tran.2D.R
##==============================================================================
## Transport in a two-dimensional finite difference grid
##==============================================================================
tran.2D <- function(C, C.x.up=C[1,], C.x.down=C[nrow(C),],
C.y.up=C[,1], C.y.down=C[,ncol(C)],
flux.x.up=NULL, flux.x.down=NULL, flux.y.up=NULL, flux.y.down=NULL,
a.bl.x.up=NULL, C.bl.x.up=NULL, a.bl.x.down=NULL, C.bl.x.down=NULL,
a.bl.y.up=NULL, C.bl.y.up=NULL, a.bl.y.down=NULL, C.bl.y.down=NULL,
D.grid=NULL, D.x=NULL, D.y=D.x, v.grid=NULL, v.x=0, v.y=0,
AFDW.grid=NULL, AFDW.x=1, AFDW.y=AFDW.x,
VF.grid=NULL,VF.x=1, VF.y=VF.x,
A.grid=NULL, A.x=1, A.y=1,
grid=NULL,dx=NULL, dy=NULL,
full.check = FALSE, full.output = FALSE)
{
if (is.null(grid))
if (is.null(dx) | is.null(dy))
stop("error: either grid or dx and dy should be specified ")
Nx <- nrow(C)
Ny <- ncol(C)
# DEFAULT INFILLING OF GRID PARAMETERS
#==============================================================================
# infilling of 2D numerical grid
#==============================================================================
if (is.null(grid)) {
DX <- if (is.list(dx)) dx$dx else rep(dx,length.out=Nx)
DXaux <- if (is.list(dx)) dx$dx.aux else 0.5*(c(0,rep(dx,length.out=Nx))+
c(rep(dx,length.out=Nx),0))
DY <- if (is.list(dy)) dy$dx else rep(dy,length.out=Ny)
DYaux <- if (is.list(dy)) dy$dx.aux else 0.5*(c(0,rep(dy,length.out=Ny))+
c(rep(dy,length.out=Ny),0))
grid <- list(
dx = DX, dx.aux= DXaux,
dy = DY, dy.aux= DYaux
)
}
#==============================================================================
# infilling of grids with x.int, y.int, x.mid, y.mid needed
#==============================================================================
gridFill <- function(G.x,G.y,Name) # define a function first
{
# check if G.x and G.y is not NULL
if (is.null(G.x) | is.null(G.y))
stop( (paste("error: ",Name,"and (",Name,".x and", Name,".y) cannot be NULL at the same time", del="")))
G.grid <- list()
# infilling of x-matrix
if (is.matrix(G.x)) {
if (sum(abs(dim(G.x) - c(Nx+1,Ny)))!=0)
stop (paste("error: ",Name,".x matrix not of correct (Nx+1) Ny dimensions", del=""))
G.grid$x.int <- G.x
G.grid$x.mid <- 0.5*(G.x[1:Nx,]+G.x[2:(Nx+1),])
} else if (class(G.x)=="prop.1D") {
G.grid$x.int <- matrix(data=G.x$int,nrow=(Nx+1),ncol=Ny)
G.grid$x.mid <- matrix(data=G.x$mid, nrow=Nx, ncol=Ny)
} else if (length(G.x) == 1) {
G.grid$x.int <- matrix(data=G.x,nrow=(Nx+1),ncol=Ny)
G.grid$x.mid <- matrix(data=G.x,nrow=Nx,ncol=Ny)
} else if (length(G.x) != Nx+1) {
stop (paste("error: ",Name,".x should be a vector of length 1 or Nx+1", del=""))
} else { # correct length
G.grid$x.int <- matrix(data=G.x,nrow=(Nx+1),ncol=Ny)
G.grid$x.mid <- matrix(data=0.5*(G.x[1:Nx] +G.x[2:(Nx+1)]),
nrow=Nx, ncol=Ny)
}
# infilling of y-matrix
if (is.matrix(G.y)) {
if (sum(abs(dim(G.y) - c(Nx,Ny+1)))!=0)
stop (paste("error: ",Name,".y matrix not of correct Nx(Ny+1)dimensions", del=""))
G.grid$y.int <- G.y
G.grid$y.mid <- 0.5*(G.y[,1:Ny]+G.y[,2:(Ny+1)])
} else if (class(G.y)=="prop.1D") {
G.grid$y.int <- matrix(data=G.y$int,nrow=Nx,ncol=(Ny+1))
G.grid$y.mid <- matrix(data=G.y$mid, nrow=Nx, ncol=Ny)
} else if (length(G.y) == 1) {
G.grid$y.int <- matrix(data=G.y,nrow=Nx,ncol=(Ny+1))
G.grid$y.mid <- matrix(data=G.y,nrow=Nx,ncol=Ny)
} else if (length(G.y) != Ny+1) {
stop (paste("error: ",Name,".y should be a vector of length 1 or Ny+1", del=""))
} else { # correct length
G.grid$y.int <- matrix(data=G.y,nrow=Nx,ncol=(Ny+1))
G.grid$y.mid <- matrix(data=0.5*(G.y[1:Nx] +G.y[2:(Nx+1)]),
nrow=Nx, ncol=Ny)
}
G.grid
}
# Need this for VF and A (volume fraction and surface
if (is.null(VF.grid)) VF.grid <- gridFill(VF.x,VF.y,"VF")
if (is.null(A.grid)) A.grid <- gridFill(A.x,A.y,"A")
