Implicit Euler subiterations meaning
 

Guys,

I have a silly doubt. I have an answer but I am not sure about it.

While solving a problem using Implicit Euler method, you come across a non-linear equation and you have to use a technique like Newton-Raphson method to solver the non-linear equation.

Now my doubt is regarding the number of subiterations (usually one gives a value of 10) that one should mention while choosing the Implicit Euler method. What is the meaning behind this number of subiterations? Is this about the number of times, the Newton-Raphson method is allowed to iterate for each time step. That is, after 10 iterations whatever might be the answer is given as the output (even if it is not right)?

Thanks

Vicky

Hello,


You understood it right. When you discretize with backward Euler a non-linear pde, you end up with solving a system of non-linear equations at each time step. At this point there are several ways to solve this non-linear system. Subiterations in this scenario probably means the number of iterations performed to solve this non-linear system.



You mention Newton's method, which is one way. In theory one may achieve quadratic convergence but in practice this needs a good intial guess and exact jacobian matrix. The initial guess is not always close to the final solution especially when the time-step is high. The jacobian matrix is rarely evaluated exactly because of complexity and/or memory resources. So in the end the convergence of the iterative scheme used to solve the non-linear problem is rarely quadratic and you need to specify a maximum number of iterations or a tolerance criterion to avoid too slow computation.


Dual-time step methods are another way to solve the non-linear system. This method is equivalent to Newton when the dual-time step tends to infinity.



The ideal number of subiterations, or the tolerance, is not easy to predict. It should be enough to ensure that the non-linear residual magnitude falls below the temporal truncation error. No need to converge the non-linear problem to machine accuracy in practice.

What you mentioned is a valid problem. I have seen that some use the Explicit Euler method to get a valid guess. But again if the time step is high this is not possible.

I am also looking for a good source regarding the local time stepping (LTS) method. I will tell, what I have understood about LTS method. Normally unsteady simulations use a global time step. When using LTS by providing a CFL number, each cell has its own time step size based on its size. The underlying meaning is that, a cell near the wall will have a low time step size when compared to the cell at a far field position, where its size will be larger comparatively. My doubt is that, by doing this I think there is a loss of physical meaning behind simulation time length. Each cell will have its own time step size. It will be difficult to estimate the total simulation time, because it will lose its significance. I don't know if I have a proper understanding on the LTS method. If I didn't, can you give some input on this?

Thanks

Vignesh

LTS is one technique to speed-up convergence of steady-state problems. As you pointed out you cannot use it directly to solve a time-dependent problem.


But you can see your unsteady discrete equations as successive steady state problems to be solved at each time step.


The dual-time-step method of Jameson is built upon this remark. By introducing a dual-time between each physical time step, the goal is to converge as fast as possible the dual step problem (the non-linear problem arising from the implicit discretization). Thus you can use accelerating techniques such as LTS, or multigrid on the dual time step or use a fully implicit scheme.


Look for dual time step on the web, you will find many references and in particular those from Jameson