
[Sponsors] 
August 9, 2007, 05:19 
Info on method of lines approach

#1 
Guest
Posts: n/a

Hi
Im not sure if this is the right place to post this so excuse me! I have a system of 3 non linear first order pde's, differentiated with respect to time and one spatial dimension. I am thinking of using a method of lines approach for removing the spatial differentiation, and then using a runge  kutta technique [or something similar], to solve the left over coupled ode's. But i don't understand how the difference formulas, that you have from the finite difference method, can be solved numerically using a RK method. Take an example > u_t = u_x. Using a central difference formula, u_x looks like; u_x = (u_i+1,j  u_i1,j)/2h => u_t = (u_i+1,j  u_i1,j)/2h Now how do i solve the above using a method from numerically solving ode's [RK method, Euler, whatever]? I'm confused about what to do when the i+1, i1 is there  what do i do with these? Can anyone suggest a good reference on the method of lines? Any comments are appreciated! Charlie 

August 9, 2007, 09:35 
Re: Info on method of lines approach

#2 
Guest
Posts: n/a

Dear Charlie,
Let me repose your example, using Dirichlet (value specified) conditions u_t = u_x u(t, x_a) = u_a u(t, x_b) = u_b u(t_0, x) = u_0 The method of lines can be applied to any of the dimensions of the problems, either t or x.. Let us go with "x" again. You discretize your interval (a,b) into a finite number of nodes separated by h, and then apply the difference formulas at each node, then u_i_t = (u_i+1,j  u_i1,j)/2h for i = 1, 2, .. N for i = 1 u = u_a i = N u = u_b You got N2 ODE's that are coupled. You can use a RK integrator. If you ODE's are linear you can find their analytical solution in closed form. Hope this helps, Opaque 

August 9, 2007, 11:06 
Re: Info on method of lines approach

#3 
Guest
Posts: n/a

Thanks Opaque very helpful  it makes sense. I have a system of PDE's that look something like;
h_t = hv_x  h_x T_t = (T/h  Tv/h  E)h_x  Tv_x  vT_x  hE_x v_t = vv_x  p_x +TE/h + v_xx + (1/h)v_x*h_x p_x = h_x + hxx + E^2 + T^2 E = f(x,h,E,v,t) I hope that is readable! So it is basically a system of PDE's (note only three involve time derivatives]. E is a function that depends on the rest of the variables, and p does not directly depend upon t. i have a set of inital conditions for the above. Now i understand how you solve one pde to get a system of ode's, but what about a system of coupled pde's looking something like above? How do you solve the pde's system, when you have multiple variables? Any help is very much appreciated, Charlie 

Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Turbulence inflow generation  recycling method  panda60  OpenFOAM Running, Solving & CFD  15  April 25, 2013 01:34 
discretizer  gmshToFoam  Andyjoe  Open Source Meshers: Gmsh, Netgen, CGNS, ...  13  March 14, 2012 05:35 
Fluent 6.3.26 vs 12.1 and partition method  Anorky  FLUENT  0  April 27, 2010 10:55 
Comparison: Finite Volume Method vs. Analytic Method  mfry  Main CFD Forum  1  April 20, 2010 14:40 
What is C.V. based finite element method  CH Kuo  Main CFD Forum  3  November 5, 1998 10:07 