Numerical analysis of a singularly perturbed convection diffusion problem with shift in space

We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.Comment: 17 pages, 1 figur

An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays

AbstractThis paper deals with singularly perturbed boundary value problem for a linear second order delay differential equation. It is known that the classical numerical methods are not satisfactory when applied to solve singularly perturbed problems in delay differential equations. In this paper we present an exponentially fitted finite difference scheme to overcome the drawbacks of the corresponding classical counter parts. The stability of the scheme is investigated. The proposed scheme is analyzed for convergence. Several linear singularly perturbed delay differential equations have been solved and the numerical results are presented to support the theory

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

A numerical scheme for singularly perturbed delay differential equations of convection-diffusion type on an adaptive grid

In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method

AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS WITH A NEGATIVE SHIFT

Abstract. In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve singularly perturbed boundary value problems for second order ordinary differential equations of reactiondiffusion type with a delay (negative shift). In this technique, the original problem of solving the second order differential equation is reduced to solving four first order singularly perturbed differential equations without delay and one algebraic equation with a delay. The singularly perturbed problems are solved by a second order hybrid finite difference scheme. An error estimate is derived by using supremum norm and it is of order O(ε + N −2 ln 2 N ), where N is a discretization parameter and ε is the perturbation parameter. Numerical results are provided to illustrate the theoretical results

Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions

Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups

Parameter-uniformly convergent numerical scheme for singularly perturbed delay parabolic differential equation via extended B-spline collocation

This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results

Numerical solution of a boundary value problem including both delay and boundary layer

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

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

This paper introduces a novel computational approach utilizing the quartic B-spline method on a uniform mesh for the numerical solution of non-linear singularly perturbed delay differential equations (NSP-DDE) of second-order with a small negative shift. These types of equations are encountered in various scientific and engineering disciplines, including biology, physics, and control theory. We are using quartic B-spline methods to solve NSP-DDE without linearizing the equation. Thus, the set of equations generated by the quartic B-spline technique is non-linear and the obtained equations are solved by Newton-Raphson method. The success of the approach is assessed by applying it to a numerical example for different values of perturbation and delay parameter parameters, the maximum absolute error (MAE) is obtained via the double mesh principle. The convergence rate of the proposed method is four. Obtained numerical results are compared with existing numerical techniques in literature and observe that the proposed method is superior with other numerical techniques. The quartic B-spline method provides the numerical solution at any point of the given interval. It is easy to implement on a computer and more efficient for handling second-order NSP-DDE