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Meshless Generalized Finite Differences for numerically solve Partial Differential Equations

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mGFD

Data and methods for numerically solving Partial Differential Equations using a meshless Generalized Finite Differences Scheme.

All the codes are distributed under MIT License on GitHub and are free to use, modify, and distribute giving the proper copyright notice.

Description 📝

This repository proposes a way to achieve approximations to various Partial Differential Equations in two dimensions on highly irregular regions.

For this, the proposed method uses a Generalized Finite Differences Method for the numerical solution on unstructured clouds of points.

Examples of solving various problems in an irregular region can be found below.

Titicaca Lake Cloud of Points Titicaca Lake Cloud of Points with Holes
Poisson Equation
$\mid\mid e\mid\mid = 2.577253005346005e-06$ $\mid\mid e\mid\mid = 4.051923736734612e-06$
Heat Equation
Solution.mp4
Solution.mp4
$\mid\mid e\mid\mid = 2.772683874643615e-07$ $\mid\mid e\mid\mid = 3.844963195258414e-07$
Advection-Diffusion Equation
Solution.mp4
Solution.mp4
$\mid\mid e\mid\mid = 8.682520100538671e-07$ $\mid\mid e\mid\mid = 5.293394861064519e-07$
Wave Equation
Solution.mp4
Solution.mp4
$\mid\mid e\mid\mid = 3.999132412126389e-06$ $\mid\mid e\mid\mid = 4.584086365945307e-06$

Data 📂

All the data were taken from Author's Cloud-Generation Github Repository. The data is free for anyone to use to compare the results using different methods with the same dataset.

The following regions were considered for this repository:

  • BAN: Banderas bay in Mexico.
  • BLU: Blue Lagoon in Iceland.
  • CUA: Unitary square.
  • CUI: Cuitzeo Lake in Mexico.
  • ENG: United Kingdom Island.
  • GIB: Strait of Gibraltar.
  • HAB: Havana bay.
  • MIC: Michoacan State in Mexico.
  • PAT: Patzcuaro Lake in Mexico.
  • TIT: Titicaca Lake in South America.
  • TOB: Toba Lake in Indonesia.
  • UCH: Uchinskoye Reservoir in Russia.
  • VAL: Valencia Lake in Spain.
  • ZIR: Zirahuen Lake in Mexico

How to 🔬

The codes are self explained and completely documented. Example on how to perform approximations can be found on the files:

  • run_Poisson.py: Execution of the methods to numerically solve Poisson equation, on all the regions in the Data folder.
  • run_Heat.py: Execution of the methods to numerically solve Heat equation, on all the regions in the Data folder.
  • run_AdvDif.py: Execution of the methods to numerically solve Advection-Diffusion equation, on all the regions in the Data folder.
  • run_Wave.py: Execution of the methods to numerically solve Wave equation, on all the regions in the Data folder.

These examples can be easily modified to perform approximations with different conditions and coefficients.

Researchers 🧑‍🔬

All the codes presented were developed by:

Students 👨‍🎓 👩‍🎓

Funding 💵

With the financing of:

  • National Council of Humanities, Sciences and Technologies, CONAHCyT (Consejo Nacional de Humanidades, Ciencias y Tecnologías, CONAHCyT), México.

  • Coordination of Scientific Research, CIC-UMSNH (Coordinación de la Investigación Científica de la Universidad Michoacana de San Nicolás de Hidalgo, CIC-UMSNH), México.

  • Aula CIMNE-Morelia, México.

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Meshless Generalized Finite Differences for numerically solve Partial Differential Equations

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