scholarly journals Sixth-order compact finite difference scheme with discrete sine transform for solving Poisson equations with Dirichlet boundary conditions

2021 ◽  
Vol 10 ◽  
pp. 100148
Author(s):  
Amanuel Hossiso Gatiso ◽  
Melisew Tefera Belachew ◽  
Getinet Alemayehu Wolle
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sixth Order ◽  
Compact Finite Difference Scheme ◽  
Poisson Equations ◽  
Sine Transform

scholarly journals Non parabolic interface motion for the 1-D Stefan problem: Dirichlet boundary conditions

2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2327-2336 ◽  
Author(s):  
Ernesto Hernandez ◽  
José Otero ◽  
Rubén Santiago ◽  
Raúl Martinez ◽  
Francisco Castillo ◽  
...  
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Large Time ◽  
Dirichlet Boundary Conditions ◽  
Time Intervals ◽  
Interface Motion ◽  
Interface Position ◽  
The Heat Balance

Over a finite 1-D specimen containing two phases of a pure substance, it has been shown that the liquid-solid interface motion exhibits parabolic behavior at small time intervals. We study the interface behavior over a finite domain with homogeneous Dirichlet boundary conditions for large time intervals, where the interface motion is not parabolic due to finite size effects. Given the physical nature of the boundary conditions, we are able to predict exactly the interface position at large time values. These predictions, which to the best of our knowledge, are not found in the literature, were confirmed by using the heat balance integral method of Goodman and a non-classical finite difference scheme. Using heat transport theory, it is shown as well, that the temperature profile within the specimen is exactly linear and independent of the initial profile in the asymptotic time limit. The physics of heat transport provides a powerful tool that is used to fine tune the numerical methods. We also found that in order to capture the physical behavior of the interface, it was necessary to develop a new non-classical finite difference scheme that approaches asymptotically to the predicted interface position. We offer some numerical examples where the predicted effects are illustrated, and finally we test our predictions with the heat balance integral method and the non-classical finite difference scheme by studying the liquid-solid phase transition in aluminum.

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