My PDE is not obeying conservation principle
in [Matlab]
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From: Torsten Hennig on 8 Jul 2010 03:37 > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > in message > <128269470.77069.1278420640088.JavaMail.root(a)gallium.m > athforum.org>... > > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> > wrote > > > in message > > > > <1394417502.76713.1278415717104.JavaMail.root(a)gallium. > > > mathforum.org>... > > > > > Torsten Hennig > <Torsten.Hennig(a)umsicht.fhg.de> > > > wrote > > > > > in message > > > > > > > > > <2044145382.75601.1278398289051.JavaMail.root(a)gallium. > > > > > mathforum.org>... > > > > > > >Hi, > > > > > > >My equation is Boltzman equation. Without > > > > > collision term >it is > > > > > > >dN/dt+v*dN/dr+F*dN/dp=0 > > > > > > >where v is velocity, F is force and p is > > > momentum > > > > > > >by rearranging few terms we are getting > this > > > > > > >dN/dt + k*c1*dN/dr-c2*r*dN/dk=0 > > > > > > >here unit of k is 1/meter and r is also > > > length. > > > > > > >so, then my procedure of calculating total > > > number > > > > > of >particles is right or I have to do in > other > > > way? > > > > > > > > > > > > > >With Regards, > > > > > > >Sunipa Som > > > > > > > > > > > > Start from the equation > > > > > > dN/dt + k*c1*dN/dr-c2*r*dN/dk=0 > > > > > > or > > > > > > dN/dt + div(k*c1*N,-c2*r*N) = 0 > > > > > > to be solved over the rectangle > > > > > > V=[r_min;r_max]x[k_min,k_max]. > > > > > > > > > > > > Integrate over the rectangle V with > boundary A > > > to > > > > > get > > > > > > d/dt int_{V} N dV = int_{V} > div(-k*c1*N,c2*r*N) > > > dV. > > > > > > Applying Gauss' integral theorem to the > right > > > hand > > > > > side > > > > > > results in > > > > > > d/dt int_{V} N dV = int_{A} > (-k*c1*N,c2*r*N)*n > > > dA > > > > > > where n is the unit normal pointing > outwards of > > > the > > > > > > > > > > > rectangle. > > > > > > So if no particles enter the rectangle over > the > > > > > boundary, > > > > > > the quantity > > > > > > int_{V} N dV = > > > > > > int_{r_min}^{r_max} int_{k_min}^{k_max} > N(r,k) > > > dr > > > > > dk > > > > > > is conserved. > > > > > > This quantity can be approximated from your > > > > > calculations > > > > > > by > > > > > > sum_{i=1}^{i=N-1} sum_{j=1}^{j=M-1} > N(r_i,k_j)* > > > > > > (r_(i+1)-r_(i))*(k_(j+1)-k_(j)). > > > > > > > > > > > > Whether this is the number of molecules in > the > > > > > sphere, > > > > > > you must decide from the physical > background of > > > > > > > > > your problem. > > > > > > > > > > > > Best wishes > > > > > > Torsten. > > > > > > > > > > Hi, > > > > > I am facing a typical problem to solve my > > > Boltzmann > > > > > equation. > > > > > My PDE > > > > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > > > > Nk(i,j)=((N(i,j+1)-N(i,j))); > > > > > > > > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > > > > *Nk(i,j)); > > > > > For my equation, when I am giving Boundary > > > condition > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > Do you mean N(1,j) = 0; N(i,nk) = 0 here ? > > > > Otherwise your boundary conditions do not make > > > sense > > > > since for a first-order differential equation, > > > > it is not allowed to prescribe dN/dr = 0 or > dN/dk = > > > 0 > > > > at a boundary. > > > > Or where did you read that such a boundary > > > condition > > > > can be applied ? > > > > > > > > > then curve is acting properly, but total > number > > > of > > > > > particles decreasing with increasing time > steps. > > > > > > > > If your discretization scheme is conservative, > > > > this can only happen because