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Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's equation over arbitrary domains

### Abstract

The finite difference method (FDM) based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.

© 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.

Received 11 March 2014
Accepted 13 June 2014
Published online 25 June 2014

Article outline:

I. INTRODUCTION
II. INTERPOLATION FINITE-DIFFERENCE METHOD
A. High-order accurate FD scheme
B. Relationship between FD scheme and polynomial interpolation
III. TWO-DIMENSIONAL CALCULATION
IV. THREE-DIMENSIONAL CALCULATION
V. CONCLUSION

/content/aip/journal/adva/4/6/10.1063/1.4885555

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