effect of order of temporal accuracy
hi,
i know that for unsteady scheme, there are euler implicit (1st order), CN2, AB2 (2nd order) etc. however, what are the actual implications when running a simulation? for e.g., if I use a smaller time step for the euler implicit compared to the CN2, can I get a "better" accuracy as compared to CN2? also, if my simulation is undergoing a periodic variation (e.g cylinder oscillating in a flow), will the time accuracy for important if i'm interested in variables such as St no., lift/drag coefficient? Tks in advance. 
Re: effect of order of temporal accuracy
Dear Zonexo,
The physical time step has a bearing on temporal accuracy, just as the grid size has an influence on spatial accuracy. In analogy with (grid size,spatial accuracy) arguments it is easy to see that a very small physical time step will give you very timeaccurate results, though it would take more time. Too high a time step, not only reflects as a loss in temporal acuracy but also could affect the stability (if chosen to be larger than the "physical time step", which is however known for a few cases, and not in general). A compromise therefore is needed from practical engineering approach, and a time step that preserves temporal accuracy, and is yet fast would be preferred. You could also make use of aceleration convergence devices to speed up the code. Euler implicit (1st order) is first order accurate in time, whereas CN2 is second order accurate in time. Accordingly, a smaller time step for Euler implicit scheme is not much helpful. An easier way is to look at "temporal refinement" in the same perspective as "spatial grid refinement" studies. On the same grid, CN has a slope of 2, when dt reduces, while EI would have a slope of 1. This means that to achieve the same error levels as CN, the dt must be lowered substantially for EI1. In apractical simulation as you have mentioned, temporal accuracy is definietly important and has influence on quantities of interest. In general, the time accuracy should be atleast equal to the spatial acuracy for stable discretisations. It is in this context, that second order time accurate schemes are employed, as second order spatial accuracy is a mrked improvement over the first order spatial accuracy. Employing a CN2 scheme also i not sufficient to ensure second order temporal accuracy, the time step delt chosen for simulations also matter. First order implicit schems for temporal discretisation are nevertheless employed by atleast a few researchers, and though the effects of temporal accuracy do not seemingly be as profound as those of spatial accuracy, they are indeed important in any transient simulation, esp. in problems such as aeroelasticity involving fluidstructure interaction. Hope this helps Regards, Ganesh 
Re: effect of order of temporal accuracy
Dear Zonexo,
The physical time step has a bearing on temporal accuracy, just as the grid size has an influence on spatial accuracy. In analogy with (grid size,spatial accuracy) arguments it is easy to see that a very small physical time step will give you very timeaccurate results, though it would take more time. Too high a time step, not only reflects as a loss in temporal acuracy but also could affect the stability (if chosen to be larger than the "physical time step", which is however known for a few cases, and not in general). A compromise therefore is needed from practical engineering approach, and a time step that preserves temporal accuracy, and is yet fast would be preferred. You could also make use of aceleration convergence devices to speed up the code. Euler implicit (1st order) is first order accurate in time, whereas CN2 is second order accurate in time. Accordingly, a smaller time step for Euler implicit scheme is not much helpful. An easier way is to look at "temporal refinement" in the same perspective as "spatial grid refinement" studies. On the same grid, CN has a slope of 2, when dt reduces, while EI would have a slope of 1. This means that to achieve the same error levels as CN, the dt must be lowered substantially for EI1. In apractical simulation as you have mentioned, temporal accuracy is definietly important and has influence on quantities of interest. In general, the time accuracy should be atleast equal to the spatial acuracy for stable discretisations. It is in this context, that second order time accurate schemes are employed, as second order spatial accuracy is a marked improvement over the first order spatial accuracy. Employing a CN2 scheme also i not sufficient to ensure second order temporal accuracy, the time step delt chosen for simulations also matter. First order implicit schems for temporal discretisation are nevertheless employed by atleast a few researchers, and though the effects of temporal accuracy do not seemingly be as profound as those of spatial accuracy, they are indeed important in any transient simulation, esp. in problems such as aeroelasticity involving fluidstructure interaction. Hope this helps Regards, Ganesh 
Re: effect of order of temporal accuracy
thank you ganesh for your detailed explanation. however, i am still wondering if the temporal order is important in getting the mean/rms of unsteady but periodic simulation. since in those cases, we are more interested in the average instead of the value at that instant of time...

Re: effect of order of temporal accuracy
Dear Zonexo,
Temporal accuracy is of great imortance in transient simulations. In many periodic simulations, you are looking at response of the system, which could deteriorate if you are simulation is not timeaccurate. An example in this regard, is the itching airfoil case. It can be seen easily that a higher physical time step would give a less accurate hysterisis loop than lower hyscial time step, an illustration of difference between first order temporal accuracy and second order temporal accuracy. Also, in problems of aeroelasticity, where there is structuralfluid coupling, the response is critically dependent on temporal accuracy. Regards, Ganesh 
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