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3 coupled non linear partial differential equations
Posted Sep 10, 2009, 3:28 p.m. EDT 2 Replies
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I have 3 variables, w(y,t), v(y,t), k(y,t). Using the standard pde notation, I want to solve
w_t=-(vg*w)_y+(gamma-D*(kx0^2+k^2))*w
v_t=(vg*w)_y-nu*v
k_t=-kx0*v_y
Subject to
w(0,y)=w0*exp(-.1*y)
v(0,y)=vx0*cos(3y)
k(0,y)=ky0*exp(-.05*y)
And
w(t,0)=w0; v(t,0)=vx0; k(t,0)=k0;
w(t,100)=0=v(t,100)=k(t,100);
Here vg=2*B*kx0*k/((kx0^2+k^2)^2)
Where I use the following values
w0=15; ky0=5=kx0; vx0=2;
B=80; kx0=5; ky0=5;gamma=7*exp(-0.1*y); D=0.1*exp(.01y);
COMSOL just gives a bunch of numerical oscillation for the w equation (which isn't physically reasonable) and the other results are full of oscillation as well. Is there a way to optimize COMSOL to give a smooth solution? I have a solution I believe is somewhat correct that I used PDEPE in Matlab to obtain.
I have tried: Matlab's PDEPE, discretizing the RHS and using the method of lines, FlexPDE, and NDSolve in Mathematica.
I have tried so many different solvers and none have been able to solve this system. It is getting quite frustrating...
w_t=-(vg*w)_y+(gamma-D*(kx0^2+k^2))*w
v_t=(vg*w)_y-nu*v
k_t=-kx0*v_y
Subject to
w(0,y)=w0*exp(-.1*y)
v(0,y)=vx0*cos(3y)
k(0,y)=ky0*exp(-.05*y)
And
w(t,0)=w0; v(t,0)=vx0; k(t,0)=k0;
w(t,100)=0=v(t,100)=k(t,100);
Here vg=2*B*kx0*k/((kx0^2+k^2)^2)
Where I use the following values
w0=15; ky0=5=kx0; vx0=2;
B=80; kx0=5; ky0=5;gamma=7*exp(-0.1*y); D=0.1*exp(.01y);
COMSOL just gives a bunch of numerical oscillation for the w equation (which isn't physically reasonable) and the other results are full of oscillation as well. Is there a way to optimize COMSOL to give a smooth solution? I have a solution I believe is somewhat correct that I used PDEPE in Matlab to obtain.
I have tried: Matlab's PDEPE, discretizing the RHS and using the method of lines, FlexPDE, and NDSolve in Mathematica.
I have tried so many different solvers and none have been able to solve this system. It is getting quite frustrating...
2 Replies Last Post Sep 19, 2009, 6:55 p.m. EDT
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Posted:
2 decades ago
Hi
to be sure I understand you, and to allow mw to play a little with your equations, is
k(t,0)=k0; then what is "k0" or was it a typo and I should read k(t,0)=ky0; ?
and what is the value of "nu" in v_t=(vg*w)_y-nu*v
And have you tried Maple as solver ?
Ivar
to be sure I understand you, and to allow mw to play a little with your equations, is
k(t,0)=k0; then what is "k0" or was it a typo and I should read k(t,0)=ky0; ?
and what is the value of "nu" in v_t=(vg*w)_y-nu*v
And have you tried Maple as solver ?
Ivar
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Posted:
2 decades ago
It should read ky0. The variable is actually k_y but I did not want to use the subscript because it can be confused with PDE notation so I switched to k as the variable name. For the parameters I was using, nu=0.1. It is the drag on the flow.
I have not tried Maple but Mathematica gives nonsensical results. Matlab is a bit better but not sure it is correct either. I have attached a PDF detailing the mathematical description of the problem in easy to ready latex.
I have not tried Maple but Mathematica gives nonsensical results. Matlab is a bit better but not sure it is correct either. I have attached a PDF detailing the mathematical description of the problem in easy to ready latex.
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