Convection-diffusion Problems Research Articles

Nonstandard finite difference method for time-fractional singularly perturbed convection–diffusion problems with a delay in time

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be ɛ-uniform convergent with order of convergence O((Δt)2) along the temporal domain and M−1 along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.

Supercloseness of the LDG method for a singularly perturbed convection–diffusion problem on Bakhvalov-type mesh

In this study, the focus is on exploring the supercloseness property of the local discontinuous Galerkin (LDG) method in the context of a singularly perturbed convection–diffusion problem on Bakhvalov-type mesh. By developing specialized local Gauss–Radau projections in the two-dimensional case, and establishing a novel interpolation premised on the special projections, supercloseness of an optimal order k+1 can be achieved on Bakhvalov-type mesh. Notably, the obtained result remains unaffected by the singular perturbation parameter ɛ, indicating that the limitations encountered in previous studies, where the supercloseness results on Bakhvalov-type mesh were always impacted by the logarithmic factor ln1ɛ are overcome.

Automatic variationally stable analysis for finite element computations: Transient convection-diffusion problems

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time “slices” in an attempt to reduce the computational cost of the space-time discretizations.We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serves as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive strategies.

Crank–Nicolson Method for Singularly Perturbed Unsteady Parabolic Problem With Multiple Boundary Turning Points

In this paper, a numerical scheme for a time‐dependent singularly perturbed parabolic convection–diffusion problem with boundary turning points is presented. The problem exhibits a left boundary layer in the spatial domain. We use the Crank–Nicolson method for temporal discretization and a nonstandard finite difference approach for spatial discretization on uniform meshes. Through rigorous error analysis, it has been shown that the scheme is stable and parameter‐uniform convergence with a second‐order accuracy in time and a first‐order accuracy in space. Three model examples are provided to show the applicability of the scheme. It is shown that the numerical results are in agreement with the theoretical findings.

A Novel Richardson Extrapolation Technique on a Shishkin and Exponentially Graded Mesh for Singularly Perturbed Convection-diffusion Problems With Integral Boundary Conditions

A Novel Richardson Extrapolation Technique on a Shishkin and Exponentially Graded Mesh for Singularly Perturbed Convection-diffusion Problems With Integral Boundary Conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

COMPARISON OF VARIATIONAL DISCRETIZATIONS FOR A

For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare three methods: the standard linear discretization, the saddle point least square and the upwinding Petrov–Galerkin methods. We conclude that the bubble upwinding the Petrov–Galerkin method is the most performant discretization for the one-dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multi-dimensional cases.

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-Diffusion Problems With A Discontinuous Initial Condition

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-Diffusion Problems With A Discontinuous Initial Condition

An upwind finite difference scheme for singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions on a piecewise-uniform mesh

An upwind finite difference scheme for singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions on a piecewise-uniform mesh

Parameter-uniform Hybrid Numerical Scheme for Singularly Perturbed Parabolic Convection-diffusion Problems With Discontinuous Initial Conditions <br>

Parameter-uniform Hybrid Numerical Scheme for Singularly Perturbed Parabolic Convection-diffusion Problems With Discontinuous Initial Conditions <br>

CONNECTIONS BETWEEN FINITE DIFFERENCE AND FINITE ELEMENT APPROXIMATIONS FOR A CONVECTION-DIFFUSION PROBLEM

We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov–Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right-hand side load vectors for the two methods. This new approach allows for improving well-known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov–Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.

A convection–diffusion problem with a large shift on Durán meshes

A convection–diffusion problem with a large shift on Durán meshes

Continuous dependence of renormalized solution for convection-diffusion problems involving a nonlocal operator

In Ouédraogo A. et al (cf. [30]), it is provided existence and uniqueness results of L1-renormalized entropy solution for the Cauchy problem associated to the following vast class of nonlinear anisotropic degenerate parabolic-hyperbolic equations involving a nonlocal diffusion term: \begin{equation*}\label{problem_(CP)} \partial_{t}u+\nabla.F(u)-\displaystyle\sum_{i,j=1}^{N}\partial^{2}_{x_{i}x_{j}}A_{ij}(u) -{\cal L}_{\mu}[u]= f(u) \hskip0.3 cm\hbox{in }\ \ Q=(0,T)\times \RR^N \ \text{with}\ T>0\ \text{and}\ N\geq 1. \end{equation*} Our goal is to complement this previous work with a continuous dependence result of the L1-solution with respect to the data set (F,a,μ,f, u0). The strategy is to follow the approach developed by Karlsen and Ulusoy in [28]. However, we must manage the difficulties due to the fact that we are working in the whole space RN with an only integrable initial datum u0 and the term source f depends on the unknown function u.