more particles > leave > > > > the domain than enter it over the four > boundaries > > > > (see my response above). > > > > > > > > > But when I am giving Boundary condition > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > N(i,1)=0; > > > > > then from second time steps total number of > > > particles > > > > > are constant but figure is not moving or not > > > working > > > > > properly. > > > > > In my case particles are within potential > trap, > > > so it > > > > > cant go out. > > > > > Can you give some idea why it is happening? > > > > > > > > > > Thank you. > > > > > With regards, > > > > > Sunipa Som > > > > > > > > Best wishes > > > > Torsten. > > > Hi, > > > Nr(1,j)=0;Nk(i,nk)=0; means dN/dr(1,j)=0 and > > > dN/dk(i,nk)=0 which I put as boundary condition > and > > > curve is working fine. > > > But particles numbers are > > > decreasing. In my case particles are within > potential > > > trap, so it can not go out, so how can it > decrease? > > > and how can i resolve it? Because in my case > total > > > number of particles should be constant. > > > > > > Thank you. > > > With regards, > > > Sunipa Som > > > > What is c1*sqrt(j)*N(i,j) for i=nr , > > -c1*sqrt(j)*N(i,j) for i=1, > > -c2*sqrt(j)*(i-1)*dr*N(i,j) for j=1 and > > c2*sqrt(j)*(i-1)*dr*N(i,j) for j=nk ? > > If the expressions are greater than zero, the > > number of particles in the volume will decrease > > in time. > > > > Best wishes > > Torsten. > > Hi, > In my case particle should be constant. So I want > ant to make a reflecting wall or reflecting sphere > which will reflect this particles(which moving out) > inside. Do you know how can I do it? > > Thank you. > > With Regards, > S Som Do you use a finite-difference or finite volume scheme to discretize your equations ? Best wishes Torsten.
From: Sunipa Som on 8 Jul 2010 08:04 Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote in message <1630764328.89371.1278589087664.JavaMail.root(a)gallium.mathforum.org>... > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > > in message > > <128269470.77069.1278420640088.JavaMail.root(a)gallium.m > > athforum.org>... > > > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> > > wrote > > > > in message > > > > > > <1394417502.76713.1278415717104.JavaMail.root(a)gallium. > > > > mathforum.org>... > > > > > > Torsten Hennig > > <Torsten.Hennig(a)umsicht.fhg.de> > > > > wrote > > > > > > in message > > > > > > > > > > > > <2044145382.75601.1278398289051.JavaMail.root(a)gallium. > > > > > > mathforum.org>... > > > > > > > >Hi, > > > > > > > >My equation is Boltzman equation. Without > > > > > > collision term >it is > > > > > > > >dN/dt+v*dN/dr+F*dN/dp=0 > > > > > > > >where v is velocity, F is force and p is > > > > momentum > > > > > > > >by rearranging few terms we are getting > > this > > > > > > > >dN/dt + k*c1*dN/dr-c2*r*dN/dk=0 > > > > > > > >here unit of k is 1/meter and r is also > > > > length. > > > > > > > >so, then my procedure of calculating total > > > > number > > > > > > of >particles is right or I have to do in > > other > > > > way? > > > > > > > > > > > > > > > >With Regards, > > > > > > > >Sunipa Som > > > > > > > > > > > > > > Start from the equation > > > > > > > dN/dt + k*c1*dN/dr-c2*r*dN/dk=0 > > > > > > > or > > > > > > > dN/dt + div(k*c1*N,-c2*r*N) = 0 > > > > > > > to be solved over the rectangle > > > > > > > V=[r_min;r_max]x[k_min,k_max]. > > > > > > > > > > > > > > Integrate over the rectangle V with > > boundary A > > > > to > > > > > > get > > > > > > > d/dt int_{V} N dV = int_{V} > > div(-k*c1*N,c2*r*N) > > > > dV. > > > > > > > Applying Gauss' integral theorem to the > > right > > > > hand > > > > > > side > > > > > > > results in > > > > > > > d/dt int_{V} N dV = int_{A} > > (-k*c1*N,c2*r*N)*n > > > > dA > > > > > > > where n is the unit normal pointing > > outwards of > > > > the > > > > > > > > > > > > > rectangle. > > > > > > > So if no particles enter the rectangle over > > the > > > > > > boundary, > > > > > > > the quantity > > > > > > > int_{V} N dV = > > > > > > > int_{r_min}^{r_max} int_{k_min}^{k_max} > > N(r,k) > > > > dr > > > > > > dk > > > > > > > is conserved. > > > > > > > This quantity can be approximated from your > > > > > > calculations > > > > > > > by > > > > > > > sum_{i=1}^{i=N-1} sum_{j=1}^{j=M-1} > > N(r_i,k_j)* > > > > > > > (r_(i+1)-r_(i))*(k_(j+1)-k_(j)). > > > > > > > > > > > > > > Whether this is the number of molecules in > > the > > > > > > sphere, > > > > > > > you must decide from the physical > > background of > > > > > > > > > > > your problem. > > > > > > > > > > > > > > Best wishes > > > > > > > Torsten. > > > > > > > > > > > > Hi, > > > > > > I am facing a typical problem to solve my > > > > Boltzmann > > > > > > equation. > > > > > > My PDE > > > > > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > > > > > Nk(i,j)=((N(i,j+1)-N(i,j))); > > > > > > > > > > > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > > > > > *Nk(i,j)); > > > > > > For my equation, when I am giving Boundary > > > > condition > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > > > Do you mean N(1,j) = 0; N(i,nk) = 0 here ? > > > > > Otherwise your boundary conditions do not make > > > > sense > > > > > since for a first-order differential equation, > > > > > it is not allowed to prescribe dN/dr = 0 or > > dN/dk = > > > > 0 > > > > > at a boundary. > > > > > Or where did you read that such a boundary > > > > condition > > > > > can be applied ? > > > > > > > > > > > then curve is acting properly, but total > > number > > > > of > > > > > > particles decreasing with increasing time > > steps. > > > > > > > > > > If your discretization scheme is conservative, > > > > > this can only happen because more particles > > leave > > > > > the domain than enter it over the four > > boundaries > > > > > (see my response above). > > > > > > > > > > > But when I am giving Boundary condition > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > N(i,1)=0; > > > > > > then from second time steps total number of > > > > particles > > > > > > are constant but figure is not moving or not > > > > working > > > > > > properly. > > > > > > In my case particles are within potential > > trap, > > > > so it > > > > > > cant go out. > > > > > > Can you give some idea why it is happening? > > > > > > > > > > > > Thank you. > > > > > > With regards, > > > > > > Sunipa Som > > > > > > > > > > Best wishes > > > > > Torsten. > > > > Hi, > > > > Nr(1,j)=0;Nk(i,nk)=0; means dN/dr(1,j)=0 and > > > > dN/dk(i,nk)=0 which I put as boundary condition > > and > > > > curve is working fine. > > > > But particles numbers are > > > > decreasing. In my case particles are within > > potential > > > > trap, so it can not go out, so how can it > > decrease? > > > > and how can i resolve it? Because in my case > > total > > > > number of particles should be constant. > > > > > > > > Thank you. > > > > With regards, > > > > Sunipa Som > > > > > > What is c1*sqrt(j)*N(i,j) for i=nr , > > > -c1*sqrt(j)*N(i,j) for i=1, > > > -c2*sqrt(j)*(i-1)*dr*N(i,j) for j=1 and > > > c2*sqrt(j)*(i-1)*dr*N(i,j) for j=nk ? > > > If the expressions are greater than zero, the > > > number of particles in the volume will decrease > > > in time. > > > > > > Best wishes > > > Torsten. > > > > Hi, > > In