Colliding particles either for the purpose of getting fresh yet unexplained experimental results or as a mean of testing proposed theories appears to be a very crude experimental method. Could area of physics that deals with elementary particles currently have mathematical model capable to simulate processes and outcomes which are happening in colliders, so the need for actual physical colliders could be eliminated?
The question we are answering in collider experiments isn't "what happens if you do all of this math." The question is "how does the world we live in behave?"
You can make a computer model that does anything you want — that's the Pixar business model.
The way that we understand our world better is to find places where different models make inconsistent predictions. Then we build an experiment and see which (if any) of those predictions is most like our actual world.
Theory and experiment have an intertwined history. Colliders have been traditionally used to study poorly-understood phenomena at unexplored energy levels, to guide the evolution of mathematical models that describe and account for those phenomena. Sometimes theory gets out ahead of experiment and predicts the existence of particles generated in collisions, where the experiments have not yet been performed- and then the experimentalists go off and set up an experiment specifically designed to produce those particles.
Sometimes, the experiments detect things that aren't predicted by the mathematical model, and the experiments are repeated- and sometimes shown to be in error. Sometimes the experiments fail to detect things that are predicted by the model and the model gets revisited.
This is a dynamic process- the back and forth pull of theory and experiment. When everyone gets it right, a Nobel prize is the result.
Colliding particles either for the purpose of getting fresh yet unexplained experimental results or as a mean of testing proposed theories appears to be a very crude experimental method.
The history of accelerator physics which collides particles is long, can be seen here , and there is nothing crude about the extensive intellectual effort that was involved and is involved in the constructions.
Of course it depends on the definition of crude, this is what you must have in mind
- : rough or inexpert in plan or execution/ a crude shelter
The sequential mathematical models used to model the data coming from accelerators and , very important for validation of a physics theory , to predict new data are also anything but crude.
Could area of physics that deals with elementary particles currently have mathematical model capable to simulate processes and outcomes which are happening in colliders, so the need for actual physical colliders could be eliminated?
That is the holy grail of particle physics research, both theoretical and experimental, the theory of everything.
Then one could say "only engineering remains", BUT
In 1900 Kelvin (26 June 1824 – 17 December 1907) addressed the British Association for the Advancement of Science with these words: “There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.”
Look at the quantum revolution that came in the twentieth century, from more precise measurements, including accelerator experiments.