Is it possible to refine your question?

Nevertheless, if you already have a multigrid code to solve Poisson's equation in 2D, then it is a very quick modification (viz. slightly different coefficients, extra time-stepping loop) to alter it for methods such as Backward Euler, Crank-Nicolson, BDF2).

First you need to pick a discrete time-stepping scheme. In practice, \Phi is just the function that you apply to the solution at time step i to get the solution at time step i+1. So write out the scheme, then write u_{i+1} as a function of u_i. If you are thinking more algebraically, here https://arxiv.org/pdf/1906.06672.pdf in (1)-(3) we show that for Runge-Kutta schemes, \Phi is pretty much the stability function of the RK method in a matrix tensor form.