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November 14, 2002, 13:58 
Time Step

#1 
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Hello
Could some one help me with the basics of ..what exactly is meant by saying "Time Steps" and the "Number of Time steps"...what do these essentially mean in calculations ? how are the results based on these values ? If someone has literature on it ..please help me with it Thanks Bharath 

November 14, 2002, 21:21 
Re: Time Step

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For a timemarching PDE, the time is discretized as t0, t1, t2, ..., t_n, t_n+1, ..., where solution is wanted at every moment. TIME STEP = t_n+1  t_n, Number of time steps = number of time intervals the solution has marched.


November 16, 2002, 08:21 
Re: Time Step

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Time stepping is required whenever you are solving an evolutiontype of pde, that is, the pde models a system which is evolving in time. This means that the pde will contain a partial derivative wrt time, usually, of the first or second order. You also need to have some appropriate initial conditions. Then you obtain a discretization of the pde; this basically allows you to "march" the solution from a known value at time t, to another value at time t + dt. Here dt is called the time step, and it is usually determined from stability considerations. The number of times you march the solution over a timestep dt is the "number of time steps". This depends on your particular problem. For example, if you require the solution at time T=10s, and your timestep is dt=0.1s, then you will need to perform 100 integrations in time. Many times the problem has a steady state solution; then you keep on marching in time till the solution change from one time level to the next is below some TOL.
The solution you get is also affected by the value of dt. Take a very large value (but within stability limits) and you might lose accuracy. The order of accuracy of the time integration scheme will also affect the final accuracy. For steady state problems the value of dt is not important (again provided it is within stability limits). These are very basic things in CFD and in numerical solution of pde. You must read some book to get a good understanding. Please go through the book list on this web site. 

November 18, 2002, 17:24 
Re: Time Step

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So does it mean that you get transient solution from initial condition by marching in time step?


November 18, 2002, 18:26 
Re: Time Step

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"So does it mean that you get transient solution from initial condition by marching in time step?"
If your initial condition is a valid solution to the NS equations and your boundary conditions are physically valid, then the solution generated is a transient solution for those conditions. More often, the transient is just a convenient way to get from an 'arbitrary' initial guess at a steady state solution to the problem that your boundary conditions define. If fact, some of the older books (Roache "Computational Fluid Dynamics" for example) draw an analogy between a relaxation factor and the time step. 

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