my case particle should be constant. So I want > > ant to make a reflecting wall or reflecting sphere > > which will reflect this particles(which moving out) > > inside. Do you know how can I do it? > > > > Thank you. > > > > With Regards, > > S Som > > Do you use a finite-difference or finite volume > scheme to discretize your equations ? > > Best wishes > Torsten. I am using finite difference. My PDE Nr(i,j)=((N(i,j)-N(i-1,j))/dr); Nk(i,j)=((N(i,j+1)-N(i,j))/dk); Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. *Nk(i,j)); Boundary condition Nr(1,j)=0;Nk(i,nk)=0; With Regards, S Som
From: Torsten Hennig on 8 Jul 2010 04:47 > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > in message > <1630764328.89371.1278589087664.JavaMail.root(a)gallium. > mathforum.org>... > > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> > wrote > > > in message > > > > <128269470.77069.1278420640088.JavaMail.root(a)gallium.m > > > athforum.org>... > > > > > Torsten Hennig > <Torsten.Hennig(a)umsicht.fhg.de> > > > wrote > > > > > in message > > > > > > > > > <1394417502.76713.1278415717104.JavaMail.root(a)gallium. > > > > > mathforum.org>... > > > > > > > Torsten Hennig > > > <Torsten.Hennig(a)umsicht.fhg.de> > > > > > wrote > > > > > > > in message > > > > > > > > > > > > > > > > <2044145382.75601.1278398289051.JavaMail.root(a)gallium. > > > > > > > mathforum.org>... > > > > > > > > >Hi, > > > > > > > > >My equation is Boltzman equation. > Without > > > > > > > collision term >it is > > > > > > > > >dN/dt+v*dN/dr+F*dN/dp=0 > > > > > > > > >where v is velocity, F is force and p > is > > > > > momentum > > > > > > > > >by rearranging few terms we are > getting > > > this > > > > > > > > >dN/dt + k*c1*dN/dr-c2*r*dN/dk=0 > > > > > > > > >here unit of k is 1/meter and r is > also > > > > > length. > > > > > > > > >so, then my procedure of calculating > total > > > > > number > > > > > > > of >particles is right or I have to do in > > > other > > > > > way? > > > > > > > > > > > > > > > > > >With Regards, > > > > > > > > >Sunipa Som > > > > > > > > > > > > > > > > Start from the equation > > > > > > > > dN/dt + k*c1*dN/dr-c2*r*dN/dk=0 > > > > > > > > or > > > > > > > > dN/dt + div(k*c1*N,-c2*r*N) = 0 > > > > > > > > to be solved over the rectangle > > > > > > > > V=[r_min;r_max]x[k_min,k_max]. > > > > > > > > > > > > > > > > Integrate over the rectangle V with > > > boundary A > > > > > to > > > > > > > get > > > > > > > > d/dt int_{V} N dV = int_{V} > > > div(-k*c1*N,c2*r*N) > > > > > dV. > > > > > > > > Applying Gauss' integral theorem to the > > > right > > > > > hand > > > > > > > side > > > > > > > > results in > > > > > > > > d/dt int_{V} N dV = int_{A} > > > (-k*c1*N,c2*r*N)*n > > > > > dA > > > > > > > > where n is the unit normal pointing > > > outwards of > > > > > the > > > > > > > > > > > > > > > rectangle. > > > > > > > > So if no particles enter the rectangle > over > > > the > > > > > > > boundary, > > > > > > > > the quantity > > > > > > > > int_{V} N dV = > > > > > > > > int_{r_min}^{r_max} int_{k_min}^{k_max} > > > N(r,k) > > > > > dr > > > > > > > dk > > > > > > > > is conserved. > > > > > > > > This quantity can be approximated from > your > > > > > > > calculations > > > > > > > > by > > > > > > > > sum_{i=1}^{i=N-1} sum_{j=1}^{j=M-1} > > > N(r_i,k_j)* > > > > > > > > (r_(i+1)-r_(i))*(k_(j+1)-k_(j)). > > > > > > > > > > > > > > > > Whether this is the number of molecules > in > > > the > > > > > > > sphere, > > > > > > > > you must decide from the physical > > > background of > > > > > > > > > > > > > your problem. > > > > > > > > > > > > > > > > Best wishes > > > > > > > > Torsten. > > > > > > > > > > > > > > Hi, > > > > > > > I am facing a typical problem to solve my > > > > > Boltzmann > > > > > > > equation. > > > > > > > My PDE > > > > > > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > > > > > > Nk(i,j)=((N(i,j+1)-N(i,j))); > > > > > > > > > > > > > > > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > > > > > > *Nk(i,j)); > > > > > > > For my equation, when I am giving > Boundary > > > > > condition > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > > > > > Do you mean N(1,j) = 0; N(i,nk) = 0 here ? > > > > > > Otherwise your boundary conditions do not > make > > > > > sense > > > > > > since for a first-order differential > equation, > > > > > > it is not allowed to prescribe dN/dr = 0 or > > > dN/dk = > > > > > 0 > > > > > > at a boundary. > > > > > > Or where did you read that such a boundary > > > > > condition > > > > > > can be applied ? > > > > > > > > > > > > > then curve is acting properly, but total > > > number > > > > > of > > > > > > > particles decreasing with increasing time > > > steps. > > > > > > > > > > > > If your discretization scheme is > conservative, > > > > > > this can only happen because more particles > > > leave > > > > > > the domain than enter it over the four > > > boundaries > > > > > > (see my response above). > > > > > > > > > > > > > But when I am giving Boundary condition > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > N(i,1)=0; > > > > > > > then from second time steps total number > of > > > > > particles > > > > > > > are constant but figure is not moving or > not > > > > > working > > > > > > > properly. > > > > > > > In my case particles are within potential > > > trap, > > > > > so it > > > > > > > cant go out. > > > > > > > Can you give some idea why it is > happening? > > > > > > > > > > > > > > Thank you. > > > > > > > With regards, > > > > > > > Sunipa Som > > > > > > > > > > > > Best wishes > > > > > > Torsten. > > > > > Hi, > > > > > Nr(1,j)=0;Nk(i,nk)=0; means dN/dr(1,j)=0 and > > > > > dN/dk(i,nk)=0 which I put as boundary > condition > > > and > > > > > curve is working fine. > > > > > But particles numbers are > > > > > decreasing. In my case particles are within > > > potential > > > > > trap, so it can not go out, so how can it > > > decrease? > > > > > and how can i resolve it? Because in my case > > > total > > > > > number of particles should be constant. > > > > > > > > > > Thank you. > > > > > With regards, > > > > > Sunipa Som > > > > > > > > What is c1*sqrt(j)*N(i,j) for i=nr , > > > > -c1*sqrt(j)*N(i,j) for i=1, > > > > -c2*sqrt(j)*(i-1)*dr*N(i,j) for j=1 and > > > > c2*sqrt(j)*(i-1)*dr*N(i,j) for j=nk ? > > > > If the expressions are greater than zero, the > > > > number of particles in the volume will decrease > > > > > in time. > > > > > > > > Best wishes > > > > Torsten. > > > > > > Hi, > > > In my case particle should be constant. So I want > > > ant to make a reflecting wall or reflecting > sphere > > > which will reflect this particles(which moving > out) > > > inside. Do you know how can I do it? > > > > > > Thank you. > > > > > > With Regards, > > > S Som > > > > Do you use a finite-difference or finite volume > > scheme to discretize your equations ? > > > > Best wishes > > Torsten. > > I am using finite difference. > My PDE > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > Nk(i,j)=((N(i,j+1)-N(i,j))/dk); > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > *Nk(i,j)); > Boundary condition > Nr(1,j)=0;Nk(i,nk)=0; > > > With Regards, > S Som And at the other two boundaries you want to implement a reflecting boundary condition for N ? Best wishes Torsten.
