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February 15, 2008, 13:53 
Questions for Trapezoidal method

#1 
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I need to solve following ODE
dU/dt=S(U) in which S(U) is geometric source vector in symmetric Euler equations. Due to instabilities resulting from stiffness, I am considering the use of an implicit time differential scheme. Among them, I first think about the Trapezoidal rule as U^n+1=U^n+(S(U^n)+S(U^n+1))/dt. The problem is that I do not know how to construct S(U^n+1). Could you anyone give me idea or direction to find books? Thanks. 

February 15, 2008, 16:30 
Re: Questions for Trapezoidal method

#2 
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You can expand S(U^n+1) using taylor series. Look up in any CFD book for implicit time marching.


February 15, 2008, 16:47 
Re: Questions for Trapezoidal method

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Do you mean that S(U^n+1)=S(U^n)+dS/dU*(U^n+1U^n)? The problem in my implementations is that S consists of geometric term and any source term to impose sponge layer effects at boundaries. Thus, it was not easy to expand dS/dU for complex relations. Anyway thanks for your reply.


February 15, 2008, 17:42 
Re: Questions for Trapezoidal method

#4 
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to solve u' = F(u,t), where functionality of F is not explicit in relation with u and t
there areseveral methods that only need time step size and a function that evaluate F at specific "u" and "t", e.g. Rungee Kuta methods (http://en.wikipedia.org/wiki/RungeKutta_methods) if you can not evaluate F at mid way of time step (RK needs), you could take its as picewise constant or use onesided interpolation from avaialble data. 

February 15, 2008, 19:53 
Re: Questions for Trapezoidal method

#5 
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Did you mention it for implicit schemes? I needed to implement an implicit scheme to continue the computation beyond a certain instant. I am currently suffering from stability problems when the compressible Euler code with a geometric source applied to simulate a wide range of fluid flows.
Thanks in advance! 

February 15, 2008, 21:36 
Re: Questions for Trapezoidal method to rt

#6 
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As you mentioned, I obtained interpolated values in implicit method as
U^n+1=U^n+1/2*dt*(S(u^n)+S(u*^n+1)) where u*^n+1 is interpolated from previous solutions. The procedure to interpolate is based on U*^n+1=U^n+(U^nU^n1). This is my idea to obtain values. Did you intend to let me know this way? Or is there any other way? I really appreciate your great advice 

February 15, 2008, 23:46 
Re: Questions for Trapezoidal method to rt

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suggested methods are explicit, but with good stability bound, note that the RK method are like predictorcorrector methods and they try to reconstruct function with higher order and lower needed data, i should say that your extrapolation is also somhow explicit and it is like AdamsBashforth method which is bit unstable than 4thRK, so try RK methods first, they may resolve. They are extremly simple to implement.
i suggest extrapolation to evaluate f(u,t) at midway. i.e.. at (u+du, t+dt), where u+du<n^{n+1} t+dt<t^{n+1} what is structure of "F", i do not understand what you mean from "geometric"? do you have IMSL library, if, it has several prepared routine inter/extrapolation and ODE solvers. 

February 16, 2008, 02:09 
Re: Questions for Trapezoidal method

#8 
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If you know S(u) and if you know U initial value why you dont'use RK?


February 16, 2008, 02:19 
Re: Questions for Trapezoidal method

#9 
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I used RK scheme but it caused instabilities when the simulation went long. At early time(within submicrosecond), it worked well. But at relatively late time(around submillisecond), the computation failed to continue after a certain instant. I guessed that this instability results from stiffness of ODE which I was solving. The ODE is a part of compressible Euler equation with a geometric source implementing symmetry spherically or cylindrically. Unfortunately, it leaded to the sudden abortion of computation when I used RK scheme to numerically solve that ODE. It is
dU/dt=S(u) where U=[rho, rho*u, rho*E]^T and S(u)=1/r[rho*u,rho*u^2m rho*(E+p)]^T. This is a simplified version of compressible Euler equation implementing symmetric flow motion. I did not know about this difficulty before. However, when I attempted to see latetime flow motion, I had to run the compressible code somewhat long and I met with sudden abortion of computation due to several reasons. I analyzed that the stiffness of ODE is a main source of this instability. To overcome this issue, I tried to implement an implicit time integration which is free from CFL restriction. If you have any other advices, could you give me that? I am looking forward to having change to talk all together. Thanks. 

February 16, 2008, 09:16 
Re: Questions for Trapezoidal method

#10 
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if you need a really implicit solver, you could perform an outer iteration within each time step and go uot when your predicted u^n+1 converge.
there are several implicit ode solver but cost is high, e.g. see Adamsâ€"Moulton methods http://en.wikipedia.org/wiki/Linear_multistep_method if you look at a classic numerical analysis book found formal information needed to solve ode e.g. this one is good: Numerical Analysis, R. Burden 

February 18, 2008, 07:52 
Re: Questions for Trapezoidal method

#11 
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