A Note on Crank-Nicolson Scheme for Burgers’ Equation

Pandey, Kanti; Verma, Lajja

doi:10.4236/am.2011.27118

Cited by 15 publications

(9 citation statements)

References 10 publications

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“…The method is proved to be consistent and is of order two in space and time. The numerical solution is calculated for two test problems with different values of constant of diffusivity k. It is observed that the method is more accurate than the existing numerical methods [9], [27], [31]. …”

Section: Discussionmentioning

confidence: 99%

“…Numerical solutions of one dimensional nonlinear Burgers equation (1) are obtained by Crank-Nicolson Type method (4) for two problems given in section 1 and results are compared with existing three methods [9], [27], [31] and exact solution given in section 1. It is observed that the the method (4) gives more accurate solution than the other methods.…”

Section: Crank-nicolson Type Methodsmentioning

confidence: 99%

“…3, No. 5, September 2013 solution is compared with the errors in numerical solution obtained in [9], [27], [31]. Fig.…”

Section: Crank-nicolson Type Methodsmentioning

confidence: 99%

“…The method is shown to be second order in time and space and consistent. The solutions of Burgers equation obtained by Crank-Nicolson type method are compared with numerical solutions obtained in [9], [27], [31].…”

Section: Introductionmentioning

confidence: 99%

“…In Section III the difference scheme is developed the method is proved to be consistent and the scheme is of second order in both space and time. In Section IV the numerical solutions of certain test problems are obtained by Crank-Nicolson type method and results are compared with analytical solutions and other numerical methods given in [9], [27], [31].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Crank-Nicolson Type Method for Burgers Equation

Wani¹,

Thakar²

2013

IJAPM

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Abstract-In this paper we presented Crank-Nicolson type scheme for numerical solution of one dimensional non linear Burgers equation with Homogeneous Dirichelets Boundary conditions. The difference scheme is shown to be consistent and is of second order in time and space. The numerical solutions are obtained for two test problems for different values of t and constant of diffusivity k. The solutions are compared with existing methods.

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Section: Discussionmentioning

confidence: 99%

Section: Crank-nicolson Type Methodsmentioning

confidence: 99%

“…3, No. 5, September 2013 solution is compared with the errors in numerical solution obtained in [9], [27], [31]. Fig.…”

Section: Crank-nicolson Type Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Crank-Nicolson Type Method for Burgers Equation

Wani¹,

Thakar²

2013

IJAPM

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Numerical Simulation of Mixing between Sequential Fluids in a Pipeline Considering Real Conditions

2022

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The mixing behavior of two sequential petroleum products resulting from convection and diffusion mechanisms was numerically modeled. To this end, the initial mixing that happens during the fluid change in the pipeline was considered. A numerical code was also developed, and two‐dimensional volume fraction variations of a 100‐m‐long, 16‐in‐diameter pipe were calculated. The model was validated by comparison with field data in a real pipeline. The results showed a significant difference in mixing length between the old and new conditions. When the actual conditions were used, the error in the results was reduced from 62 to 12 %. Overall, it was discovered that if available correlations are modified on the basis of the initial mixing, they can also produce good results in short pipelines.

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On a weakly L‐stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations

Rawani

Verma

et al. 2021

Math Methods in App Sciences

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In the present paper, we establish an efficient numerical scheme based on weakly L‐stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burgers' equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomial of order five and backward explicit Taylor's series approximation of order six to derive the numerical integration formula for the initial value problem (IVP) y′false(tfalse)=gfalse(t,yfalse),0.1emyfalse(t0false)=ρ0. We combine this method with the NSFD scheme and convert the problem into the system of algebraic equations. We discuss the convergence, truncation error, and stability of the developed method. To demonstrate the efficiency of the developed method, we compare the numerical results with some existing numerical results and exact solutions.

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A Note on Crank-Nicolson Scheme for Burgers’ Equation

Cited by 15 publications

References 10 publications

Crank-Nicolson Type Method for Burgers Equation

Crank-Nicolson Type Method for Burgers Equation

Numerical Simulation of Mixing between Sequential Fluids in a Pipeline Considering Real Conditions

On a weakly L‐stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations

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