2D Poisson's Equation

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Introduction: Poisson's Equation

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• Poisson's Equation is a Partial Differential Equation (PDE) that describes how a Scalar Field (like Electric Potential, Temperature, or Gravitational Potential) varies in Space due to a Source Term.
• The mathematical form of this PDE is:
  • del * (del u) = f(x, y).
  • del * del is the Laplace Operator.
  • u is the unknown Scalar Field.
  • f(x, y) is the Source Term in 2D Space; note: 3D case: f(x, y, z).
• What does this equation tell us?
• Poisson’s Equation tells you how sources (e.g., charge, mass, heat) generate or influence a Potential Field.
• In terms of Fluid Dynamics, u represents the Pressure Field inside an Incompressible Flow and f represents Negative Divergence of Velocity.
• What this means is How Velocity Divergence relates to Pressure.
• Note: this project/software doesn't look any physical scenario.

Poisson's Equation and OpenMP

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• OpenMP helps solve Poisson’s equation more efficiently by Parallelizing the computational workload, which is often the most time-consuming part in numerical methods.
• In methods like Finite Difference or Finite Element, you Discretize the Domain into a Grid and iteratively solve for u(x,y) using an algorithm like Jacobi, Gauss-Seidel, or SOR.
• For a 2D grid of size N×N, each iteration involves updating every interior grid point using neighboring values.
• This is computationally expensive, especially for:
  • Fine grids (large N).
  • Many iterations (to converge).

How OpenMP Helps

1. Parallelizes Grid Updates

• Each point in the grid update is independent (especially in Jacobi iteration), so OpenMP can safely split the workload among multiple threads.
• This means multiple points are computed simultaneously, reducing total computation time.

2. Speeds Up Iterations

• Since each iteration involves updating the entire grid, OpenMP allows multiple CPU cores to work in parallel, reducing the time per iteration. This is especially effective for high-resolution grids.

3. Improves Scalability

• OpenMP is scalable across many cores, so performance improves with hardware — making it a simple yet powerful way to scale up simulations.

Requirements

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• Compiler: gcc 13.1.0; needs to support OpenMP.
• OS: Ubuntu 20.04.
• A `text editor for any changes.
• Make.

Getting and Using the Software

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• $ git clone git@github.com:MRLintern/2D_Poisson_Equation_OpenMP.git
• $ cd 2D_Poisson_Equation_OpenMP
• If you want to generate the results and produce the plot yourself:
• $ make clean
• Unit Testing:
• $ make test
• All test cases (should) pass. Now create the executable:
• $ make
• $ ./main
• Plot the Results:
• You don't need to be in the results directory to create the plot; just stay inside 2D_Poisson_Equation_OpenMP. Note: the plot, solution_and_error_surface.png, will be saved inside results.
• $ python3 plot_results.py
• An image will be created and saved in the results directory.