Hybrid hopscotch Crank-Nicholson-Du Fort and Frankel (HP-CN-DF) method for solving two dimensional system of Burgers' equation

Maritim, Simeon K.; Rotich, John K.; Bitok, Jacob

doi:10.12988/ams.2018.8798

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…The initial and boundary conditions for u x, y, t and v x, y, t were implemented numerical solutions [14]. The analytical solution the system of Navier-Stokes equations (39) at any point x, y, t is given by the following equations u x, y, t = (64)…”

Section: Neumann Boundary Conditionsmentioning

confidence: 99%

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Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

Charles¹,

John²,

Daniel³

2021

ACM

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The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t) and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.

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Section: Neumann Boundary Conditionsmentioning

confidence: 99%

“…Neumann boundary conditions were preferably implemented in. Numerical solution of particularly the flow problem because it produces accurate and reliable numerical results [14].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

Charles¹,

John²,

Daniel³

2021

ACM

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New Stable, Explicit, Shifted-Hopscotch Algorithms for the Heat Equation

Nagy

Saleh

Omle

et al. 2021

MCA

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Our goal was to find more effective numerical algorithms to solve the heat or diffusion equation. We created new five-stage algorithms by shifting the time of the odd cells in the well-known odd-even hopscotch algorithm by a half time step and applied different formulas in different stages. First, we tested 105 = 100,000 different algorithm combinations in case of small systems with random parameters, and then examined the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. These tests helped us find the top five combinations, and showed that these new methods are, indeed, effective since quite accurate and reliable results were obtained in a very short time. After this, we verified these five methods by reproducing a recently found non-conventional analytical solution of the heat equation, then we demonstrated that the methods worked for nonlinear problems by solving Fisher’s equation. We analytically proved that the methods had second-order accuracy, and also showed that one of the five methods was positivity preserving and the others also had good stability properties.

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Hybrid hopscotch Crank-Nicholson-Du Fort and Frankel (HP-CN-DF) method for solving two dimensional system of Burgers' equation

Cited by 2 publications

References 18 publications

Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

New Stable, Explicit, Shifted-Hopscotch Algorithms for the Heat Equation

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