Is there a way to disprove the universe is a simulation? Is there a way to disprove the theory which says the universe is a simulation?
 A: You cannot disprove that the universe is a simulation by just doing (simulated) observations of course, as long as the simulation is self-consistent.  However, the real point is that it doesn't matter !  If the universe is a perfectly self-consistent simulation, well, then if you find out its laws, that is reality.  Reality is the intellectual construct (the mental hypothesis) that allows us to understand the relationships between observed phenomena.  
This is somewhat similar to two different physical theories (different in their mathematical formulation), that are provably observationally identical.  Well, by definition, you cannot distinguish them by observation (by doing experimental science).  They're both "just as good".  They may represent totally different mental "reality hypotheses", but that doesn't matter.  They work both just as well.  We cannot distinguish the "reality hypotheses" of these two theories.  In principle, we would use Occam's Razor to prefer the conceptually simplest one.
In that sense, if we would have "a theory of the universe" that describes a "reality", and another, observationally equivalent theory of the universe that tells us that the first one is a running simulation, then the second one is holding more conceptual elements, and would be rejected by Occam's Razor.   This brings me back to my initial point: if reality is a simulation, we do away with the "simulation", and define reality as whatever are the rules of the simulation, without postulating it is a simulation.
A: There is no way to disprove it using physics.  You can make some philosophical arguments (which you could search for on the Philosophy Stack Exchange).  However, fundamentally all science does is develop predictive models of how the universe works.  It does not make any statements about how the universe actually works, which is the philosophical branch called Ontology.
There really is no way to tell the difference between a simulation artifact and a law of physics, using just empirical approaches like science.
A: How to see if you can do it:


*

*Use logic and pure thought to figure out all possible universes.

*Identify all the possible universes that contain an entity which has your experiences.
3a. See if there are experiments that can distinguish between the universes where your experience is a simulation, and where it's real. 
3b. If that's not possible, you may still be able to assign a probability to not being in a simulation, e.g. by assuming that all possible worlds are real, or that they have an equal probability of being real.
A: Depends on exactly how you define 'simulation'. If you are asking is there any way to disprove that some "Programmer" has set up the laws of nature to result in exactly what we see, then the answer is "no" because such a hypothesis is indistinguishable from religion.
There are however various specific forms of simulation that you can rule out. A particularly common idea is for example that the universe operates just like a computer, meaning it executes discrete steps of computation. This assumption is however incompatible with special relativity in which time has to be continuous. If it's discrete, this leads to what is known as Lorentz-invariance-violation. This can, and has been searched for. It has not been seen. Hence you can rule out this type of simulation (up to some precision). That's the idea behind papers like this: https://arxiv.org/abs/1210.1847
Frankly, I think that's just a way to make headlines with Lorentz-invariance-violation, but maybe that's just me. 
There are other things you can look at with physics. For example people are still arguing whether a simulation could conceivably be run by a classical computer (as opposed to a quantum computer.) The problem being you'd have to find a way to create quantum effects from a classical system, which is generally thought to be impossible. That is, unless you allow some kind of non-locality, in which case you can ask, well, then what is space. And so on.
There is also the problem that you'd have to hide whatever the "universe computer" is made of itself. Ie, if you have clever simulated physicists they might come up with a way to probe the qbits of the underlying simulation and this might reveal what's going on. 
The bottomline is that the idea that we live in a simulation in and by itself is unscientific because it's too vague to be falsifiable. Once you try to make it concrete, it turns out to be very, very hard to come up with an actual model that is compatible with the laws of nature we have observed. 
A: I agree with Cort. However, if some pseudo-solipsist asserts to me that the universe is a simulation rather than being real, I would challenge them to calculate the information rate in bits/second required to  model the operation of the simulated universe in such a manner that it would be indistinguishable from the real thing. Then I would ask them to estimate the size of the hardware required to furnish that bit rate and specify the technology used to convey the simulation to every conscious being within the model, without their being aware of it. Then I would require them to estimate the odds of that actually happening...
And this does not even get into the question of how those conscious beings themselves are modeled, and how they can be tricked into believing they have free will within the context of the Grand Simulation. 
the idea that we are living in someone else's simulation is, in my opinion, not even wrong.  
A: Yes there is one very compelling and convincing argument to why the world is not a simulation. And that is irrational numbers. 
If you know a bit about computer science and how data is represented, you know you cant actually have computers store irrational values. You can only story very accurate approximations of the actual value, but never the exact irrational. 
But irrational numbers are real, we do have them in our number system. 
Now if the universe was a simulation, irrational numbers wouldn't exist cause they cannot in a simulation. 
