| Author | Message | |||
     ecmo New member Username: ecmo Post Number: 1 Registered: 07-2010 |
I was trying to solve the following BVPs, but there was an error that I could not resolve in FlexPDE. Could you please give me a direction on how to get it solved? f''' - R*((f')^2 - ff'') + R = 0 h'' + R*f*h'+ 1 = 0 theta'' + P*f*theta' = 0 f(0)=f'(0)=0, f(1)=f'(1)=0 h(0)=h(1)=0 theta(0), theta(1)=1 I used the following relations to reduce the high order of f. f'=g f''=g' f'''=g''
| |||
| rgnelson Moderator Username: rgnelson Post Number: 1388 Registered: 06-2003 |
1. Your system is over-specified: A third-order equation has three integration constants, and therefore three boundary conditions, not four. Of your four conditions on f, one must be discarded. 2. You have applied your substitutions incorrectly. The square term is g*g, not dx(g)*dx(g). 3. The system is strongly nonlinear. In order for Newton's method to find a solution, you must have a very close initial guess. In particular, with an initial guess of g=0, the g equation wants to assign dxx(g)=-1000, which calls for very sharp peaks in g and leads to oscillation. You do not have an inital estimate anywhere near close enough to solve this system by Newtons method. 4. A useful approach with this kind of problem is to re-cast it as an evolution system in pseudo-time. This allows the timestep control to adapt to stop oscillations. We have implemented this method in the attached script, which runs successfully. However, nonlinear systems can have multiple solutions, and the final result may depend on the starting values. 5. Another way to address this system might be to introduce a strong diffusion (smoothing) on f, and STAGE the diffusion coefficient down to zero. We have not tried this.
| |||
| ecmo New member Username: ecmo Post Number: 2 Registered: 07-2010 |
I tried to solve a more complecated problem using flexpde, but my effort did not turn out to be fruitful. I tried two ways using artifical time terms and divergence terms. Could you please give me your help. Thanks in advance.
| |||
| rgnelson Moderator Username: rgnelson Post Number: 1390 Registered: 06-2003 |
1) your equation for nu reduces to 0=0, which renders the system unsolvable. 2) You have provided initial values for U and V which are discontinuous at the ends (e.g. -0.6 vs 0 BC for V at the left). With a strongly nonlinear equation this is a recipe for disaster. 3) Terms like dxx(nu)*dx(nu) are ambiguous as to sign, and allow multiple solutions and thrashing of the solver. You should start with continuous initial conditions and a simpler equation for nu and proceed from there. |
