§ Forward Euler as System of Linear Equations

• 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.