Application of diffeomorphic transform to solve PDEs with curvilinear domains
DOI:
https://doi.org/10.5902/2179460X90584Keywords:
Flow, Numerical, Curvilinear, Diffusion, SimulationAbstract
Understanding natural phenomena often requires the application of complex mathematical models. In many cases, to accurately and comprehensively capture these phenomena, it is essential to resort to partial differential equations (PDEs), which are powerful tools in describing physical and natural processes. PDEs cover a wide variety of phenomena and have different characteristics, requiring different approaches for their resolution. However, traditional PDE solving techniques often operate under the assumption that the domain in which they are defined is rectangular. This assumption significantly simplifies the process of solving equations, facilitating the use of traditional mathematical techniques. However, such an assumption can be limiting when we deal with phenomena that occur in domains with more complex geometries, such as curvilinear domains. This study therefore focuses on solving PDEs defined in curvilinear domains, aiming to expand the possibilities of applying classical techniques. To achieve this objective, we resort to the principles of Differential Geometry. Under specific conditions, it is possible to develop a diffeomorphic transformation, which establishes a new coordinate system. This transformation allows the curvilinear domain to be represented in an equivalent way in a rectangular domain, allowing the application of classical PDE resolution techniques.
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