scratch
§ Forward Euler as System of Linear Equations
created 2024-10-06 · last edited 2024-12-26
- Suppose we want to solve
df/dt = grad f on a grid of size 3, for 5 time steps. - I give boundary conditions
f(x=1, t=- = 42, and f(x=3, t=-) = 100, and initial condition f(x=-, t=0) = [42, 80, 100]. - The idea is to note that
df/dt = grad f discretizes as f(x, t+1) - f(x, t) = f(x + 1, t) - f(x, t). - This simplifies to
f(x, t+1) = f(x, t). - We also need to impose the boudary conditions.
- We do this by writing a gigantic system of equations, of the form
A(p, q, x, t) f(x, t) - b(p, q) = 0. - Here,
f(x, t) is a 15x1 size vector. Essentially, time just becomes another dimension, and we vectorize everything to get a gigantic matrix.
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