# Inner non linear iterations and Newton method

 Register Blogs Members List Search Today's Posts Mark Forums Read

 March 11, 2010, 01:31 Inner non linear iterations and Newton method #1 New Member   Isabel Join Date: Mar 2009 Posts: 4 Rep Power: 8 Hello: Not obtaining the desired results in two different problems, I would like to clarify a concept that is the “Newton method”. In both cases I want to simulate Navier Stokes equations: the first one is a time dependent problem (which has an analytical solution) and the second one is a steady state problem which uses time as a way to get the steady state, so I will always have a temporal term in the equation. In both cases, the NS equations are: du/dt+(u*grad)u-viscosity(grad²)u+(grad)p=0; (div)u=0; To implement these equations I use different schemes: coupled method, pressure projection method, consistent pressure method... and I am using Finite Element Method (FEM). The way I advance in time is the following one: Assemble the unchanged matrices (i.e. mass matrix, diffusion matrix, gradient matrix, divergence matrix) for(unsigned int i=0; i

 March 11, 2010, 11:14 #2 Senior Member   Join Date: Jul 2009 Posts: 191 Rep Power: 8 Don't know if this will help, but I'll try - it's difficult to express some concepts on a message board. I find it very useful to think of the Newton iteration (which occurs inside the time step loop) as a search over solution space. If you discretize the governing equations at the new time level, you can write the system as R(Q(N+1)) = 0 This is prior to any application of a Newton scheme, so you have essentially a semi-discrete form of the equation. Apply a Newton iterative scheme to this system, i.e. we solve for DQ where A*DQ = -R(Q) Here A is a system Jacobian, and Q is a possible solution out of a solution space, call it Q(N+1,M). The next Newton iterate of Q, call it Q(N+1,M+1) = Q(N+1,M) + DQ. If we drive DQ to zero (which is the purpose of the Newton scheme) then at convergence we have the "correct" solution at the new time step N+1. One of the first applications I remember of this type of scheme was in conjunction with a Beam-Warming approximate factorization scheme. In the original scheme the splitting error completely destroys time accuracy in 3D for reasonable timestep sizes, but by using a Newton iterative solver the factorization error is contained within the Newton iteration and does not affect the converged value of Q(N+1). So it is not necessary, but it is a very useful approach for creating a scheme that is stable and accurate at very large timesteps. Hope that helps.

 March 11, 2010, 11:42 #3 New Member   Isabel Join Date: Mar 2009 Posts: 4 Rep Power: 8 Agd,tThank-you very much! But even being usefual approach, calculating the system Jacobian (A) for a NS equations must be really computationally expensive, mustn't it? Best Isa

 March 11, 2010, 12:19 #4 Senior Member   Join Date: Jul 2009 Posts: 191 Rep Power: 8 Depending on the way you construct the left-hand side, the system Jacobian is a combination of the inviscid flux Jacobians and/or viscous flux Jacobians. These have to be determined for an implicit scheme anyway. So the overall cost of using the Newton method versus a standard temporal marching approach is just roughly the product of the FLOPS for the temporal stepping and the number of Newton iterations. But the expense is offset by the improved convergence, the fact that you can impose boundary conditions within the Newton loop to get quasi-implicit BC implementation using simple BCs, and greatly enhanced time-accurate behavior.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules

 Similar Threads Thread Thread Starter Forum Replies Last Post kus CFX 9 April 21, 2013 01:54 Lixian CFX 3 November 27, 2008 08:57 Mohan CFX 3 November 24, 2008 04:53 Chetan Prashant CFX 4 May 17, 2007 00:58 ioana CFX 0 March 2, 2005 03:57

All times are GMT -4. The time now is 21:09.