Critical Role of Boundary Conditions in Sorption Kinetics Measurements.

In order to characterize the hygroscopic properties of cellulose-based materials, which can absorb large amounts of water from vapor in ambient air, or the adsorption capacity of pollutants or molecules in various porous materials, it is common to rely on sorption-desorption dynamic tests. This consists of observing the mass variation over time when the sample is placed in contact with a fluid containing the elements to be absorbed or adsorbed. Here, we focus on the case of a hygroscopic material in contact with air at a relative humidity (RH) differing from that at which it has been prepared. We show that the vapor mass flux going out of the sample follows from the solution of a vapor convection-diffusion problem along the surface and is proportional to the difference between the RH of the air flux and that along the surface with a multiplicative factor (δ) depending only on the characteristics of the air flux and the geometry of the system, including the surface roughness. This factor may be determined from independent measurements in which the RH along the surface is known while keeping all other variables constant. Then we show that the apparent sorption or desorption kinetics critically depend on the competition between boundary conditions and transport through the material. For sufficiently low air flux intensities or small sample thicknesses, the moisture distribution in the sample remains uniform and evolves toward the equilibrium with a kinetics depending on the value of δ and the material thickness. For sufficiently high air flux intensities or large sample thicknesses, the moisture distribution is highly inhomogeneous, and the kinetics reflect the ability of water transport by diffusion through the material. We illustrate and validate this theoretical description on the basis of magnetic resonance imaging experiments on drying cellulose fiber stacks.

Saddle point least squares discretization for convection-diffusion

We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models.

Fitted mesh scheme for singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers

In this paper, fitted mesh numerical scheme is presented for solving singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers. To approximate the solution, we discretize the temporal variable on uniform mesh and discretize the spatial one on piecewise uniform mesh of the Shishkin mesh type. The resulting scheme is shown to be almost first order convergent that accelerated to almost second order convergent by applying the Richardson extrapolation technique. Stability and consistency of the proposed method are established very well in order to guarantee the convergence of the method. Further, the theoretical investigations are confirmed by numerical experiments. Moreover, the present scheme is stable, consistent and gives more accurate solution than existing methods in the literature.

A variable stepsize hybrid block optimized technique for integrating a class of singularly perturbed parabolic problems

This paper presents and successfully applies an optimized hybrid block technique using a variable stepsize implementation to integrate a type of singularly perturbed parabolic convection–diffusion problems. The problem under consideration is semi-discretized by utilizing the method of lines. A few numerical experiments have been presented to ascertain the proposed error estimation and adaptive stepsize strategy. Furthermore, the comparison of the proposed method with other techniques in the literature is conducted via numerical experiments, and the results show that our method outperforms other existing methods.

Richardson Extrapolation Technique for Singularly Perturbed Convection-diffusion Problem with Non-local Boundary Conditions

This study presents the Richardson extrapolation techniques for solving singularly perturbed convection -diffusion problems (SPCDP) with non-local boundary conditions. A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) mesh. To handle the non-local boundary conditions, the trapezoidal rule is applied. The study establishes an error bound for numerical solutions and determines the numerical approximation for scaled derivatives. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from firs-order to second order convergence

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems.

Fitted finite element methods for singularly perturbed elliptic problems of convection-diffusion type

Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform numerical method. These schemes satisfy a discrete maximum principle. If the diffusivity parameter is bounded below by some fixed positive constant, the numerical approximations converge, in L∞, at a rate of second order. Moreover, the numerical approximations converge at a rate of first order for all values of the singular perturbation parameter.