We then start with an initial guess at the potential everywhere inside the box. The way relaxation works is the following: we start with our fixed boundary condition on the outside of our simulation "box". This is our "known" starting point, and we need to then find the solutions in the middle that solve the Laplace (or the Poisson) equation. The problem we are looking at is a boundary-value problem: to be able to solve for the potential on our grid, we need to know the value of the potential on the boundaries of our simulation. Jacobi method ¶įortunately, there is a simpler method for solving these 10,000 equations, which is known as "relaxation". How many equations will we have? If we choose a grid of 100x100 points in our $x,y$ plane, we will have 10,000 equations! That's a lot! Coding this into a matrix is quite daunting. This equation is hopefully well known to you from your course in electrostatics: is is the equation that determines the electrostatic potential $\phi(\vec \approxĪll we need to do now is to equate this to zero and find the solutions of the resulting coupled (linear) equations. We will look specifically in this course at the solutions to a specific, linear partial differential equation known as Poisson's equation:
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