An efficient robust numerical method for singularly perturbed Burgers’ equation

Gowrisankar, S.; Natesan, Srinivasan

doi:10.1016/j.amc.2018.10.049

Cited by 24 publications

(23 citation statements)

References 22 publications

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“…To obtain a semi-discretization system for (4), we use finite difference weight matrices ( (17) and (18)) for diffusion (or Laplace) and dispersion terms in (3), respectively. We set ∇ 3 = ∂ 3 ∂x 3 and ∇ 3 = ∂ 3 ∂x 3 + ∂ 3 ∂y 3 for description simplicity and obtain the following systems depending on the problem dimensionality.…”

Section: Semi-discretization System Of Diffusion-dispersion Problemmentioning

confidence: 99%

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

Bak

2020

Adv Differ Equ

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In this work, we develop a high-order composite time discretization scheme based on classical collocation and integral deferred correction methods in a backward semi-Lagrangian framework (BSL) to simulate nonlinear advection-diffusion-dispersion problems. The third-order backward differentiation formula and fourth-order finite difference schemes are used in temporal and spatial discretizations, respectively. Additionally, to evaluate function values at non-grid points in BSL, the constrained interpolation profile method is used. Several numerical experiments demonstrate the efficiency of the proposed techniques in terms of accuracy and computation costs, compare with existing departure traceback schemes.

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Section: Semi-discretization System Of Diffusion-dispersion Problemmentioning

confidence: 99%

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

Bak

2020

Adv Differ Equ

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“…To obtain such a mesh, we use the idea of equidistribution principle which has been applied to a wide range of practical problems (see, e.g. [6,9,16,19,20]). A mesh Ω K is said to be equidistributed, if…”

Section: Adaptive Time Meshes Via Equidistributionmentioning

confidence: 99%

“…This type of monitor function has been used in some literature; see e.g., Das and Vigo-Aguiar [6], Gowrisankar and Natesan [9] and Kopteva et al [16].…”

mentioning

confidence: 99%

An adaptive moving mesh method for a time-fractional Black–Scholes equation

2019

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In this paper we study the numerical method for a time-fractional Black-Scholes equation, which is used for option pricing. The solution of the fractional-order differential equation may be singular near certain domain boundaries, which leads to numerical difficulty. In order to capture the singular phenomena, a numerical method based on an adaptive moving mesh is developed. A finite difference method is used to discretize the time-fractional Black-Scholes equation and error analysis for the discretization scheme is derived. Then, an adaptive moving mesh based on an a priori error analysis is established by equidistributing monitor function. Numerical experiments support these theoretical results. MSC: 65M06; 65M12; 65M15

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“…It has interesting properties of nonlinear advection [8]. The Burgers equation belongs to the class of Navier-Stokes equation [5], [7] is a fundamental partial differential equation from fluid mechanics [7]. Therefore, studying the solution of the Burgers equation will be helpful to solve the Navier-Stokes equations [8].…”

Section: Introductionmentioning

confidence: 99%

“…With pseudo-spectral methods care must be taken with the round-off error issue when higher derivatives or a large several points is involved. Gowrisankar, S., and Natesan, S. [7] present the numerical solution of singularly perturbed initial-boundary Burgers" equation by using an efficient robust numerical method. They provide an e-uniformly convergent numerical method for the singularly perturbed Burger equation.…”

Section: Introductionmentioning

confidence: 99%

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

Koroche

2021

IJAM

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In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

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An efficient robust numerical method for singularly perturbed Burgers’ equation

Cited by 24 publications

References 22 publications

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

An adaptive moving mesh method for a time-fractional Black–Scholes equation

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

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