scholarly journals Numerical comparison of nonstandard schemes for the Airy equation

2017 ◽  
Vol 6 (4) ◽  
pp. 141
Author(s):  
Oluwaseye Adekanye ◽  
Talitha Washington
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Difference Scheme ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Numerical Comparison ◽  
Central Difference ◽  
Central Difference Scheme ◽  
Airy Equation

This paper considers the Airy ordinary differential equation (ODE) and different ways it can be discretized. We first consider a standard discretization using the central difference scheme. We then consider two difference schemes which were created using a nonstandard methodology. Finally, we compare the different schemes and how well they approximate solutions to the Airy ODE.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

The role and application of entropy terms for acoustic wave modeling by the finite‐difference method

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1457-1465 ◽  
Author(s):  
M. A. Dablain
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Acoustic Wave Propagation ◽  
Central Difference ◽  
Central Difference Scheme ◽  
One Dimensional ◽  
The Finite Difference Method

The significance of entropy‐like terms is examined within the context of the finite‐difference modeling of acoustic wave propagation. The numerical implications of dissipative mechanisms are tested for performance within two very distinct differencing algorithms. The two schemes which are reviewed with and without dissipation are (1) the standard central‐difference scheme, and (2) the Lax‐Wendroff two‐step scheme. Numerical results are presented comparing the short‐wavelength response of these schemes. In order to achieve this response, the linearized version of an exploding one‐dimensional source is used.

scholarly journals Criticality dependence on data and parameters for a problem in combustion theory

1986 ◽  
Vol 27 (4) ◽  
pp. 416-441 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Boundary Data ◽  
Main Part ◽  
Original Problem ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Central Problem ◽  
Good Agreement

AbstractA central problem in the theory of combustion, consisting of a nonlinear parbolic equation together with initial and boundary conditions, is considered. The influence of the initial and boundary data examined. In the main part of the study, a two-step linearization is developed such that the interesting features of the original problem are given by the solution of a non-liner and ordinary differential equation. Approximate solutions are obtained and upper and lower solutions are used to assess the validity of the approximations. Whenever possible, results are compared with those obtained previously and there is good agreement in all cases.

scholarly journals On the determination of the dissipation rate of turbulence kinetic energy

2021 ◽  
Vol 62 (7) ◽  
Author(s):  
James M. Lewis ◽  
Timothy W. Koster ◽  
John C. LaRue
Keyword(s):  
Kinetic Energy ◽  
Difference Scheme ◽  
Dissipation Rate ◽  
Hot Wire ◽  
Central Difference ◽  
Time Resolved ◽  
Central Difference Scheme ◽  
Derivative Spectra ◽  
Velocity Derivative ◽  
Velocity Spectra

Abstract The paper presents a comparison of the dissipation rate obtained from numerical differentiation of the time-resolved velocity, analog differentiation of the hot-wire signal, integration of the velocity derivative spectra obtained from the velocity spectra, and the application of a power decay law. Hot-wire measurements downstream of an active-grid provide the time-resolved velocity with a Taylor Reynolds number in the range of 200–470, turbulence intensities in the range of 5.8–11%, and nominal mean velocities of 4, 6, and 8 m s$$^{-1}$$ - 1 . The dissipation rate calculated using a ninth-order central-difference scheme differs at most by $${\pm }$$ ±  4% from the value obtained by analog differentiation. For comparison, a 23rd-order central-difference scheme offers negligible (0.02%) difference relative to the ninth-order scheme. Correction for an apparent uncertainty in the calibration of the analog differentiator reduces the difference to $${\pm }$$ ±  2.5%. In contrast, integration of the velocity derivative spectra obtained from the velocity spectra leads to a dissipation rate 14–45% larger than the corresponding values obtained using analog differentiation. Results obtained from the application of a power decay law of turbulence kinetic energy with a nonzero virtual origin to determine the dissipation rate deviate by 1.7%, 1.6%, and 3.6% relative to the corresponding values obtained from the analog differentiator based on the ensemble average of downstream locations with a $${\pm }$$ ± 5.6% scatter about the ensemble average. Graphic abstract

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