High-order finite difference schemes for the solution of the generalized Burgers-Fisher equation

Sarı, Murat; Gürarslan, Gürhan; Zeytinoğlu, Asuman

doi:10.1002/cnm.1360

Cited by 7 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years, various numerical methods were used by many researchers to solve GBHE andGBFE. In 2011, Bratsos (2011) and Sari et al (2011) have obtained the numerical solution for GBHE. In the same year, Mohammadi (2011) proposed a spline method for GBFE.…”

Section: Ec 378mentioning

confidence: 99%

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

Mohanty

Sharma

2020

View full text Add to dashboard Cite

Purpose This paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number. Design/methodology/approach In this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method. Findings The paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs. Research limitations/implications There are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable. Practical implications Physical problems with singular and non-singular coefficients are directly solved by this method. Originality/value The paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

show abstract

Section: Ec 378mentioning

confidence: 99%

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

Mohanty

Sharma

2020

View full text Add to dashboard Cite

show abstract

“…Spatial derivatives are computed by the FD6 scheme based on the Taylor series expansion [43]. In order to use finite difference approximation, it is started by defining a uniform grid consisting of N points satisfying 0…”

Section: Spatial Discretizationmentioning

confidence: 99%

A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

2019

Self Cite

View full text Add to dashboard Cite

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes.

show abstract

“…Many researchers have studied it theoretically and numerically. Because of its strong nonlinearity, it is often used as a model problem to test various numerical methods, where, among others, finite-difference methods [16,23,24], the Adomian decomposition method [17,19] and differential quadrature [22] have been used. Powerful mathematical methods such as the tanh [11,28], extended tanh [10], tanh-coth [29,30], exp-function [31], variational iteration [21], homotopy analysis [2], factorization [12] and spectral collocation [14,18] methods have also been used for this equation.…”

Section: Introductionmentioning

confidence: 99%

An Improved Second-Order Numerical Method for the Generalized Burgers–fisher Equation

Bratsos

2013

ANZIAM J.

View full text Add to dashboard Cite

A second-order in time finite-difference scheme using a modified predictor-corrector method is proposed for the numerical solution of the generalized Burgers-Fisher equation. The method introduced, which, in contrast to the classical predictor-corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a singlekink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the twokink wave arising from the collision of two single-kink waves are examined.2010 Mathematics subject classification: primary 35K57; secondary 35Q51, 65M06, 65Y10.

show abstract

High-order finite difference schemes for the solution of the generalized Burgers-Fisher equation

Cited by 7 publications

References 20 publications

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

An Improved Second-Order Numerical Method for the Generalized Burgers–fisher Equation

Contact Info

Product

Resources

About