Many years ago what I am about to say would have been science fiction, today we have sufficiently advanced computers to do this.
To help illustrate what I am trying to say, for example, we have a fluid in a container which is moving. Let u(x,y,t) represent the x-th and y-th coordinate of the location of the surface of the fluid at time t. Say the fluid starts at some initial state u(x,y,0)=f(x,y) and given motion by an inital speed u_t(x,y,0)=g(x,y). And say these boundary values satisfy some differencial equation which represents the position of the surface (I have no idea, maybe, Navier-Stokes). But anyway, we manage to solve this equation somehow. And have this unique function u(x,y,t). Is there such a thing when we draw surfaces by keeping "t" fixed and varying continously say from t>=0 till the end of the process. What I am saying at t=0 we have u(x,y,0) a surface, at t=.01 we have u(x,y,.01) a different surface, at t=.02 we have u(x,y,.02) and so one. And we create an animation, that is we create a program which shows all these surfaces as the time is varying continously. Which should produce an animation of exactly the fluid behaves.
Is there such a program?