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This subroutine solves a one-dimensional advection equation. There are two ! options, depending whether the advection should be conservative or not. ! Conservative advection has to be applied when settling of sediment or ! rising of phytoplankton is considered. In this case the advection is of ! the form ! $$! \label{Yadvection_cons} ! \partder{Y}{t} = - \partder{F}{z} ! \comma !$$ ! where $F=wY$ is the flux caused by the advective velocity, $w$. ! ! Non-conservative advective transport has to be applied, when the water ! has a non-zero vertical velocity. In three-dimensional applications, ! this transport would be conservative, since vertical divergence would be ! compensated by horizontal convergence and vice versa. However, the ! key assumption of one-dimensional modelling is horizontal homogeneity, ! such that we indeed have to apply a vertically non-conservative method, ! which is of the form ! $$! \label{Yadvection_noncons} ! \partder{Y}{t} = - w\partder{Y}{z} ! = - \left(\partder{F}{z} - Y\partder{w}{z} \right). !$$ ! ! The discretized form of \eq{Yadvection_cons} is ! $$! \label{advDiscretized_cons} ! Y_i^{n+1} = Y_i^n ! - \dfrac{\Delta t}{h_i} ! \left( F^n_{i} - F^n_{i-1} \right) ! \comma !$$ ! where the integers $n$ and $i$ correspond to the present time and space ! level, respectively. ! ! For the non-conservative form \eq{Yadvection_noncons}, ! an extra term needs to be included: ! $$! \label{advDiscretized_noncons} ! Y_i^{n+1} = Y_i^n ! - \dfrac{\Delta t}{h_i} ! \left( F^n_{i} - F^n_{i-1} -Y_i^n \left(w_k-w_{k-1} \right)\right). !$$ ! ! Which advection method is applied is decided by the flag {\tt mode}, ! which gives conservative advection \eq{advDiscretized_cons} ! for {\tt mode=1} and ! non-conservative advection \eq{advDiscretized_noncons} for {\tt mode=0}. ! ! Fluxes are defined at the grid faces, the variable $Y_i$ is defined at the ! grid centers. The fluxes are computed in an upstream-biased way, ! $$! \label{upstream} ! F^n_{i} = \dfrac{1}{\Delta t} ! \int_{z^\text{Face}_{i} - w \Delta t}^{z^\text{Face}_{i}} Y(z') dz' ! \point !$$ ! For a third-order polynomial approximation of $Y$ (see \cite{Pietrzak98}), ! these fluxes can be written the in so-called Lax-Wendroff form as ! $$! \label{fluxDiscretized} ! \begin{array}{rcll} ! F_{i} &=& w_{i} \left(Y_i + \dfrac{1}{2} \Phi^+_{i} ! \left(1-\magn{c_{i}} \right) \left( Y_{i+1} - Y_i \right) \right) ! \quad & \text{for} \quad w_{i} > 0 ! \comma \\[5mm] ! F_{i} &=& w_{i} \left(Y_{i+1} + \dfrac{1}{2} \Phi^-_{i} ! \left(1-\magn{c_{i}} \right) \left( Y_i - Y_{i+1} \right) \right) ! \quad & \text{for} \quad w_{i} < 0 ! \comma ! \end{array} !$$ ! where $c_{i} = 2 w_{i} \Delta t / (h_i+h_{i+1})$ is the Courant number. ! The factors appearing in \eq{fluxDiscretized} are defined as ! $$! \label{phiDiscretized} ! \Phi^+_{i} = \alpha_{i} + \beta_{i} r^+_{i} ! \comma ! \Phi^-_{i} = \alpha_{i} + \beta_{i} r^-_{i} ! \comma !$$ ! where ! $$! \label{alphaDiscretized} ! \alpha_{i} = \dfrac{1}{2} ! + \dfrac{1}{6} \left( 1- 2 \magn{c_{i}} \right) \comma ! \beta_{i} = \dfrac{1}{2} ! - \dfrac{1}{6} \left( 1- 2 \magn{c_{i}} \right) ! \point !$$ ! The upstream and downstream slope parameters are ! $$! \label{slopeDiscretized} ! r^+_{i} = \dfrac{Y_i - Y_{i-1}}{Y_{i+1}-Y_{i}} \comma ! r^-_{i} = \dfrac{Y_{i+2} - Y_{i+1}}{Y_{i+1}-Y_{i}} ! \point !