Layer-adapted methods for a singularly perturbed singular problem

2011 ◽  
Vol 11 (2) ◽  
pp. 192-205
Author(s):  
Christian Grossmann ◽  
Lars Ludwig ◽  
Hans-Görg Roos
Keyword(s):  
Boundary Layer ◽  
Finite Elements ◽  
Boundary Value Problem ◽  
Bessel Functions ◽  
Basis Functions ◽  
Singularly Perturbed ◽  
Singular Problem ◽  
Dimensional Problem ◽  
Modified Bessel Functions ◽  
One Dimensional

Abstract In the present paper we analyze linear finite elements on a layer adapted mesh for a boundary value problem characterized by the overlapping of a boundary layer with a singularity. Moreover, we compare this approach numerically with the use of adapted basis functions, in our case modified Bessel functions. It turns out that as well adapted meshes as adapted basis functions are suitable where for our one-dimensional problem adapted bases work slightly better.

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

scholarly journals Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

2017 ◽  
Vol 8 (1-2) ◽  
pp. 52
Author(s):  
Samir Karasuljic ◽  
Enes Duvnjakovic ◽  
Vedad Pasic ◽  
Elvis Barakovic
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Uniform Convergence ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
One Dimensional ◽  
The One ◽  
Theoretical Results

We consider an approximate solution for the one--dimensional semilinear singularly--perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.

scholarly journals Ingularly perturbed equations in critical cases

2021 ◽  
Vol 84 (4) ◽  
pp. 69-75
Author(s):  
Zh.K. Daniyarova ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Ordinary Differential Equations ◽  
Asymptotic Expansions ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Singular Problem ◽  
Time Interval

Singularly perturbed partial differential equations with small parameters with higher derivatives deserve special attention, which often arise in a variety of applied problems and are used in describing mathematical models of diffusion processes, absorption taking into account small diffusion, filtration of liquids in porous media, chemical kinetics, chromatography, heat and mass transfer, hydrodynamics and many other fields. It is necessary to consider the creation of an asymptotic classification of solutions of singularly perturbed equations using a well-known approach to solving the boundary value problem. In this case, the singular problem is understood as the problem of constructing the asymptotics of the solution of the Cauchy problem for a system of ordinary differential equations with a small parameter with a large derivative. The asymptotics of the solution in all cases is based on the last time interval or the construction of a boundary value problem for a system with a weak clot in an asymptotically large time interval. Purpose - to construct and substantiate the asymptotics of solving a singular initial problem for a system of two nonlinear ordinary differential equations with a small parameter; To date, a number of methods have been developed for constructing asymptotic expansions of solutions to various problems. This is the method of boundary functions developed in the works of A.B. Vasilyeva, M.I. Vishik, L.A. Lusternik, V.F. Butuzov; the regularization method of S. A. Lomov, methods of averaging, VKB, splicing of asymptotic decompositions of A.M. Ilyin and others. All the above methods allow us to obtain asymptotic expansions of solutions for wide classes of equations. At the same time, such singularly perturbed problems often arise, to which ready-made methods are not applicable or do not allow to obtain an effective result. Therefore, the development of methods for solving equations remains a very urgent problem. As a result of the study, an algorithm for constructing an asymptotic classification of the initial solution of the problem with a singular perturbation is given, and approaches to estimating the residual term are also shown.

scholarly journals NUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM INCLUDING BOTH DELAY AND BOUNDARY LAYER

2018 ◽  
Vol 23 (4) ◽  
pp. 568-581 ◽  
Author(s):  
Erkan Cimen
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Delay Differential Equation ◽  
Order Parameter ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
First Order ◽  
Delay Differential

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.

Modifications of the Godunov method for computing disturbance propagation in an elastic body

2016 ◽  
Vol 11 (1) ◽  
pp. 119-126 ◽  
Author(s):  
A.A. Aganin ◽  
N.A. Khismatullina
Keyword(s):  
Elastic Body ◽  
Godunov Method ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
One Dimensional ◽  
Disturbance Propagation ◽  
Linear Waves ◽  
Longitudinal And Shear Waves ◽  
Contact Discontinuities

Numerical investigation of efficiency of UNO- and TVD-modifications of the Godunov method of the second order accuracy for computation of linear waves in an elastic body in comparison with the classical Godunov method is carried out. To this end, one-dimensional cylindrical Riemann problems are considered. It is shown that the both modifications are considerably more accurate in describing radially converging as well as diverging longitudinal and shear waves and contact discontinuities both in one- and two-dimensional problem statements. At that the UNO-modification is more preferable than the TVD-modification because exact implementation of the TVD property in the TVD-modification is reached at the expense of “cutting” solution extrema.

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