# Doubt on Implicit Methods

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

 February 12, 2007, 23:17 Doubt on Implicit Methods #1 analyse In India Guest   Posts: n/a Dear All, I have a doubt related to Implicit Methods in CFD? What does this implicit means? As per my understanding when you want to solve the set of linear equations implicitly, you would end solving the matrix using any one the direct methods. If my understanding is correct does the solver user "direct solver" while solving the problem implicitly. If the solver is using implicit methods, then why still it is going on in ieterations ... with all these confusions I have assumed the following steps in the solver, I need to know whether my understnading is correct 1. The solver goes iterative because of to convert the non liner problem to linear problem, thus each iteration will become a linear problem. The set of algebric equations formed for this linear problem (one iteration) can be solved implicitly (direct matrix inversion) or explicitly ( iterative method eg : Gauss Jacobi) .. is my understanding is correct ....

 February 14, 2007, 11:04 Re: Doubt on Implicit Methods #2 Gustavo Guest   Posts: n/a OK...first let me say that I'm a student then what I write is my understanding of the stuff.... Using method of lines in finite volume approach makes it possible to separate time from space integration. Explicit time approaches uses properties that you already have (let's say on step n) to calculate properties on next time step (n+1). We could do (theoreticaly) this also without marching on steps, using just the properties we know in the begining (initial conditions) but that would mean solving analitically the NS equations what as far as I know no one managed to do (for any case I mean). Then we linearize them from step to step in time; the smaller the step the better the linearization... The difference to implicit methods is that these (implicit) need not only information from the step n to advance; i also needs information from the surrounding control volumes on the following step to advance the solution on the control volume you are in; but on step n we do not have that information yet...then we write the solution for that point implicitly as a function of the properties of the others on the following time; but when you do this for all points you get to a closed system, from where you get the system matrix; the algoritm you are using to invert this matrix (LUSGS, ADI, krylov subspaces or whatever) is another problem; but this is still a implicit approach; the necessity for iterations is the same as for the explicits....the bigger the advance the bigger the error linearizing.... that's my view...

 February 15, 2007, 08:39 Re: Doubt on Implicit Methods #3 Paolo Lampitella Guest   Posts: n/a Hi, i'm just a student but the following is absolutely right. For example, consider the incomp. N.S. eqns. in their conservative form dropping out the pressure term (this is another story) and take on the left just the time derivative. Considering V as the vector variable for the velocity field, they will have the form(d/dt denoting partial derivative in time): dV/dt=(1/Re)Div(Grad(V))-Div(VV)=c1-c2 where the secon equality is just for convenience. If we now integrate in time for a single time step from t to t+dt we obtain: V(t+dt)-V(t)=Int(c1,t,t+dt)-Int(c2,t,t+dt) which is still an exact relation. Now we'll introduce the approximation, that is, we have to express the two integrals numerically. Two simple examples: 1st order Explicit Euler scheme V(t+dt)=V(t)+dt*c1(t)-dt*c2(t) this means that for every point in the flowfield the velocity vector at the next time can be expressed only in terms of known quantities, so no system of equations to be solved, no iterations involved, just a simple algebraic relation. 1st order Implicit Euler scheme V(t+dt)+dt*c2(t+dt)-c1(t+dt)=V(t) As you can see, two problems arised in this new approach: 1)You have to solve a system of equations to obtain the velocity field at the next time step because in the left side of the equation ( the time t+dt) appeared the discretized form of the spatial derivatives involved in c1 and c2, both expressed in term of V(t+dt). 2)The system is nonlinear due to the presence of c2=Div(VV) So, to obtain the new velocity field you have to linearize the equations and solve the system, at each time step, iteratively. So, why to use a so much complicated scheme instead of a very simple explicit one? This is due to the stability costraints of the explicit schemes which are much more severe than those of the implicit ones (that is how much small must be the time step dt in function of the grid size dx). Obviously this was just an example, i think that the most widely used scheme is the Crank-Nicholson, that is an implicit 2° order scheme. Note that, due to its easy stability costraints, the implicit scheme is also used for steady calculations (it must be used!) because you can choose a very large time step to reach in the fastest way the steady solution.

