A higher order scheme for singularly perturbed delay parabolic turning point problem

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Swati Yadav ◽  
Pratima Rai
Keyword(s):  
Turning Point ◽  
Higher Order ◽  
Time Variable ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Diffusion Type ◽  
Content Type ◽  
Convection Diffusion ◽  
Order Scheme ◽  
Higher Order Scheme

Purpose The purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point. Design/methodology/approach The authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme. Findings The proposed method has a convergence rate of order O(N−2L2+Δt). Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor. Originality/value A class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

Numerical analysis and simulation of delay parabolic partial differential equation involving a small parameter

2019 ◽  
Vol 37 (1) ◽  
pp. 289-312 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Content Type ◽  
Convection Diffusion ◽  
Initial Boundary Value

Purpose The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem. Design/methodology/approach The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh. Findings The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature. Originality/value A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.

Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

2011 ◽  
Vol 9 (4) ◽  
pp. 897-916 ◽  
Author(s):  
Y. V. S. S. Sanyasiraju ◽  
Nachiketa Mishra
Keyword(s):  
Coefficient Matrix ◽  
Higher Order ◽  
Diffusion Equations ◽  
Wave Number ◽  
Two Dimensional ◽  
Thomas Algorithm ◽  
Convection Diffusion ◽  
Numerous Model ◽  
Order Scheme ◽  
Higher Order Scheme

AbstractThis paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.

A Compact Higher-Order Scheme for Two-Dimensional Unsteady Convection–Diffusion Equations

2019 ◽  
Vol 17 (07) ◽  
pp. 1950025
Author(s):  
Yon-Chol Kim
Keyword(s):  
Stability Analysis ◽  
Numerical Study ◽  
Higher Order ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Convection Diffusion ◽  
Order Scheme ◽  
Higher Order Scheme ◽  
Convection Dominated ◽  
Diffusion Problems

In this paper, we study a compact higher-order scheme for the two-dimensional unsteady convection–diffusion problems using the nearly analytic discrete method (NADM), especially, focusing on the convection dominated-diffusion problems. The numerical scheme is constructed and the stability analysis is implemented. We find the order of accuracy of scheme is [Formula: see text], where [Formula: see text] is the size of time steps and [Formula: see text] the size of spacial steps, especially, making clear the relation between [Formula: see text] and [Formula: see text] is according to the different values of diffusion parameter [Formula: see text] through von Neumann stability analysis. The obtained numerical results for several benchmark problems show that our method makes progress in the numerical study of NADM for convection–diffusion equation and is to be effective and helpful particularly in computations for the convection dominated-diffusion equations and, furthermore, valuable in the numerical treatment of many real-world problems such as MHD natural convection flow.

A second order numerical method for singularly perturbed delay parabolic partial differential equation

2019 ◽  
Vol 36 (2) ◽  
pp. 420-444 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Second Order ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Content Type ◽  
Convection Diffusion ◽  
Initial Boundary Value

Purpose The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Design/methodology/approach For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme. Findings The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time. Originality/value A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.

A Parallel HOFD Scheme for Direct Numerical Simulation of Turbulent Magnetically Driven Flows

2001 ◽  
Author(s):  
X. Ai ◽  
B. Q. Li
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Direct Numerical Simulation ◽  
Turbulent Flows ◽  
Parallel Machines ◽  
Higher Order ◽  
Electrically Conducting ◽  
Wide Range ◽  
Order Scheme ◽  
Higher Order Scheme

Abstract Turbulent magnetically flows occur in a wide range of material processing systems involving electrically conducting melts. This paper presents a parallel higher order scheme for the direct numerical simulation of turbulent magnetically driven flows in induction channels. The numerical method is based on the higher order finite difference algorithm, which enjoys the spectral accuracy while minimizing the computational intensity. This, coupled with the parallel computing strategy, provides a very useful means to simulate turbulent flows. The higher order finite difference formulation of magnetically driven flow problems is described in this paper. The details of the parallel algorithm and its implementation for the simulations on parallel machines are discussed. The accuracy and numerical performance of the higher order finite difference scheme are assessed in comparison with the spectral method. The examples of turbulent magnetically driven flows in induction channels and pressure gradient driven flows in regular channels are given, and the computed results are compared with experimental measurements wherever possible.

Sign in / Sign up

Export Citation Format

Share Document