From: Sunipa Som on 8 Jul 2010 10:23 Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote in message <949632433.89696.1278593304386.JavaMail.root(a)gallium.mathforum.org>... > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > > in message > > <1630764328.89371.1278589087664.JavaMail.root(a)gallium. > > mathforum.org>... > > > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> > > wrote > > > > in message > > > > > > <128269470.77069.1278420640088.JavaMail.root(a)gallium.m > > > > athforum.org>... > > > > > > Torsten Hennig > > <Torsten.Hennig(a)umsicht.fhg.de> > > > > wrote > > > > > > in message > > > > > > > > > > > > <1394417502.76713.1278415717104.JavaMail.root(a)gallium. > > > > > > mathforum.org>... > > > > > > > > Torsten Hennig > > > > <Torsten.Hennig(a)umsicht.fhg.de> > > > > > > wrote > > > > > > > > in message > > > > > > > > > > > > > > > > > > > > <2044145382.75601.1278398289051.JavaMail.root(a)gallium. > > > > > > > > mathforum.org>... > > > > > > > > > > > > > > > > > > Best wishes > > > > > > > > > Torsten. > > > > > > > > > > > > > > > > Hi, > > > > > > > > I am facing a typical problem to solve my > > > > > > Boltzmann > > > > > > > > equation. > > > > > > > > My PDE > > > > > > > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > > > > > > > Nk(i,j)=((N(i,j+1)-N(i,j))); > > > > > > > > > > > > > > > > > > > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > > > > > > > *Nk(i,j)); > > > > > > > > For my equation, when I am giving > > Boundary > > > > > > condition > > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > > > > > > > Do you mean N(1,j) = 0; N(i,nk) = 0 here ? > > > > > > > Otherwise your boundary conditions do not > > make > > > > > > sense > > > > > > > since for a first-order differential > > equation, > > > > > > > it is not allowed to prescribe dN/dr = 0 or > > > > dN/dk = > > > > > > 0 > > > > > > > at a boundary. > > > > > > > Or where did you read that such a boundary > > > > > > condition > > > > > > > can be applied ? > > > > > > > > > > > > > > > then curve is acting properly, but total > > > > number > > > > > > of > > > > > > > > particles decreasing with increasing time > > > > steps. > > > > > > > > > > > > > > If your discretization scheme is > > conservative, > > > > > > > this can only happen because more particles > > > > leave > > > > > > > the domain than enter it over the four > > > > boundaries > > > > > > > (see my response above). > > > > > > > > > > > > > > > But when I am giving Boundary condition > > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > N(i,1)=0; > > > > > > > > then from second time steps total number > > of > > > > > > particles > > > > > > > > are constant but figure is not moving or > > not > > > > > > working > > > > > > > > properly. > > > > > > > > In my case particles are within potential > > > > trap, > > > > > > so it > > > > > > > > cant go out. > > > > > > > > Can you give some idea why it is > > happening? > > > > > > > > > > > > > > > > Thank you. > > > > > > > > With regards, > > > > > > > > Sunipa Som > > > > > > > > > > > > > > Best wishes > > > > > > > Torsten. > > > > > > Hi, > > > > > > Nr(1,j)=0;Nk(i,nk)=0; means dN/dr(1,j)=0 and > > > > > > dN/dk(i,nk)=0 which I put as boundary > > condition > > > > and > > > > > > curve is working fine. > > > > > > But particles numbers are > > > > > > decreasing. In my case particles are within > > > > potential > > > > > > trap, so it can not go out, so how can it > > > > decrease? > > > > > > and how can i resolve it? Because in my case > > > > total > > > > > > number of particles should be constant. > > > > > > > > > > > > Thank