$$ ! ! To obtain monotonic and positive schemes also in the presence of strong ! gradients, so-called slope limiters are aplied for the factors $\Phi^+_{i}$ ! and $\Phi^-_{i}$. The two most obvious cases are ! the first-order upstream discretisation with $\Phi^+_{i}=\Phi^-_{i}=0$ ! and the Lax-Wendroff scheme with $\Phi^+_{i}=\Phi^-_{i}=1$. ! The subroutine {\tt adv\_center.F90} provides six different slope-limiters, ! all discussed in detail by \cite{Pietrzak98}: ! ! \begin{itemize} ! \item first-order upstream ({\tt method=UPSTREAM}) ! \item second-order upstream-biased polynomial scheme ({\tt method=P1}, ! not yet implemented) ! \item third-order upstream-biased polynomial scheme ({\tt method=P2}) ! \item third-order scheme (TVD) with Superbee limiter ({\tt method=Superbee}) ! \item third-order scheme (TVD) with MUSCL limiter ({\tt method=MUSCL}) ! \item third-order scheme (TVD) with ULTIMATE QUICKEST limiter ! ({\tt method=P2\_PDM}) ! \end{itemize} ! ! ! If during a certain time step the maximum Courant number is larger ! than one, a split iteration will be carried out which guarantees that the ! split step Courant numbers are just smaller than 1. ! ! Several kinds of boundary conditions are implemented for the upper ! and lower boundaries. They are set by the integer values {\tt Bcup} ! and {\tt Bcdw}, that have to correspond to the parameters defined ! in the module {\tt util}, see \sect{sec:utils}. The ! following choices exist at the moment: ! ! For the value {\tt flux}, the boundary values {\tt Yup} and {\tt Ydw} are ! interpreted as specified fluxes at the uppermost and lowest interface. ! Fluxes into the boundary cells are counted positive by convention. ! For the value {\tt value}, {\tt Yup} and {\tt Ydw} specify the value ! of $Y$ at the interfaces, and the flux is computed by multiplying with ! the (known) speed at the interface. For the value {\tt oneSided}, ! {\tt Yup} and {\tt Ydw} are ignored and the flux is computed ! from a one-sided first-order upstream discretisation using the speed ! at the interface and the value of $Y$ at the center of the boundary cell. ! For the value {\tt zeroDivergence}, the fluxes into and out of the ! respective boundary cell are set equal. ! This corresponds to a zero-gradient formulation, or to zero ! flux divergence in the boundary cells. ! ! Be careful that your boundary conditions are mathematically well defined. ! For example, specifying an inflow into the boundary cell with the ! speed at the boundary being directed outward does not make sense. ! ! ! !USES: use util IMPLICIT NONE ! ! !INPUT PARAMETERS: ! number of vertical layers integer, intent(in) :: N ! time step (s) REALTYPE, intent(in) :: dt ! layer thickness (m) REALTYPE, intent(in) :: h(0:N) ! old layer thickness (m) REALTYPE, intent(in) :: ho(0:N) ! vertical advection speed REALTYPE, intent(in) :: ww(0:N) ! type of upper BC integer, intent(in) :: Bcup ! type of lower BC integer, intent(in) :: Bcdw ! value of upper BC REALTYPE, intent(in) :: Yup ! value of lower BC REALTYPE, intent(in) :: Ydw ! type of advection scheme integer, intent(in) :: method ! advection mode (0: non-conservative, 1: conservative) integer, intent(in) :: mode ! ! !INPUT/OUTPUT PARAMETERS: REALTYPE :: Y(0:N) ! ! !DEFINED PARAMETERS: REALTYPE, parameter :: one6th=1.0d0/6.0d0 integer, parameter :: itmax=100 ! ! !REVISION HISTORY: ! Original author(s): Lars Umlauf ! ! $Log: adv_center.F90,v$ ! Revision 1.4 2006-11-06 13:36:46 hb ! Option for conservative vertical advection added to adv_center ! ! Revision 1.3 2006-03-20 09:06:38 kbk ! removed explicit double precission dependency ! ! Revision 1.2 2005/11/18 10:59:34 kbk ! removed unused variables - some left in parameter lists ! ! Revision 1.1 2005/06/27 10:54:33 kbk ! new files needed ! ! ! !EOP ! ! !LOCAL VARIABLES: integer :: i,k,it REALTYPE :: x,r,Phi,limit=_ZERO_ REALTYPE :: Yu,Yc,Yd REALTYPE :: c,cmax REALTYPE :: cu(0:N) ! !