scholarly journals A finite difference method for 3D incompressible flows in cylindrical coordinates

2005 ◽  
Vol 34 (8) ◽  
pp. 950-971 ◽  
Author(s):  
E. Barbosa ◽  
O. Daube
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Incompressible Flows ◽  
Cylindrical Coordinates

On the Analytical Inversion of Solver Matrices Used in Numerical Approximations for the Diffusion Equation

2021 ◽  
Vol 413 ◽  
pp. 3-18
Author(s):  
Till Glage ◽  
Axel von der Weth ◽  
Frederik Arbeiter ◽  
Daniela Piccioni Koch
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Difference Method ◽  
Analytical Approach ◽  
Computational Effort ◽  
Gas Release ◽  
Cylindrical Coordinates ◽  
Optimal Range

The goal of this paper is to introduce an analytical approach for the inversion of nxn solver matrices, which are typically used in Finite Difference Method approximations. In the present case, they are used to solve the Diffusion Equation numerically, since in many physics and engineering fields, partial differential equations cannot be solved analytically. The method presented in this work is primarily formulated for cylindrical coordinates, which are often used in Gas Release Experiments as those described in [8]. However, it is possible to introduce a generalized method, which also allows solutions for Cartesian solvers. The advantage of having the explicit inverse is considerable, since the computational effort is reduced. In this paper we also carry out an investigation on the eigenvalues of the backward and forward solver matrix in order to determine an optimal range for the discretization parameters.

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