Explicit difference approximations of Du Fort-Frankel type of the one-dimensional diffusion equation

1971 ◽  
Vol 18 (1) ◽  
pp. 18-25 ◽  
Author(s):  
G. R. Joubert
Keyword(s):  
Diffusion Equation ◽  
One Dimensional ◽  
Difference Approximations ◽  
Dimensional Diffusion ◽  
The One

Numerical Solution of the Differential Diffusion Equation for a Steel Carburizing Process

2018 ◽  
Vol 284 ◽  
pp. 1230-1234
Author(s):  
Mikhail V. Maisuradze ◽  
Alexandra A. Kuklina
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Differential Diffusion ◽  
One Dimensional ◽  
The Third ◽  
Dimensional Diffusion ◽  
Simplified Algorithm ◽  
The One ◽  
Carburizing Process ◽  
Precise Assessment

The simplified algorithm of the numerical solution of the differential diffusion equation is presented. The solution is based on the one-dimensional diffusion model with the third kind boundary conditions and the finite difference method. The proposed approach allows for the quick and precise assessment of the carburizing process parameters – temperature and time.

Gravity-induced wrinkle lines in vertical membranes

1981 ◽  
Vol 375 (1762) ◽  
pp. 307-325 ◽  
Keyword(s):  
Heat Flow ◽  
Diffusion Equation ◽  
Vertical Plane ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Prime Objective ◽  
Flow Through ◽  
Flexible Membranes ◽  
The One

A theory is presented for the behaviour under self-weight of inextensible but perfectly flexible membranes supported in a vertical plane. Slack in the membrane manifests itself in the formation of (curved) wrinkle lines whose determination is the prime objective. The equilibrium and strain conditions are derived and solutions are given for several simple cases. It is shown that the wrinkle lines satisfy the one-dimensional diffusion equation and hence there are analogies, for example, with heat flow through a slab.

scholarly journals A New Approach for Error Reduction in the Volume Penalization Method

2014 ◽  
Vol 16 (5) ◽  
pp. 1181-1200 ◽  
Author(s):  
Wakana Iwakami ◽  
Yuzuru Yatagai ◽  
Nozomu Hatakeyama ◽  
Yuji Hattori
Keyword(s):  
Diffusion Equation ◽  
Stokes Equations ◽  
Error Reduction ◽  
Penalization Method ◽  
New Approach ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Mask Function ◽  
The One ◽  
Volume Penalization Method

AbstractA new approach for reducing error of the volume penalization method is proposed. The mask function is modified by shifting the interface between solid and fluid by toward the fluid region, where v and η are the viscosity and the permeability, respectively. The shift length is derived from the analytical solution of the one-dimensional diffusion equation with a penalization term. The effect of the error reduction is verified numerically for the one-dimensional diffusion equation, Burgers’ equation, and the two-dimensional Navier-Stokes equations. The results show that the numerical error is reduced except in the vicinity of the interface showing overall second-order accuracy, while it converges to a non-zero constant value as the number of grid points increases for the original mask function. However, the new approach is effectivewhen the grid resolution is sufficiently high so that the boundary layer,whose width is proportional to , is resolved. Hence, the approach should be used when an appropriate combination of ν and η is chosen with a given numerical grid.

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