----------------------------------------------------------------------- !BOC #ifdef DEBUG integer, save :: Ncall = 0 Ncall = Ncall+1 write(*,*) 'adv_center # ',Ncall #endif ! initialize interface fluxes with zero cu = _ZERO_ ! initialize maximum Courant number cmax = _ZERO_ ! compute maximum Courant number do k=1,N-1 c=abs(ww(k))*dt/(0.5*(h(k)+h(k+1))) if (c.gt.cmax) cmax=c enddo it=min(itmax,int(cmax)+1) !#ifdef DEBUG if (it .gt. 1) then STDERR 'In adv_center():' STDERR 'Maximum Courant number is ',cmax STDERR it,' iterations used for vertical advection' endif !#endif ! splitting loop do i=1,it ! vertical loop do k=1,N-1 ! compute the slope ration if (ww(k) .gt. _ZERO_) then ! compute Courant number c=ww(k)/float(it)*dt/(0.5*(h(k)+h(k+1))) if (k .gt. 1) then Yu=Y(k-1) ! upstream value else Yu=Y(k) end if Yc=Y(k ) ! central value Yd=Y(k+1) ! downstream value ! compute slope ration if (abs(Yd-Yc) .gt. 1e-10) then r=(Yc-Yu)/(Yd-Yc) else r=(Yc-Yu)*1.e10 end if ! negative speed else ! compute Courant number c=-ww(k)/float(it)*dt/(0.5*(h(k)+h(k+1))) if (k .lt. N-1) then Yu=Y(k+2) ! upstream value else Yu=Y(k+1) end if Yc=Y(k+1) ! central value Yd=Y(k ) ! downstream value ! compute slope ratio if (abs(Yc-Yd) .gt. 1e-10) then r=(Yu-Yc)/(Yc-Yd) else r=(Yu-Yc)*1.e10 end if end if ! compute the flux-factor phi x = one6th*(1.-2.0*c) Phi = (0.5+x)+(0.5-x)*r ! limit the flux according to different suggestions select case (method) case (UPSTREAM) limit=_ZERO_ case (P1) FATAL "P1 advection method not yet implemented, choose other method" stop "adv_center.F90" case ((P2),(P2_PDM)) if (method.eq.P2) then limit=Phi else limit=max(_ZERO_,min(Phi,2./(1.-c),2.*r/(c+1.e-10))) end if case (Superbee) limit=max(_ZERO_, min(_ONE_, 2.0*r), min(r,2.*_ONE_) ) case (MUSCL) limit=max(_ZERO_,min(2.*_ONE_,2.0*r,0.5*(1.0+r))) case default LEVEL3 method FATAL 'unkown advection method in adv_center()' stop end select ! compute the limited flux cu(k)=ww(k)*(Yc+0.5*limit*(1-c)*(Yd-Yc)) end do ! do the upper boundary conditions select case (Bcup) case (flux) cu(N) = - Yup ! flux into the domain is positive case (value) cu(N) = ww(N)*Yup case (oneSided) if (ww(N).ge._ZERO_) then cu(N) = ww(N)*Y(N) else cu(N) = _ZERO_ end if case (zeroDivergence) cu(N) = cu(N-1) case default FATAL 'unkown upper boundary condition type in adv_center()' stop end select ! do the lower boundary conditions select case (Bcdw) case (flux) cu(0) = Ydw ! flux into the domain is positive case (value) cu(0) = ww(0)*Ydw case (oneSided) if(ww(0).le._ZERO_) then cu(0) = ww(0)*Y(1) else cu(0) = _ZERO_ end if case (zeroDivergence) cu(0) = cu(1) case default FATAL 'unkown lower boundary condition type in adv_center()' stop end select ! do the vertical advection step which will be used for prescribed ! vertical flow velocity and for settling of suspended matter. if (mode.eq.0) then ! non-conservative do k=1,N Y(k)=Y(k)-1./float(it)*dt*((cu(k)-cu(k-1))/ & h(k)-Y(k)*(ww(k)-ww(k-1))/h(k)) enddo else ! conservative do k=1,N Y(k)=Y(k)-1./float(it)*dt*((cu(k)-cu(k-1))/h(k)) enddo end if end do ! end of the iteration loop #ifdef DEBUG write(*,*) 'Leaving adv_center()' write(*,*) #endif return end subroutine adv_center !EOC !----------------------------------------------------------------------- ! Copyright by the GOTM-team under the GNU Public License - www.gnu.org !-----------------------------------------------------------------------