 March 5, 2007, 17:41 Re: Doubt on Implicit Methods #4 Venkatesh V Guest   Posts: n/a Thanks to Gustavo and Paolo. My doubt is related to steady state problem. Any CFD commercial software will have to options Explicit and Implicit. I have doubt on the implicit scheme. If we want to implement 100% implicit scheme for solving set off algebric linear equations, we will endup in inverting the matrix. Which is computationaly expensive and requires more memory.? My doubt is really the copmmercial softwares do invert the matrix? If they do so? why there is huge difference between the number of "discretization points" handled by FEA based codes and FVM based codes. FEA based codes cannot handle large number of "discretization points" (whether cell center or node) handled by FVM based codes. I thought this may be due to FEA based solvers does matrix inversion at each iteration. If FVM based code also does matrix inversion for implicit method, the maxumum number of "discretization points" handled by both codes on a same hardware should be of same order even though not same. But why there is difference

 March 6, 2007, 05:27 Re: Doubt on Implicit Methods #5 Paolo Lampitella Guest   Posts: n/a Hello Venkatesh, i don't really know how the FEA works in CFD but for FVM i'm sure: no matrix inversion occur in any commercial solver and this should be avoided in any ownmade code too. The reason is a simple one, the number of operations grows up more then linearly with the number of unknowns of the system. Moreover, with an iterative solver you have some control on the error but with an exact solver and 1 million unknowns, even if there is the time to solve it (and this is NOT the case), there is no control on the error growth with the enormous number of operations required, so just don't has any sense to TRY to solve exactly a big system of equations. But, we have to distinguish the cases in which could there be the necessity to solve a system of equations in incompressible CFD. 1)Poisson eqn. for pressure: this is always the case. It can be a pressure correction in a steady solution; in this case you will solve it different times per each iterationand because you are just trying to reach the steadyness it not necessary to solve exactly the system every iteration. Or it can be an exact one as in the projection method, in this case (usually unsteady) you will solve it once per time step and ...yes, you will solve it iteratively. 2)Implicit approach: in this case you will have to solve a system to obtain the velocity unknowns per each time step. The implicit approach is nearly always used: in unsteady cases for stability constraints or in steady cases too, in which you have to drive the solution, as fast as you can, to the steady state, so stability constraints too. If the discretization is Fully Implicit, that is convective and diffusive terms are both discretized implicitly, than the system will be nonlinear and you will not have choice: some kind of linearization and solve iteratively. If the convective terms are discretized explicitly and the diffusive terms implicitly, you still have to solve a system, but it is linear. Differents approachs emerges in the fully implicit cases, depending if the equations are solved in a coupled or uncoupled manner, but these are the main features of the problem. About the FEA, if it stands for Finite Element, i don't really know. In structural mechanics, an uncoupled system of linear equations, i'm not sure but, i think that depending of the number of unknowns the system switch between exact and iterative solver. Usually, if the number of equations to be solved is less than 2000, an exact solver could be a good choice. I hope i've been useful.

 March 6, 2007, 10:02 Re: Doubt on Implicit Methods #6 Venkatesh V Guest   Posts: n/a Hi Paolo, Still I am not sure what does implicit solvers means in Commercial Solvers. Let as assume we will explicitly discretize the convective terms and implicitly dizcretize diffusive terms for Steady state X-momentum equations. But if you solve the set of algeberic liner equations iteratively, then when you solve the equation Ui(U velocity at i th cell) you will use the Uin (U velocity of i the cells neighbours) from previously. THat means you are solving the equation explicitly. This is equivalent to discretizing the equation explicitly. For make it simpler(to remove nonlineraity) we can take only steady state heat conduction equations (poisson equation). If you solve iteratively then it becomes explicit method. How can we have implicit iterative method.for steady state heat conduction equation???

 March 6, 2007, 15:37 Re: Doubt on Implicit Methods #8 Venkatesh V Guest   Posts: n/a Hi Paolo, It looked like I was getting clear finally I aam confused. Thanks for your efforts. I will pose the question from different angle .. * I am interested in "STEADY STATE" linear equation. That is equivallent of "One step" linearzied non-linear problem. As per your previous explanation, if I have chosen Implicit Discretization and And I have some how got some initial guess( it may be random guess or solution in previous iteration or solution). Now I have end up in system of linear equation. If I chosen to solve using SOR, I will use the values from previous iteration or initial guess. This is explicitness in the solution procedure. On the other hand if you solve gaussian elimination you will never use values from previous time step or initial condition. This is implicitness of the solution procedure. So whenever I think of some iterative method, some explicitness is entering into the solution procedure. So how can commercial softwares have implicit iterative solver

 March 8, 2007, 21:01 Re: Doubt on Implicit Methods #10 analyse In India Guest   Posts: n/a Hi Paolo, That is great. My doubt is clear now. Thank you very much. Thanks Venkatesh V

 March 9, 2007, 04:01 Re: Doubt on Implicit Methods #11 Paolo Lampitella Guest   Posts: n/a You're welcome. Paolo

 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 fsaltara OpenFOAM 8 December 28, 2012 05:16 Dave Rudolf Main CFD Forum 10 January 29, 2007 11:13 Runge_Kutta Main CFD Forum 3 March 4, 2005 18:36 CFD-junior Main CFD Forum 0 January 27, 2005 10:07 Yogesh Talekar Main CFD Forum 3 July 28, 1999 09:54

All times are GMT -4. The time now is 22:09.