Numerical Experiments with a Linear Convection–Diffusion Problem Containing a Time-Varying Interior Layer

2015 ◽  
pp. 221-231
Author(s):  
Eugene O’Riordan ◽  
Jason Quinn
Keyword(s):  
Numerical Experiments ◽  
Diffusion Problem ◽  
Time Varying ◽  
Interior Layer ◽  
Convection Diffusion

scholarly journals A STABILIZING SUBGRID FOR CONVECTION–DIFFUSION PROBLEM

2006 ◽  
Vol 16 (02) ◽  
pp. 211-231 ◽  
Author(s):  
ALI I. NESLITURK
Keyword(s):  
Numerical Experiments ◽  
A Priori ◽  
Diffusion Problem ◽  
Triangular Element ◽  
Discrete Solution ◽  
Convergence Properties ◽  
Small Diffusion ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Priori Error Estimates

A stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection–diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.

scholarly journals The Sdfem for a Convection-diffusion Problem with Two Small Parameters

2003 ◽  
Vol 3 (3) ◽  
pp. 443-458 ◽  
Author(s):  
Hans-Görg Roos ◽  
Zorica Uzelac
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Small Parameters ◽  
Perturbation Parameters ◽  
Theoretical Results

AbstractA singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

scholarly journals A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Musa Çakır ◽  
Gabil M. Amiraliyev
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Convection Diffusion ◽  
First Order ◽  
Type Boundary

The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameterε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.

scholarly journals Supercloseness Result of Higher Order FEM/LDG Coupled Method for Solving Singularly Perturbed Problem on S-Type Mesh

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shenglan Xie ◽  
Huonian Tu ◽  
Peng Zhu
Keyword(s):  
Numerical Experiments ◽  
Diffusion Problem ◽  
Higher Order ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Piecewise Polynomial ◽  
Coupled Method ◽  
Convection Diffusion ◽  
Polynomial Approximations ◽  
Theoretical Results

we present a first supercloseness analysis for higher order FEM/LDG coupled method for solving singularly perturbed convection-diffusion problem. Based on piecewise polynomial approximations of degreek  (k≥1), a supercloseness property ofk+1/2in DG norm is established on S-type mesh. Numerical experiments complement the theoretical results.

Numerical Experiments for a Singularly Perturbed Parabolic Problem with Degenerating Convective Term and Discontinuous Source

2012 ◽  
Vol 12 (2) ◽  
pp. 139-152 ◽  
Author(s):  
Carmelo Clavero ◽  
Jose L. Gracia ◽  
Grigorii Shishkin ◽  
Lidia Shishkina
Keyword(s):  
Numerical Experiments ◽  
Parabolic Problem ◽  
Singularly Perturbed ◽  
Interior Layer ◽  
Convective Term ◽  
Convective Flux ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Uniform Grids ◽  
Singularly Perturbed Parabolic Problem

Abstract A finite difference scheme on special piecewise-uniform grids condensing in the interior layer is constructed for a singularly perturbed parabolic convection-diffusion equation with a discontinuous right-hand side and a multiple degenerating convective term (the convective flux is directed into the domain). When constructing the scheme, monotone grid approximations, similar to those developed and justified earlier by authors for a problem with a simple degenerating convective term, are used. Using the known technique of numerical experiments on embedded meshes, it is numerically verified that the constructed scheme converges ε-uniformly in the maximum norm at the convergence rate close to one.

Finite Difference Schemes for Convection-diffusion Problems with a Concentrated Source and a Discontinuous Convection Field

2001 ◽  
Vol 2 (1) ◽  
pp. 41-49 ◽  
Author(s):  
Torsten Linß
Keyword(s):  
Numerical Experiments ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Convection Diffusion ◽  
Concentrated Source ◽  
General Meshes ◽  
Theoretical Results ◽  
Diffusion Problems

AbstractA singularly perturbed convection-diffusion problem with a concentrated source is considered. The problem is solved numerically using two upwind difference schemes on general meshes. We prove convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm on Shishkin and Bakhvalov meshes. Numerical experiments complement our theoretical results.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

scholarly journals A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 769-782
Author(s):  
Amiya K. Pani ◽  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Second Order ◽  
Time Step ◽  
Convection Diffusion ◽  
First Order ◽  
Backward Euler ◽  
Strang Splitting ◽  
Spatially Periodic

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

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