Introduction of Curvilinear Coordinates into Numerical Analysis

TL;DR

This paper proposes using curvilinear coordinates in numerical analysis to better handle complex boundaries, discontinuities, and mesh generation, offering a practical alternative to tensor notation limitations in computer programming.

Contribution

It introduces an elementary approach to incorporate curvilinear coordinates into numerical methods, facilitating improved boundary handling and mesh generation without tensor notation complexities.

Findings

Enhanced treatment of fixed and moving discontinuities

Ability to transform non-square regions into square ones

Improved approximation of curved boundaries

Abstract

Introduction of curvilinear coordinates might be very convenient in many cases. Theoretically, tensor analysis would be most suited. However, tensor notation can't be used in numerical procedure. For example, the strict discrimination of upper and lower suffices is impossible in the present computer languages. In the present paper, a rather elementary approach more suited to write codes of computer programming is adapted. If we introduce curvilinear coordinates, we would be able to treat in more natural way to handle fixed discontinuity, moving discontinuity and curved boundary. We could generate fine mesh in the neighborhood of fixed and moving discontinuities. Furthermore, if we introduce curvilinear coordinates, we can transform a non-square region into square one and can use a regular mesh. Usually in numerical calculation, a curved boundary is approximated by a jagged or non-smooth boundary. An introduction of curvilinear coordinates could solve many problems in the numerical analyses.