Non linear PDE - Three point centered finite difference
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From: Kamel on 14 Apr 2010 15:06 Hello, I would like to resolve a non linear PDE (used in porous media) which looks like: H*d2u/dz2=dU/dt, with H = (k1/Y)*dsig/dY ; Sig = k2*(Y^2-1)/(Y^2+1)*exp((Y^2-1) ; Y=1+du/dz ; with one initial condition: U(z=0, t)=0 and 2 BC : U(0,t)=0 and sig(h,t)=sig0 k1, k2, sig0 are constants. Do you know what is the best way to resolve this kind of problem using finite difference method? I know how to resolve a simple linear parabolic PDE using a finite difference scheme (2nd order spatial discretization + backward euler time integration) but in this case, the problem is more complex. I guess that an implicit three-point centered finite difference method should work but I don't know how to discretize the equations and the boundary conditions. I would be very grateful if you could help me or guide me! Thank you Kmel
From: Torsten Hennig on 14 Apr 2010 22:58 > Hello, > > I would like to resolve a non linear PDE (used in > porous media) which looks like: > > H*d2u/dz2=dU/dt, > with H = (k1/Y)*dsig/dY ; > Sig = k2*(Y^2-1)/(Y^2+1)*exp((Y^2-1) ; > Y=1+du/dz ; > Are u and U identical ? Do you know whether H = k1/Y*dsig/dY > 0 throughout ? > with one initial condition: U(z=0, t)=0 > and 2 BC : U(0,t)=0 and sig(h,t)=sig0 > > k1, k2, sig0 are constants. > > Do you know what is the best way to resolve this kind > of problem using finite difference method? > I know how to resolve a simple linear parabolic PDE > using a finite difference scheme (2nd order spatial > discretization + backward euler time integration) but > in this case, the problem is more complex. I guess > that an implicit three-point centered finite > difference method should work but I don't know how to > discretize the equations and the boundary conditions. > > > I would be very grateful if you could help me or > guide me! > > Thank you > > Kmel Best wishes Torsten.
From: Kamel on 15 Apr 2010 04:17 Hello Torsten, yes u and U are identical (it's just a mistake). Otherwie, Sig is a solid stress and in most of the cases that I want to simulate (stress relxation, cycling loading), I think that H can also be < 0. Regards, Kamel Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote in message <1558865423.15632.1271314738298.JavaMail.root(a)gallium.mathforum.org>... > > Hello, > > > > I would like to resolve a non linear PDE (used in > > porous media) which looks like: > > > > H*d2u/dz2=dU/dt, > > with H = (k1/Y)*dsig/dY ; > > Sig = k2*(Y^2-1)/(Y^2+1)*exp((Y^2-1) ; > > Y=1+du/dz ; > > > > Are u and U identical ? > Do you know whether H = k1/Y*dsig/dY > 0 throughout ? > > > with one initial condition: U(z=0, t)=0 > > and 2 BC : U(0,t)=0 and sig(h,t)=sig0 > > > > k1, k2, sig0 are constants. > > > > Do you know what is the best way to resolve this kind > > of problem using finite difference method? > > I know how to resolve a simple linear parabolic PDE > > using a finite difference scheme (2nd order spatial > > discretization + backward euler time integration) but > > in this case, the problem is more complex. I guess > > that an implicit three-point centered finite > > difference method should work but I don't know how to > > discretize the equations and the boundary conditions. > > > > > > I would be very grateful if you could help me or > > guide me! > > > > Thank you > > > > Kmel > > Best wishes > Torsten.
From: Torsten Hennig on 15 Apr 2010 01:19 > Hello Torsten, > > yes u and U are identical (it's just a mistake). > Otherwie, Sig is a solid stress and in most of the > cases that I want to simulate (stress relxation, > cycling loading), I think that H can also be < 0. > > Regards, > > Kamel > > I think you should solve in u and Y. Write your equations as eps*dY/dt - du/dz = 1-Y du/dt -1/H*dY/dz = 0 for small eps>0 and use CLAWPACK available under http://www.amath.washington.edu/~claw/ to solve. The system as above should be used for the case that H>0 throughout ; in the case H<0 throughout use eps*dY/dt + du/dz = Y-1 du/dt - 1/H*dY/dz = 0 for small eps>0. The system will be hard to solve by just using standard discretization methods - so you should use professional software (as CLAWPACK cited above). Best wishes Torsten. > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > in message > <1558865423.15632.1271314738298.JavaMail.root(a)gallium. > mathforum.org>... > > > Hello, > > > > > > I would like to resolve a non linear PDE (used in > > > porous media) which looks like: > > > > > > H*d2u/dz2=dU/dt, > > > with H = (k1/Y)*dsig/dY ; > > > Sig = k2*(Y^2-1)/(Y^2+1)*exp((Y^2-1) ; > > > Y=1+du/dz ; > > > > > > > Are u and U identical ? > > Do you know whether H = k1/Y*dsig/dY > 0 throughout > ? > > > > > with one initial condition: U(z=0, t)=0 > > > and 2 BC : U(0,t)=0 and sig(h,t)=sig0 > > > > > > k1, k2, sig0 are constants. > > > > > > Do you know what is the best way to resolve this > kind > > > of problem using finite difference method? > > > I know how to resolve a simple linear parabolic > PDE > > > using a finite difference scheme (2nd order > spatial > > > discretization + backward euler time integration) > but > > > in this case, the problem is more complex. I > guess > > > that an implicit three-point centered finite > > > difference method should work but I don't know > how to > > > discretize the equations and the boundary > conditions. > > > > > > > > > I would be very grateful if you could help me or > > > guide me! > > > > > > Thank you > > > > > > Kmel > > > > Best wishes > > Torsten.
From: Torsten Hennig on 15 Apr 2010 22:33
> > > <1558865423.15632.1271314738298.JavaMail.root(a)gallium. > > > mathforum.org>... > > > > Hello, > > > > > > > > I would like to resolve a non linear PDE (used > in > > > > porous media) which looks like: > > > > > > > > H*d2u/dz2=dU/dt, > > > > with H = (k1/Y)*dsig/dY ; > > > > Sig = k2*(Y^2-1)/(Y^2+1)*exp((Y^2-1) ; > > > > Y=1+du/dz ; > > > > > > > > > > Are u and U identical ? > > > Do you know whether H = k1/Y*dsig/dY > 0 > throughout > > ? > > > > > > > with one initial condition: U(z=0, t)=0 > > > > and 2 BC : U(0,t)=0 and sig(h,t)=sig0 > > > > > > > > k1, k2, sig0 are constants. > > > > > > > > Do you know what is the best way to resolve > this > > kind > > > > of problem using finite difference method? > > > > I know how to resolve a simple linear > parabolic > > PDE > > > > using a finite difference scheme (2nd order > > spatial > > > > discretization + backward euler time > integration) > > but > > > > in this case, the problem is more complex. I > > guess > > > > that an implicit three-point centered finite > > > > difference method should work but I don't know > > how to > > > > discretize the equations and the boundary > > conditions. > > > > > > > > > > > > I would be very grateful if you could help me > or > > > > guide me! > > > > > > > > Thank you > > > > > > > > Kmel > > > I thought about your problem again and came to the following more direct approach: You can use MATLAB's pdepe to solve the following system: du/dt = H*d^2u/dz^2 0 = du/dz + (1-Y) with boundary conditions u(z=0,t) = 0 du/dz(z=0,t) = Y - 1 du/dz(z=h,t) = Y - 1 Y(z=h,t) = Y^(-1)(sig0) Best wishes Torsten. |