you. > > > > > > With regards, > > > > > > Sunipa Som > > > > > > > > > > What is c1*sqrt(j)*N(i,j) for i=nr , > > > > > -c1*sqrt(j)*N(i,j) for i=1, > > > > > -c2*sqrt(j)*(i-1)*dr*N(i,j) for j=1 and > > > > > c2*sqrt(j)*(i-1)*dr*N(i,j) for j=nk ? > > > > > If the expressions are greater than zero, the > > > > > number of particles in the volume will decrease > > > > > > > in time. > > > > > > > > > > Best wishes > > > > > Torsten. > > > > > > > > Hi, > > > > In my case particle should be constant. So I want > > > > ant to make a reflecting wall or reflecting > > sphere > > > > which will reflect this particles(which moving > > out) > > > > inside. Do you know how can I do it? > > > > > > > > Thank you. > > > > > > > > With Regards, > > > > S Som > > > > > > Do you use a finite-difference or finite volume > > > scheme to discretize your equations ? > > > > > > Best wishes > > > Torsten. > > > > I am using finite difference. > > My PDE > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > Nk(i,j)=((N(i,j+1)-N(i,j))/dk); > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > *Nk(i,j)); > > Boundary condition > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > With Regards, > > S Som > > And at the other two boundaries you want to implement a > reflecting boundary condition for N ? > > Best wishes > Torsten. Yes, if it is possible by putting boundary condition, I can do that. But what will be these conditions? So, particles can not move out from the boundary rather they will reflect back to the rgion. With Regards, Sunipa Som
From: Torsten Hennig on 11 Jul 2010 23:03
> Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > in message > <949632433.89696.1278593304386.JavaMail.root(a)gallium.m > athforum.org>... > > > Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> > wrote > > > in message > > > > <1630764328.89371.1278589087664.JavaMail.root(a)gallium. > > > mathforum.org>... > > > > > Torsten Hennig > <Torsten.Hennig(a)umsicht.fhg.de> > > > wrote > > > > > in message > > > > > > > > > <128269470.77069.1278420640088.JavaMail.root(a)gallium.m > > > > > athforum.org>... > > > > > > > Torsten Hennig > > > <Torsten.Hennig(a)umsicht.fhg.de> > > > > > wrote > > > > > > > in message > > > > > > > > > > > > > > > > <1394417502.76713.1278415717104.JavaMail.root(a)gallium. > > > > > > > mathforum.org>... > > > > > > > > > Torsten Hennig > > > > > <Torsten.Hennig(a)umsicht.fhg.de> > > > > > > > wrote > > > > > > > > > in message > > > > > > > > > > > > > > > > > > > > > > > > > <2044145382.75601.1278398289051.JavaMail.root(a)gallium. > > > > > > > > > mathforum.org>... > > > > > > > > > > > > > > > > > > > > Best wishes > > > > > > > > > > Torsten. > > > > > > > > > > > > > > > > > > Hi, > > > > > > > > > I am facing a typical problem to > solve my > > > > > > > Boltzmann > > > > > > > > > equation. > > > > > > > > > My PDE > > > > > > > > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > > > > > > > > Nk(i,j)=((N(i,j+1)-N(i,j))); > > > > > > > > > > > > > > > > > > > > > > > > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > > > > > > > > *Nk(i,j)); > > > > > > > > > For my equation, when I am giving > > > Boundary > > > > > > > condition > > > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > > > > > > > > > Do you mean N(1,j) = 0; N(i,nk) = 0 > here ? > > > > > > > > Otherwise your boundary conditions do > not > > > make > > > > > > > sense > > > > > > > > since for a first-order differential > > > equation, > > > > > > > > it is not allowed to prescribe dN/dr = > 0 or > > > > > dN/dk = > > > > > > > 0 > > > > > > > > at a boundary. > > > > > > > > Or where did you