Exponential Boundary Layers Research Articles

Exponential time difference methods with spatial exponential approximations for solving boundary layer problems

AbstractIn this work, an efficient and stable exponential time difference method is presented for solving boundary layer problems. By combining exponential time difference schemes with spatial direct discontinuous Galerkin discretization based on exponential boundary layer approximations, the proposed algorithm not only may admit large time step sizes but also could provide good spatial approximations even if on rather coarse spatial grids in the boundary layer. Some energy stabilities of the numerical scheme are rigorously derived. Numerical examples illustrate the accuracy, stability and efficiency of the algorithm.

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme. The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Parameter‐Uniform Convergent Numerical Approach for Time‐Fractional Singularly Perturbed Partial Differential Equations With Large Time Delay

In this study, we consider a parameter‐uniform convergent numerical approach for a class of time‐fractional singularly perturbed partial differential equations (TF‐SPDPDEs) with large delay in time that exhibits a regular exponential boundary layer on the right side of the spatial domain. An arbitrary very small parameter ε(0 &lt; ε &lt; &lt;1) multiplies the highest‐order derivative term of these singularly perturbed problems. The time‐fractional derivative is considered in the Caputo sense with order α ∈ (0, 1). The numerical scheme comprises the L1 scheme and nonstandard finite difference method (FDM) for discretizing the time and space variables, respectively, on a uniform mesh. To show the parameter uniform convergence of the proposed method, the truncation error and stability analysis are discussed. The method is shown to be parameter‐uniform convergent of order O((Δt)2−α + Δx), where Δt and Δx are the step sizes in the time and space directions, respectively. In order to confirm the theoretical predictions, two numerical examples are presented, and the numerical results support the theoretical concepts discussed. Finally, to show the advantage of the proposed scheme, we made comparisons with the existing numerical methods in the literature, and the numerical results reveal that the present scheme is more accurate.

A robust higher-order finite difference technique for a time-fractional singularly perturbed problem

A higher-order finite difference method is developed to solve the variable coefficients convection–diffusion singularly perturbed problems (SPPs) involving fractional-order time derivative with the order α∈(0,1). The solution to this problem class possesses a typical weak singularity at the initial time t=0 and an exponential boundary layer at the right lateral surface as the perturbation parameter ɛ→0. Alikhanov’s L2−1σ approximation is applied in the temporal direction on a suitable graded mesh, and the spatial variable is discretized on a piecewise uniform Shishkin mesh using a combination of midpoint upwind and central finite difference operators. Stability estimates and the convergence analysis of the fully discrete scheme are provided. It is shown that the fully discrete scheme is uniformly convergent with a rate of O(M−p+N−2(logN)2), where p=min{2,rα}, r is the graded mesh parameter and M,N are the number of mesh points in the time and space direction, respectively. Two numerical examples are taken in counter to confirm the sharpness of the theoretical estimates.

A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis.

Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem

A singularly perturbed convection-diffusion problem posed on the unit square in R 2 \mathbb {R}^2 , whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most k &gt; 0 k&gt;0 on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than O ( N − ( k + 1 / 2 ) ) O(N^{-(k+1/2)}) accurate in the energy norm induced by the bilinear form of the weak formulation, where N N mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish O ( N − ( k + 1 ) ) O(N^{-(k+1)}) energy-norm superconvergence on all three types of mesh for the difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space. This supercloseness property implies a new N − ( k + 1 ) N^{-(k+1)} bound for the L 2 L^2 error between the LDG solution on each type of mesh and the true solution of the problem; this bound is optimal (up to logarithmic factors). Numerical experiments confirm our theoretical results.

A robust uniformly convergent finite difference scheme for the time-fractional singularly perturbed convection-diffusion problem

The design and analysis of a finite difference method for solving a class of time-fractional singularly perturbed convection-diffusion problems is the subject of this paper. The time-fractional derivative is taken in the Caputo sense with the order α∈(0,1). The solution to this class of problems possesses weak singularity in the vicinity of the initial time t=0 and an exponential boundary layer at the right lateral surface as the singular perturbation parameter ε→0. To tackle the initial layer at time t=0, a classical L1 finite difference scheme is employed on a graded mesh to discretize the time-fractional derivative. Further, a standard upwinding procedure in the spatial direction is used on a piecewise uniform Shishkin mesh. It is shown that the resultant fully discrete scheme is parameter uniformly convergent with order of convergence O(M−p+N−1log⁡N), where p=min⁡{2−α,rα}, r is the graded mesh exponent and N,M are the number of mesh subintervals in space and time direction respectively. Lastly, numerical results illustrate that the proposed scheme is simple, efficient, robust and accurate as well as validate the error estimates presented in the paper.

