What is it about?

FDM (finite difference method) was first proposed as a numerical calculation of partial differential equations. The prevailing theory is that this method is not applicable in 2D calculations unless the calculation area is square or rectangular. But this dogma is now being completely overcome. Although influential research results by many researchers have been published, the main point of my research is to formulate the calculation method of FDM using interpolation polynomials. This is the origin of the IFDM (interpolation FDM) naming.

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Why is it important?

Numerical calculations in the Cartesian coordinate system are still very important. In IFDM, it is possible to define the calculation area analytically, but the mesh configuration (calculation point setting) is instantly determined only by inputting the polygonal line coordinates of the boundary shape, and the calculation result is fully automated. By creating a general-purpose system, many numerical calculation examples can be output instantly just by changing the parameters.

Perspectives

Traditionally, numerical calculations have been said to be approximate calculations. However, in view of the theory of IFDM calculation based on interpolation polynomials, the proposed method may develop into a method of calculating an infinitely exact solution. I will report it in the future.

PH.D. Tsugio Fukuchi

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This page is a summary of: Higher order difference numerical analyses of a 2D Poisson equation by the interpolation finite difference method and calculation error evaluation, AIP Advances, December 2020, American Institute of Physics, DOI: 10.1063/5.0018915.
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