# Questions for Trapezoidal method

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 February 15, 2008, 13:53 Questions for Trapezoidal method #1 jinwon park Guest   Posts: n/a 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 Harish Guest   Posts: n/a 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 #3 jinwon park Guest   Posts: n/a Do you mean that S(U^n+1)=S(U^n)+dS/dU*(U^n+1-U^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 rt Guest   Posts: n/a 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/Runge-Kutta_methods) if you can not evaluate F at mid way of time step (RK needs), you could take its as picewise constant or use one-sided interpolation from avaialble data.

 February 15, 2008, 19:53 Re: Questions for Trapezoidal method #5 jinwon park Guest   Posts: n/a 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 jinwon park Guest   Posts: n/a 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^n-U^n-1). 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 #7 rt Guest   Posts: n/a suggested methods are explicit, but with good stability bound, note that the R-K method are like predictor-corrector 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 Adams-Bashforth method which is bit unstable than 4th-R-K, so try R-K 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

 February 16, 2008, 02:09 Re: Questions for Trapezoidal method #8 Giovanni Lanzillo Guest   Posts: n/a If you know S(u) and if you know U initial value why you dont'use R-K?

 February 16, 2008, 02:19 Re: Questions for Trapezoidal method #9 jinwon park Guest   Posts: n/a I used RK scheme but it caused instabilities when the simulation went long. At early time(within sub-microsecond), it worked well. But at relatively late time(around sub-millisecond), 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 late-time 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 rt Guest   Posts: n/a 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 rt Guest   Posts: n/a you may find this paper interesting: http://www.global-sci.com/freedownload/v3_479.pdf

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