Perturbation methodology for electromagnetic radiative fluxing of chemical reactive Casson fluid flow under heat source (sink) effectiveness

In this paper, magnetohydrodynamics of a Casson fluid flow is inspected with the presence of thermal radiation and chemical reaction. Employing the perturbation procedure, the modeling equations are tenacious; the graphs are acquired to illustrate the results. The Casson fluid velocity increases as the perturbation parameter increases. Grashof values for heat and mass transport enhanced Casson fluid velocity. Increasing Casson, magnetic, heat source, and radiation parameters reduce the flow velocity. Prandtl number, heat source, and radiation parameter all reduced the temperature profiles. Chemical reaction parameters lowered the concentration profiles. The skin friction enhances with Casson parameter impact. However, the skin-friction coefficient, Sherwood and Nusselt numbers reduce with an increment in the perturbation parameter. In certain cases, this study’s answers agreed well with the previous literature. Casson liquid with a magnetic region using mixed convection by an exponential vertical boundary layer is the novelty of the work.

Numerical treatment of a singularly perturbed 2-D convection-diffusion elliptic problem with Robin-type boundary conditions

We consider a singularly perturbed two-dimensional steady-state convection-diffusion problem with Robin boundary conditions. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ϵ, is a positive perturbation parameter, and so it may be arbitrarily small. Solutions to such problems present regular (exponential) boundary layers as well as corner layers. In this article, a numerical approach is carried out using a finite-difference technique with an appropriate layer-adapted piecewise-uniform Shishkin mesh to provide a good approximation of the exact solution. Some numerical examples are presented that show that the approximations obtained are accurate and that they are in agreement with the theoretical results.

Analysis of Numerical Differential Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

The paper considers numerical differentiation of functions with large gradients in the region of an exponential boundary layer. This topic is important, since the application of classical polynomial difference formulas for derivatives to such functions in the case of a uniform mesh leads to unacceptable errors if the perturbation parameter is comparable with the mesh size. The numerical differentiation formula with a given number of nodes in the difference stencil is built on subintervals covering the original interval. The accuracy of numerical differentiation formulas on a Bakhvalov mesh, which is widely used in the construction of difference schemes for singularly perturbed problems, is analyzed. For the original function of one variable, a representation in the form of a sum of regular and boundary-layer components, based on the Shishkin decomposition, is used to solve a singularly perturbed problem. Previously, such a decomposition was used to prove the convergence of difference schemes. An estimate of the error of classical polynomial formulas for numerical differentiation on a Bakhvalov mesh is obtained. The error estimate on a Bakhvalov mesh is obtained in the general case, when a derivative of an arbitrarily given order is calculated and the difference stencil for this derivative contains a given number of nodes. The error estimate depends on the order of the calculated derivative and the number of nodes in difference stencil and takes into account the uniformity in the parameter . The results of numerical experiments are presented, which are consistent with the error estimates obtained.

Optimal pointwise convergence of the LDG method for singularly perturbed convection–diffusion problem

A local discontinuous Galerkin (LDG) method is considered for a one-dimensional singularly perturbed convection–diffusion problem with an exponential boundary layer. Based on the technique of discrete Green’s function, we establish optimal pointwise convergence (up to a logarithmic factor) of the LDG method on three typical families of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type and Bakhvalov-type. Numerical experiments are also given.

AN EFFICIENT PARAMETER UNIFORM SPLINE-BASED TECHNIQUE FOR SINGULARLY PERTURBED WEAKLY COUPLED REACTION-DIFFUSION SYSTEMS

A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction-diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic $B$-spline basis functions with an exponentially graded mesh are used to solve a $\ell\times\ell$ system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh-generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numerically.

Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation

This paper considers the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation. The diffusion term of the equation is multiplied by a small perturbation parameter (ε), which takes an arbitrary value in the interval (0, 1]. For small values of ε, the solution of the equation exhibits an exponential boundary layer which makes it difficult to solve it analytically or using classical numerical methods. We proposed numerical schemes using the Crank–Nicolson method in time derivative discretization and the nonstandard finite difference method (exact finite difference method) in space derivative discretization on a uniform and piecewise uniform Shishkin mesh. The existence of unique discrete solutions and the stability of the schemes are discussed and proved. Uniform convergence of the schemes is proved. The formulated schemes converge uniformly with linear order of convergence. The method on Shishkin mesh possesses boundary layer resolving property. We validated the methods by considering two numerical examples for different values of ε and mesh length.

Convergence of a finite element method on a Bakhvalov‐type mesh for a singularly perturbed convection–diffusion equation in 2D

AbstractA finite element method of any order is applied on a Bakhvalov‐type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.

A new upwind difference analysis of an exponentially graded Bakhvalov-type mesh for singularly perturbed elliptic convection–diffusion problems

We are concerned with numerical methods for singularly perturbed elliptic convection–diffusion differential equations in two dimensions. Due to the presence of exponential boundary layers in these problems, special meshes – which are dense in layer regions – are designed to robustly resolve the layers. Recently, the so-called exponentially graded mesh has been extensively studied by Xenophontos et al. (see, e.g., (Xenophontos et al., 2016)) for which the authors classify the mesh as a Shishkin-type mesh. In this paper, we propose a modification of the exponentially graded mesh as a Bakhvalov-type mesh. The modified mesh is constructed by incorporating Bakhvalov’s idea that allows the mesh not only to be gradually graded in the layer regions, but also to preserve the favorable properties of the original Bakhvalov mesh. Our numerical experiments show an improvement in the accuracy of the computed solution when an upwind difference scheme is discretized on our modified mesh. More importantly, the main purpose of the paper is to propose a novel finite-difference error analysis, which is based on the truncation error and barrier function approach, for a class of Bakhvalov-type meshes; that is although such analysis for two-dimensional singularly perturbed convection–diffusion problems on Shishkin-type meshes was derived two decades ago, but no similar results are known for Bakhvalov-type meshes (Roos and Stynes, 2015).

