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I want to understand how does the simulation identifies and models when two bodies are colliding with each other. For example car crash simulation. The car moves towards the wall or another crash. Until they collide some equations of moving works, but how do they change and what are the methods of dealing with the situation when the collision occures? Do the algorithms for collision detection for games work the same way (physical engines etc)?

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Could you possibly be more specific as to the application? The only place I can think of collisions between objects as complicated as cars occurring is, as you stated, video games. How it is handled in video games, needing only to fool us sufficiently with realism, would be much different than object collisions simulated for scientific purposes. – Godric Seer Oct 15 '12 at 11:27
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Previously on Physics.SE: physics.stackexchange.com/questions/39541/… – dmckee Oct 15 '12 at 13:26
@dmckee: Thanks for the heads up! – Geoff Oxberry♦ Oct 15 '12 at 18:23

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Collision is typically modeled through equations like the obstacle problem, i.e., you have a partial differential equation with an inequality constraint. This leads to variational inequalities. An important step is indeed to find out whether two objects are in contact, and there you do use similar techniques as for games.

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What are the steps after collision is detected? Change in inequality constraints to fit a new obstacle? – maximus Oct 15 '12 at 17:25
The boundary conditions change. So if you hit a solid (infinitely stiff) wall, then the leading edge of the object is constrained such that it cannot move into the wall. – tpg2114 Oct 15 '12 at 18:20
What if car hits another car? Do you know some books to read about simulation of collisions? – maximus Oct 16 '12 at 2:11
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Car hits car is no different than car hits wall -- you need to find whether the two objects have any contact or intersection. As for books -- I haven't read it but I hear that Peter Wriggers' book about contact mechanics is pretty good. – Wolfgang Bangerth Oct 16 '12 at 12:38
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@maximus: Books on collision detection by Gino van der Bergen (ISBN 9781558608016) and Christer Ericson (ISBN 9781558607323) may be interesting to you... – Deer Hunter Oct 27 '12 at 18:18
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