Reaction-diffusion Problem Research Articles

Pointwise error estimate of the LDG method for 2D singularly perturbed reaction-diffusion problem

Pointwise error estimate of the LDG method for 2D singularly perturbed reaction-diffusion problem

Modeling catalyst effectiveness factor with space-fractional derivative using Haar wavelet collocation method

Abstract Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive.

Invariant Measures for Stochastic Reaction–Diffusion Problems on Unbounded Thin Domains Driven by Nonlinear Noise

Invariant Measures for Stochastic Reaction–Diffusion Problems on Unbounded Thin Domains Driven by Nonlinear Noise

Adaptive least-squares methods for convection-dominated diffusion-reaction problems

This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.

A pore-scale lattice Boltzmann model for solute transport coupled with heterogeneous surface reactions and mineral dissolution

In this paper, we propose a pore-scale lattice Boltzmann model to simulate fluid flow coupled with heterogeneous surface reactions and mineral dissolution. The primary innovation lies in the transformation of surface reactions, originally treated as a boundary condition, into a volume-source term through dimensional augmentation within the framework of sharp liquid–solid interfaces. This significantly simplifies the implementation, particularly for reactions occurring in porous media with intricate geometric structures. Several benchmark tests were performed to validate the accuracy of this model, including a reaction–diffusion problem in a rectangular domain, a two-dimensional reaction and dissolution of a circular grain, as well as a three-dimensional calcite crystal dissolution in a micro-channel. All the obtained simulation results agree well with the reference solutions. In addition, a dissolution problem in a three-dimensional porous medium built with the sand pack is then investigated. Cases with different Péclet numbers (Pe) and Damköhler numbers (Da) were simulated, and five dissolution modes were obtained, which were finally summarized in a diagram of Pe and Da.

Drug polymer conjugates: Average release time from thin films

The reaction–diffusion problem describing the release of drugs conjugated through labile bonds to polymeric thin films has a known analytical solution, when the reaction kinetics is of first order. Using this solution, an exact formula is derived for the average release time of the system. This simple expression provides the characteristic time of release tav as the sum of the corresponding average diffusion time plus the inverse reaction rate constant: tav=(1/12)⋅(L2/D)+(1/k), where L is the slab thickness, D the diffusion coefficient, and k the reaction rate constant. The former term dominates in a diffusion-controlled release, while the latter one in a reaction-controlled delivery. The crossover regime is exactly described by their sum. The obtained result for the average release time is verified by direct numerical integration through the drug release profiles of the analytical solution. The value of fractional drug release at the characteristic average time is between 60–64%. These results can be used for the design of polymer-drug conjugates with a desired delivery time scale, as well as for the experimental determination of the values of microscopic parameters D and k in a conjugated system of interest.

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

A numerical approach based on Vieta–Fibonacci polynomials to solve fractional order advection–reaction diffusion problem

A numerical approach based on Vieta–Fibonacci polynomials to solve fractional order advection–reaction diffusion problem

Exploring the Elzaki Transform: Unveiling Solutions to Reaction-Diffusion Equations with Generalized Composite Fractional Derivatives

This article investigates the use of the Elzaki transform on a generalized composite fractional derivative. To establish the framework for this inquiry, numerous essential lemmas about the Elzaki transform are presented. We successfully extract the solution to the reaction-diffusion problem using both the Elzaki and Fourier transforms, which include a generalized composite fractional derivative. We also look at special examples of the generalized equation, which helps us understand its applications and consequences better. The results show that the Elzaki transform is successful in dealing with complicated fractional differential equations, introducing new analytical approaches and solutions to the subject of fractional calculus and its applications in reaction-diffusion systems.

