scholarly journals On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions

2013 ◽  
Vol 51 (4) ◽  
pp. 2470-2490 ◽  
Author(s):  
Sean O. Settle ◽  
Craig C. Douglas ◽  
Imbunm Kim ◽  
Dongwoo Sheen
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Poisson Equation ◽  
Dirichlet Boundary ◽  
Difference Schemes ◽  
Dirichlet Boundary Conditions ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
Compact Finite Difference ◽  
The One

scholarly journals Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2508
Author(s):  
Jesús Amo-Navarro ◽  
Ricardo Vinuesa ◽  
J. Alberto Conejero ◽  
Sergio Hoyas
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Difference Schemes ◽  
Laplacian Operator ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
Compact Finite Difference ◽  
Compact Finite Difference Schemes

In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows.

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