ICVMLS) method is applied to obtain the required approximation shape functions, and the penalty method is used to apply the essential boundary conditions, thus we can obtain the 2D discretized equations by using the improved complex variable element-free Galerkin (ICVEFG) method. The finite difference method (FDM) is used in one-dimensional splitting direction. Thus, the final solution formula about the numerical solution for 3D convection–diffusion–reaction problem can be obtained. Three numerical examples show that the HCVEFG algorithm can cut down the CPU cost of the improved element-free Galerkin (IEFG) method greatly for solving 3D steady convection–diffusion–reaction problems.

A SUPG formulation augmented with shock-capturing for solving convection-dominated reaction–convection–diffusion equations

A SUPG formulation augmented with shock-capturing for solving convection-dominated reaction–convection–diffusion equations

On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.

Convergence of numerical schemes for convection–diffusion–reaction equations on generic meshes

Using the gradient discretisation method (GDM), we provide a generic convergence of numerical approximations for convection diffusion model with non linear reaction term. The convergence of numerical schemes is proved based on a limited number of natural assumptions and properties. The validity of the proposed gradient scheme is examined using the finite volume method performed on different general meshes. We note that the numerical results indicate that a super-convergence property can be expected for the finite volume methods.

Uniform asymptotic stability for convection–reaction–diffusion equations in the inviscid limit towards Riemann shocks

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter \varepsilon . The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale \varepsilon -dependent topology reduces to the classical W^{1, \infty} -topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine–Hugoniot conditions and the nonuniform shift arising merely from phasing out the nondecaying 0-mode, as in the classical stability analysis for fronts of reaction–diffusion equations.

Nonconforming modified Quasi-Wilson finite element method for convection–diffusion-reaction equation

The nonconforming modified Quasi-Wilson finite element method (FEM) is employed to investigate the unconditional superconvergence behavior of the convection–diffusion-reaction equation for the linearized energy stable Backward-Euler (BE) and the second order Backward differentiation formula (BDF2) fully discrete schemes on arbitrary quadrilateral meshes. Then, the energy stabilities of the discrete solutions for the proposed schemes are demonstrated through the mathematical induction, which is crucial to proving the unique solvabilities with Brouwer fixed point theorem. Based on the above achievements and the two typical characteristics of the element: one is that its consistency error in the broken H1-norm can be estimated as order O(h2) when the exact solution belongs to H3(Ω), which is one order higher than its interpolation error, and the other is that the nonconforming part of the function in the finite element space is orthogonal to the bilinear polynomials on each element of the subdivision in the integration sense, the superclose estimates without any restriction between spatial partition parameter h and the time step τ are obtained. Furthermore, the superconvergence properties for the aforementioned schemes are derived by the interpolation post-processing technique. Finally, the theoretical analysis is justified by the provided numerical experiments.

A linear stability analysis of instabilities with reactive flows in porous medium

Convection, diffusion, and reaction dynamics of radial displacement of reactive fluids undergoing second-order chemical reaction in a porous medium are modeled and understood numerically. In the case of iso-viscous reactants and products, reaction dynamics are examined to understand the effect of reaction rate after solving a system of convection–diffusion–reaction equations using a method of lines. Various temporal scalings for reaction characteristics like the total amount of product and width of reaction front are obtained in terms of the Damköhler number (Da) for the first time in this work. The generation of the product having different viscosity than the reactants results in a hydrodynamic instability called viscous fingering. The numerical technique based on hybrids of compact finite difference and pseudo-spectral methods is utilized, for the first time, for the linear stability analysis (LSA) of miscible viscous fingering induced by chemical reaction. The onset time of instability (ton) is found to depend on the reaction rate, and we obtain a stable zone sandwiched between two unstable zones in the Mc,ton plane for a fixed Péclet number and Damköhler number, where Mc is the log-mobility ratio. The results agree with existing numerical studies validating the novel LSA technique utilized.

Near-Ordinary Periodic Waves of a Generalized Reaction–Convection–Diffusion Equation

Near-Ordinary Periodic Waves of a Generalized Reaction–Convection–Diffusion Equation

Numerical modeling of reactive flow in porous media at the pore scale

AbstractDifferent numerical algorithms for the solution of a class of unsteady convection‐diffusion‐reaction (CDR) equations are presented and compared in this paper. The fully implicit in time discretizations are usually preferred because they are unconditionally stable for linear problems. However, when implicit discretization is used for nonlinear problems, iterations over the nonlinearity have to be performed. Picard (simple linearization) or Newton's methods can be used for this purpose. An alternative to the fully implicit discretization is fractional time‐step methods, e.g. splitting with respect to physicochemical processes. The study of the latter class of discretization is especially interesting in the case when only the reactive term contains nonlinearity, while the convection and diffusion operators are linear. The CDR models used to describe processes in catalytic filters belong to this class. Numerical experiments for CDR equation with controllable stiffness of reaction term for different transport regimes, which are described by Peclet and Damkohler numbers are demonstrated and analyzed.

A NumericalInvestigation for a Class of Transient-State Variable Coefficient DCR Equations

In this paper, a combined Laplace transform (LT) and boundary element method (BEM) is used to find numerical solutions to problems of anisotropic functionally graded media that are governed by the transient diffusion–convection–reaction equation. First, the variable coefficient governing equation is reduced to a constant coefficient equation. Then, the Laplace-transformed constant coefficients equation is transformed into a boundary-only integral equation. Using a BEM, the numerical solutions in the frame of the Laplace transform may then be obtained from this integral equation. Then, the solutions are inversely transformed numerically back to the original time variable using the Stehfest formula. The numerical solutions are verified by showing their accuracy and steady state. For symmetric problems, the symmetry of solutions is also justified. Moreover, the effects of the anisotropy and inhomogeneity of the material on the solutions are also shown, to suggest that it is important to take the anisotropy and inhomogeneity into account when performing experimental studies.

Existence of weak solutions for a nonhomogeneous incompressible cell–fluid Navier–Stokes model with chemotaxis

This work is concerned with the mathematical analysis of a general cell–fluid Navier–Stokes model with the inclusion of chemotaxis. This general model relays on a mixture theory multiphase formulation. It consists of two mass balance equations and two general momentum balance equations, respectively, for the cell and fluid phase, combined with a convection–diffusion–reaction equation for oxygen. We investigate the existence of weak solutions in a two‐ or three‐dimensional bounded domain when the fluids are assumed to be incompressible with constant volume fraction.