Stability analysis of a multilayer diffusion-reaction heat transfer problem with a very large number of layers

The stability of thermal conduction problems is of much practical interest in the context of Li-ion batteries, power electronic devices, chemical reactors and other engineering systems. While past work in predicting the thermal stability of a multilayer diffusion-reaction problem has been carried out using pole analysis or by determining imaginary eigenvalues of the problem, such techniques are impractical when the number of layers in the problem is very large. This work presents an exact analysis of the stability of a multilayer diffusion-reaction problem, in which the number of layers may be very large. Using homogenization and Heaviside functions based representation of thermophysical properties and parameters, it is shown that the stability of such a system may be easily predicted by computing the determinant of a matrix containing contributions from each layer/interface. Results from the present work are shown to correctly reduce to results from past work for pure-diffusion and single-layer special cases, as well as agree well with numerical simulations. Threshold for thermal stability of a representative Li-ion battery pack containing 1000 Li-ion cells is determined. It is shown that stability of the system is governed by a balance between temperature-dependent heat generation, diffusion and heat removal from the boundary. A key result of practical interest is that even small improvement in thermal diffusivity of Li-ion cells may greatly help reduce the risk of thermal instability. Additionally, practical guidelines for the maximum permissible operating time of a battery pack to avoid thermal runaway are discussed. This work advances the theoretical understanding of multilayer diffusion-reaction problems, and helps understand the thermal stability of Li-ion battery packs containing a very large number of cells, which is very difficult to analyze using traditional techniques.

An adaptive mesh generation and higher-order difference approximation for the system of singularly perturbed reaction–diffusion problems

The paper presents a hybrid difference method to solve a coupled system of singularly perturbed reaction–diffusion problems over an adaptive mesh. The solution to the problem manifests a distinctive multi-scale nature, characterized by localized narrow regions where the solution undergoes exponential changes. The mesh-generating procedure relies upon equidistributing a non-negative monitor function. The method utilizes a suitable combination of the cubic and exponential spline difference schemes over a layer adaptive mesh. The proposed method is unconditionally stable and uniformly convergent. The convergence obtained is optimal, being free from logarithmic factors. Moreover, it is free from directional bias. Numerical experiments have been performed and presented for two model problems. The results obtained are in agreement with the theoretical estimates.

A novel second-order ADI Scheme for solving epidemic models with cross-diffusion

Phenomena in life sciences can be modeled using systems of reaction–diffusion partial differential equations with cross-diffusion. These equations, nonlinear in nature, exhibit complex spatial behavior. Within this framework, we propose an SIR model with cross-diffusion to depict the dynamic interaction between susceptible and infectious individuals in the presence of public policies. Achieving accurate solutions requires fine space discretization, leading to high computational costs. In addition, we propose a second-order semi-implicit method based on an Alternating Direction Implicit (ADI) scheme, called SSIADI, suitable for treating nonlinear reaction and linear diffusion problems.

Solving reaction-diffusion problems with explicit Runge–Kutta exponential methods without order reduction

In this paper a technique is given to recover the classical order of the method when explicit exponential Runge–Kutta methods integrate reaction-diffusion problems. In the literature, methods of high enough stiff order for problems with vanishing boundary conditions have been constructed, but that implies restricting the coefficients and thus, the number of stages and the computational cost may significantly increase with respect to other methods without those restrictions. In contrast, the technique which is suggested here is cheaper because it just needs, for any method, to add some terms with information only on the boundaries. Moreover, time-dependent boundary conditions are directly tackled here.

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters

Abstract This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O ( Δ τ s + M − 1 2 ) {\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s = 1 , 2 s=1,2 , where Δ τ \Delta \tau is the time step and M M is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

Efficient hybridized numerical scheme for singularly perturbed parabolic reaction–diffusion equations with Robin boundary conditions

An efficient numerical technique for a singularly perturbed parabolic reaction–diffusion problem with Robin type boundary conditions is presented in this work. The governing problem is discretized using the implicit Euler technique in time direction and a hybrid numerical technique that comprises a central finite difference method in the outer region and a cubic spline in compression method in the boundary layer regions in space direction. We use Shishkin-type meshes in the space domain to resolve the layers. The Robin type boundary conditions are handled using the second-order method. The totally discretized problem is well examined for stability and convergence. The use Bakhvalov–Shishkin and Vulanović–Shishkin meshes yields a more efficient and second-order convergence whereas Shishkin mesh produces almost second-order convergent solutions. Two test examples are computed. The current method has been compared to other methods in the scientific community.

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

An orthogonal spline collocation method for singularly perturbed parabolic reaction–diffusion problems with time delay

An orthogonal spline collocation method for singularly perturbed parabolic reaction–diffusion problems with time delay