Poisson - Invisible Fields
Modify the intensity and number of sources to see how they affect their environment in a stationary way.
Easy Science 🧪
For the curious!Power sources! Imagine placing a hot radiator or a magnet in the water: its heat or force spreads throughout the lake until everything becomes permanently still. Here you control where to place them and how strong they are!
Scientific Domain Monitor
Region: PatzcuaroClick on the monitor to set initial conditions
What is Poisson's Equation? 🧐
This equation describes how a field or potential distributes in space when the system has reached a permanent (stationary) equilibrium state. There are no changes over time; everything is still.
Physical Analogies and Applications:
- Heat in equilibrium: Imagine you have a metal plate and place heaters (sources) at certain points. After a long time, the temperature across the entire plate will stop changing. That final temperature distribution obeys Poisson's equation.
- Magnetic or electric fields: Electric charges or magnets generate a continuous force field around them. The force at each point is calculated with this equation.
- Groundwater pressure: Helps calculate how water filters and stabilizes in wells and aquifers underground.
Poisson Mathematics and mGFD Method 🧠
The Differential Equation:
$$\nabla^2 u(\mathbf{x}) = f(\mathbf{x}) \implies \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x, y)$$
- \(u(x, y)\): The physical potential (e.g., final temperature).
- \(\nabla^2\): The Laplacian operator, measuring the divergence of the potential's gradient (i.e., the local "curvature" of the field).
- \(f(x, y)\): The source or charge density function (click positions with their intensity).
Meshless Numerical Resolution (mGFD):
Since lakes have irregular boundaries, the Meshless Generalized Finite Difference (mGFD) computes the solution over an unstructured point cloud:
- For each point \(\mathbf{x}_0\), a support neighborhood (nearest nodes) is defined.
- A second-order Taylor series expansion approximation is performed: \(u(\mathbf{x}_i) \approx u(\mathbf{x}_0) + h_i \frac{\partial u}{\partial x} + k_i \frac{\partial u}{\partial y} + \frac{h_i^2}{2} \frac{\partial^2 u}{\partial x^2} + h_i k_i \frac{\partial^2 u}{\partial x \partial y} + \frac{k_i^2}{2} \frac{\partial^2 u}{\partial y^2}\).
- The coefficients/weights are solved by minimizing the error norm through weighted least squares (\(WLS\)) with a distance weight function.
- The global linear system is solved to obtain the potential value at each node while respecting the boundary condition on the lake's irregular shores.