read that such a > boundary > > > > > > > condition > > > > > > > > can be applied ? > > > > > > > > > > > > > > > > > then curve is acting properly, but > total > > > > > number > > > > > > > of > > > > > > > > > particles decreasing with increasing > time > > > > > steps. > > > > > > > > > > > > > > > > If your discretization scheme is > > > conservative, > > > > > > > > this can only happen because more > particles > > > > > leave > > > > > > > > the domain than enter it over the four > > > > > boundaries > > > > > > > > (see my response above). > > > > > > > > > > > > > > > > > But when I am giving Boundary > condition > > > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > > N(i,1)=0; > > > > > > > > > then from second time steps total > number > > > of > > > > > > > particles > > > > > > > > > are constant but figure is not moving > or > > > not > > > > > > > working > > > > > > > > > properly. > > > > > > > > > In my case particles are within > potential > > > > > trap, > > > > > > > so it > > > > > > > > > cant go out. > > > > > > > > > Can you give some idea why it is > > > happening? > > > > > > > > > > > > > > > > > > Thank you. > > > > > > > > > With regards, > > > > > > > > > Sunipa Som > > > > > > > > > > > > > > > > Best wishes > > > > > > > > Torsten. > > > > > > > Hi, > > > > > > > Nr(1,j)=0;Nk(i,nk)=0; means dN/dr(1,j)=0 > and > > > > > > > dN/dk(i,nk)=0 which I put as boundary > > > condition > > > > > and > > > > > > > curve is working fine. > > > > > > > But particles numbers are > > > > > > > decreasing. In my case particles are > within > > > > > potential > > > > > > > trap, so it can not go out, so how can it > > > > > decrease? > > > > > > > and how can i resolve it? Because in my > case > > > > > total > > > > > > > number of particles should be constant. > > > > > > > > > > > > > > Thank you. > > > > > > > With regards, > > > > > > > Sunipa Som > > > > > > > > > > > > What is c1*sqrt(j)*N(i,j) for i=nr , > > > > > > -c1*sqrt(j)*N(i,j) for i=1, > > > > > > -c2*sqrt(j)*(i-1)*dr*N(i,j) for j=1 and > > > > > > c2*sqrt(j)*(i-1)*dr*N(i,j) for j=nk ? > > > > > > If the expressions are greater than zero, > the > > > > > > number of particles in the volume will > decrease > > > > > > > > > in time. > > > > > > > > > > > > Best wishes > > > > > > Torsten. > > > > > > > > > > Hi, > > > > > In my case particle should be constant. So I > want > > > > > ant to make a reflecting wall or reflecting > > > sphere > > > > > which will reflect this particles(which > moving > > > out) > > > > > inside. Do you know how can I do it? > > > > > > > > > > Thank you. > > > > > > > > > > With Regards, > > > > > S Som > > > > > > > > Do you use a finite-difference or finite volume > > > > > scheme to discretize your equations ? > > > > > > > > Best wishes > > > > Torsten. > > > > > > I am using finite difference. > > > My PDE > > > Nr(i,j)=((N(i,j)-N(i-1,j))/dr); > > > Nk(i,j)=((N(i,j+1)-N(i,j))/dk); > > > > Nt(i,j)=-(c1*sqrt(j).*Nr(i,j))+(c2*sqrt(j).*(i-1).*dr. > > > *Nk(i,j)); > > > Boundary condition > > > Nr(1,j)=0;Nk(i,nk)=0; > > > > > > > > > With Regards, > > > S Som > > > > And at the other two boundaries you want to > implement a > > reflecting boundary condition for N ? > > > > Best wishes > > Torsten. > > Yes, if it is possible by putting boundary condition, > I can do that. But what will be these conditions? So, > particles can not move out from the boundary rather > they will reflect back to the rgion. > > With Regards, > Sunipa Som You could set the velocities normal to the boundary to zero: c1 = 0 for i=1 and i=nr ; c2 = 0 for j=1 and j=nr. Best wishes Torsten. |