An exponential time differencing method of lines for the Burgers and the modified Burgers equations

Bratsos, A. G.

doi:10.1002/num.22273

Cited by 9 publications

(3 citation statements)

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“…They applied quasilinearization and the Crank-Nicolson method for time integration to achieve an unconditionally stable scheme. A second-order exponential time differencing scheme was used by [23] to solve Burgers' equation and its modified form. A discussion on quintic spline collocation for the modified Burgers' equation was carried out in [24], while [25] examined the modified Burgers' equation using the quintic B-spline collocation method.…”

Section: Introductionmentioning

confidence: 99%

Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions

2023

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This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.

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Section: Introductionmentioning

confidence: 99%

Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions

2023

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“…A local discontinuous Galerkin technique has been utilized for obtaining solutions for the modi ed form of the Burger equation by Zhang et al in [19]. Bratsos and Abdul [20] adapted an exponential time di erencing scheme for simulating the Burger and modi ed the Burger equation. Seydao glu [21] employed an algorithm based on the combination of implicit-explicitnite di erence schemes for solving the Burger equation.…”

Section: Introductionmentioning

confidence: 99%

On numerical solutions of Telegraph, viscous, and modified Burgers equations via Bernoulli collocation method

Adel,

Rezazadeh,

Inc

2023

Scientia Iranica

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The presented work aims to develop a novel technique for obtaining the solution of linear and nonlinear Partial Di erential Equations (PDEs). This technique is based on applying a collocation method with the aid of Bernoulli polynomials and the use of such an algorithm to solve di erent types of PDEs. The method applies the regular nite di erence scheme to the main problem and transforms it into an algebraic system. The obtained system is then solved, the unknown coe cient is acquired, and an approximate solution for the problems is achieved. Some test results of famous equations, including the telegraph, viscous Burger, and modi ed Burger equations, are tested to ensure that the provided algorithm is e ective and robust. In addition, a comparison is provided with other recent techniques from the literature. The current technique proves to have high accuracy concerning the error measure and through graphical representation of the solution.

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“…The Burgers' equation admits some difficulties in obtaining numerical solutions under sufficiently high Reynolds numbers wherein shocks in the solution may form. Many authors have obtained numerical solutions to the Burgers' equation using finite elements, Galerkin methods, finite differences, operator splitting methods, energy‐preserving schemes, exponentially fitted methods, spectral methods, cubic B‐splines, Adomian decomposition methods, homotopy perturbation method, differential quadrature, wavelets, compact schemes, and method of lines .…”

Section: Introductionmentioning

confidence: 99%

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Angstmann

Henry

Jacobs

et al. 2019

Numerical Methods Partial

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Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.

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An exponential time differencing method of lines for the Burgers and the modified Burgers equations

Cited by 9 publications

References 38 publications

Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions

Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions

On numerical solutions of Telegraph, viscous, and modified Burgers equations via Bernoulli collocation method

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

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