Suppose we wanted to simulate the measurement of a qubit using a classical computer. We could, in theory, simulate the time evolution of the many body (many qubit) Schrodinger equation, perhaps for a small system even in practice.
Yes, we do this all the time as a matter of routine.
If you look at most superconducting qubit papers, you'll see that the experimental data are compared against computer simulations.
Simulating a few interacting qubits can be done on a conventional CPU in a laptop.
Using the world's largest supercomputing resources, we can simulate up to around 49 interacting qubits.
Beyond that, present day classical computing resources cannot do simulations in a reasonable amount of time.
Note the comment about time: simulating a large quantum system isn't impossible in principle, it's just inefficient and becomes intolerably slow (and requires astronomical amounts of memory) as the quantum system's size increases.
For a rough estimate, note that an $N$-qubit system's state requires $2^N$ real numbers.
If we use 1-byte floats, then a 64GB system can hold the state of only
$$ N = \log_2(64 \times 10^9) = 32$$
qubits, and we haven't even talked about the operators that act on those qubits.
What would we see if we would simulate the measurement? In particular, would we see the mysterious wave function collapse?
We don't need to speculate, this has already been done!
Wave function collapse is already pretty well understood theoretically, and there have been experiments showing exactly what happens during the measurement-induced collapse, along with computer simulations.
Here are a few examples:
There was even an experiment showing collapse during simultaneous measurement of non-commuting observables!
It would be nice if an answer could point out what would be the simplest (physical) system that we would need to simulate in order to see (in detail) what happens when we measure a qubit (when a the function collapses).
All of the experiments and simulations linked above used a single qubit.
That's all you need in order to study wave function collapse.
Now, having said all of this, I think we still haven't really answered your question.
I think you might be trying to ask how we get wave function collapse if we're just simulating the system by numerically solving the Schrodinger equation.
That's a good question, and the answer is that we don't.
The Schrodinger equation is not a complete description of quantum mechanics.
In order to understand measurement and wave function collapse, we need a more complete and more modern expansion of the theory.
This theory is very well studied and understood.
It's not new or secret or magic, it just hasn't really made it into standard textbooks, university courses, or public knowledge yet.
In order to understand measurement and wave function collapse, we have to extend quantum mechanics to the case where we only have access to a sub-part of the total system.
You see, when we measure, we're interacting something, photons, electrons, or whatever else our measurement apparatus uses, with the system under study.
An oscilloscope, for example, contains a lot of atoms, which all become entangled with the observed system.
We don't have access to the states of all those atoms, so in the end we have to find a way to describe a sub-part of the whole thing.
This is done with the density matrix, which is nowadays a crucial part of quantum theory.