A Numerical Method for Singularly Perturbed Convection-diffusion Problems Posed on Smooth Domains

A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A domain decomposition method is used, which uses a rectangular grid outside the boundary layer and a Shishkin mesh, aligned to the curvature of the outflow boundary, near the boundary layer. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.

Over-reflection of acoustic waves by supersonic exponential boundary layer flows

The two-dimensional acoustic wave equation for inviscid compressible boundary layer flows, i.e. the Pridmore-Brown equation with an exponential velocity profile for homentropic flows, is studied for the reflection and over-reflection of acoustic waves based on the exact solution in terms of the confluent Heun function. The reflection coefficient $R$ , which is the ratio of the amplitude of the reflected to that of the incoming acoustic wave, is determined as a function of the streamwise wavenumber $\alpha$ , the Mach number $M$ and the incident angle $\phi$ of the acoustic waves. Over-reflection refers to $R&gt;1$ , i.e. the reflected wave has a larger amplitude than the incident wave. We prove that, in the supersonic context, energy is always transferred from the base flow to the reflected wave, i.e. $R&lt;1$ does not exist. Meanwhile, this fact is intimately linked to the critical layer. We show that the presence of the critical layer leads to an optimal energy exchange from the base flow into the acoustic wave, i.e. the critical layer ensures $R&gt;1$ . In our analysis, we observe a special phenomenon, resonant over-reflection, which is proven to be closely related to resonant frequencies $\omega _r$ of unstable modes of the temporal stability of the base flow. At resonant frequencies of the first unstable mode, the over-reflection coefficient exhibits an unusual peak in an extremely narrow frequency interval. The maximum values of these peaks are largely synchronized with the variation of the growth rate $\omega _i$ of the unstable modes. In addition, resonant over-reflection appears also at resonant frequencies of other higher unstable modes, but their peaks of the over-reflection coefficient are always smaller than that induced by the first unstable mode.

Analysis of finite element method in balanced norms for two-parameter singularly perturbed problems

We consider the two-parameter singularly perturbed problems, and there are two exponential boundary layers in its solution. In general, the energy norm is used in error estimations of finite element method for singularly perturbed problems. However, it is difficult to capture each layer simultaneously in this norm. In this paper, a special balanced norm is defined, which can capture each exponential layer well. Based on the balanced norm, we obtain the uniform convergence of any order finite element method with respect to both parameters on a Shishkin mesh. Moreover, the optimal order is proved and some discussions in this balanced norm are presented. Finally, we give some numerical examples to verify our theoretical findings.

Stability and error analysis of a fully-discrete numerical method for system of 2D singularly perturbed parabolic PDEs

This article deals with a fully-discrete numerical method for the system of 2D singularly perturbed parabolic convection-diffusion problems. The method is based on the combination of the implicit Backward-Euler scheme for the temporal derivative and the streamline-diffusion finite element method (SDFEM) for the spatial derivatives. Due to the existence of exponential boundary layers in the solution of this problem, a piecewise-uniform layer-adapted mesh is used here for the triangulation of the spatial domain. The stability of the present method has been discussed based on an appropriate stabilization parameter. The convergency of the fully-discrete SDFEM on a discrete L2(0,T;L2)-norm using the piecewise continuous linear finite element space is analyzed in a new theoretical framework. Finally, the justification of the theoretical results is done by some numerical experiments.

An adaptive variational multiscale element free Galerkin method based on the residual-based a posteriori error estimators for convection–diffusion–reaction problems

As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from non-physical oscillations near the boundary or interior layers when solving the convection–diffusion–reaction problems with strong convection-dominated. To overcome this shortcoming, we consider incorporating an adaptive algorithm based on the residual error estimations into the VMEFG, which formed the adaptive VMEFG (AVMEFG) method. With respect to the adaptive technique adopted, two residual-based a posteriori error estimators in the H1-semi norm and energy norm are respectively used to locate and remark the high-gradient numerical solution regions. Several stationary convection–diffusion–reaction problems are solved to verify the effectiveness of the proposed method. Among them, the first example makes a comparison between the proposed method and the adaptive element free Galerkin method, while the other remaining examples compare two different residual-based a posteriori error estimators for the proposed method. Numerical examples illustrate that the proposed method is effective and efficient in solving the convection-dominated problem involving various layers. Moreover, the AVMEFG methods with the two residual-based a posteriori error estimators are generally equivalent except in the case that the solution has an exponential boundary layer, where the energy norm error estimator can achieve significantly better results than the other one.