Parabolic Convection-diffusion Research Articles

Solution of singularly perturbed system of 2D parabolic convection diffusion differential-difference equations using alternating direction method

Solution of singularly perturbed system of 2D parabolic convection diffusion differential-difference equations using alternating direction method

Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems

AbstractWe are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

PurposeSingular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.Design/methodology/approachWe design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.FindingsConsistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.Originality/valueParabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.

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.

Asymptotic expansion for convection-dominated transport in a thin graph-like junction

We consider for a small parameter [Formula: see text] a parabolic convection–diffusion problem with Péclet number of order [Formula: see text] in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order [Formula: see text] connected through a domain (node) of diameter [Formula: see text] Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as [Formula: see text] i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems [Formula: see text] are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.

FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions

This paper aims to present in a systematic form the stability and convergence analysis of a numerical method defined in nonuniform grids for nonlinear elliptic and parabolic convection–diffusion–reaction equations with Neumann boundary conditions. The method proposed can be seen simultaneously as a finite difference scheme and as a fully discrete piecewise linear finite element method. We establish second convergence order with respect to a discrete H1-norm which shows that the method is simultaneously supraconvergent and superconvergent. Numerical results to illustrate the theoretical results are included.

Absolutely stable fitted mesh scheme for singularly perturbed parabolic convection diffusion equations

Absolutely stable fitted mesh scheme for singularly perturbed parabolic convection diffusion equations

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

A numerical investigation of singularly perturbed 2D parabolic convection–diffusion problems of delayed type based on the theory of reproducing kernels

A numerical investigation of singularly perturbed 2D parabolic convection–diffusion problems of delayed type based on the theory of reproducing kernels

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

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

Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

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>

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.

Hybrid method for singularly perturbed Robin type parabolic convection–diffusion problems on Shishkin mesh

This work presents a numerical solution to singularly perturbed Robin-type parabolic convection–diffusion problems. A hybrid method that combines the central difference scheme in the inner region and the midpoint of the upwind scheme in the outer region on a Shishkin mesh is utilized in the space direction, and an implicit Euler technique is used in the time direction. The analysis of convergence demonstrates that the present method converges with O(N−2ln2N+Δt). The theoretical findings are supported by two numerical examples. The upwind method proposed in the literature is improved by the present method.

Numerical Analysis of Singularly Perturbed System of parabolic convection-diffusion IBVP system with boundary layers

In this paper, we get the numerical solution of a singularly perturbed system of boundary layer-exhibiting parabolic convection-diffusion problems. The backward-Euler method and an upwind finite difference scheme make up the suggested numerical approach for the time and spatial derivatives, respectively. For the spatial discretization, we analyse the scheme on a piecewise uniform Shishkin mesh in order to establish uniform convergence with regard to the perturbation parameters. The stability analysis for the suggested method is provided, and a parameter-uniform error estimate is generated. We have conducted some numerical tests to verify the theoretical findings.