#==============================================================================
# infilling of other grids with only x.int and y.int needed
#==============================================================================
gridInt <- function(G.x,G.y,Name) # define a function first
{
# check if G.x and G.y is not NULL
if (is.null(G.x) | is.null(G.y))
stop( (paste("error: ",Name,"and (",Name,".x and", Name,".y) cannot be NULL at the same time", del="")))
G.grid <- list()
# infilling of x-matrix
if (is.matrix(G.x)) {
if (sum(abs(dim(G.x) - c(Nx+1,Ny)))!=0)
stop (paste("error: ",Name,".x matrix not of correct (Nx+1) Ny dimensions", del=""))
G.grid$x.int <- G.x
} else if (class(G.x)=="prop.1D") {
G.grid$x.int <- matrix(data=G.x$int,nrow=(Nx+1),ncol=Ny)
} else if (length(G.x) == 1) {
G.grid$x.int <- matrix(data=G.x,nrow=(Nx+1),ncol=Ny)
} else if (length(G.x) != Nx+1) {
stop (paste("error: ",Name,".x should be a vector of length 1 or Nx+1", del=""))
} else { # correct length
G.grid$x.int <- matrix(data=G.x,nrow=(Nx+1),ncol=Ny)
}
# infilling of y-matrix
if (is.matrix(G.y)) {
if (sum(abs(dim(G.y) - c(Nx,Ny+1)))!=0)
stop (paste("error: ",Name,".y matrix not of correct Nx(Ny+1)dimensions", del=""))
G.grid$y.int <- G.y
} else if (class(G.y)=="prop.1D") {
G.grid$y.int <- matrix(data=G.y$int,nrow=Nx,ncol=(Ny+1))
} else if (length(G.y) == 1) {
G.grid$y.int <- matrix(data=G.y,nrow=Nx,ncol=(Ny+1))
} else if (length(G.y) != Ny+1) {
stop (paste("error: ",Name,".y should be a vector of length 1 or Ny+1", del=""))
} else { # correct length
G.grid$y.int <- matrix(data=G.y,nrow=Nx,ncol=(Ny+1))
}
G.grid
}
# Need this for AFDW , D and v
if (is.null(AFDW.grid)) AFDW.grid <- gridInt(AFDW.x,AFDW.y,"AFDW")
if (is.null(D.grid)) D.grid <- gridInt(D.x,D.y,"D")
if (is.null(v.grid)) v.grid <- gridInt(v.x,v.y,"v")
#==============================================================================
# INPUT CHECKS
#==============================================================================
if (full.check) {
## check dimensions of input concentrations
if (!is.null(C.x.up)) {
if (!((length(C.x.up)==1) || (length(C.x.up)==(Ny))))
stop("error: C.x.up should be a vector of length 1 or ncol(C)")
}
if (!is.null(C.x.down)) {
if (!((length(C.x.down)==1) || (length(C.x.down)==(Ny))))
stop("error: C.x.down should be a vector of length 1 or ncol(C)")
}
if (!is.null(C.y.up)) {
if (!((length(C.y.up)==1) || (length(C.y.up)==(Nx))))
stop("error: C.y.up should be a vector of length 1 or nrow(C)")
}
if (!is.null(C.y.down)) {
if (!((length(C.y.down)==1) || (length(C.y.down)==(Nx))))
stop("error: C.y.down should be a vector of length 1 or nrow(C)")
}
if (!is.null(C.bl.x.up)) {
if (!((length(C.bl.x.up)==1) || (length(C.bl.x.up)==(Ny))))
stop("error: C.bl.x.up should be a vector of length 1 or ncol(C)")
}
if (!is.null(C.bl.x.down)) {
if (!((length(C.bl.x.down)==1) || (length(C.bl.x.down)==(Ny))))
stop("error: C.bl.x.down should be a vector of length 1 or ncol(C)")
}
if (!is.null(C.bl.y.up)) {
if (!((length(C.bl.y.up)==1) || (length(C.bl.y.up)==(Nx))))
stop("error: C.bl.y.up should be a vector of length 1 or nrow(C)")
}
if (!is.null(C.bl.y.down)) {
if (!((length(C.bl.y.down)==1) || (length(C.bl.y.down)==(Nx))))
stop("error: C.bl.y.down should be a vector of length 1 or nrow(C)")
}
# check dimensions of input fluxes
if (!is.null(flux.x.up)) {
if (!((length(flux.x.up)==1) || (length(flux.x.up)==(Ny))))
stop("error: flux.x.up should be a vector of length 1 or ncol(C)")
}
if (!is.null(flux.x.down)) {
if (!((length(flux.x.down)==1) || (length(flux.x.down)==(Ny))))
stop("error: flux.x.down should be a vector of length 1 or ncol(C)")
}
if (!is.null(flux.y.up)) {
if (!((length(flux.y.up)==1) || (length(flux.y.up)==(Nx))))
stop("error: flux.y.up should be a vector of length 1 or nrow(C)")
}
if (!is.null(flux.y.down)) {
if (!((length(flux.y.down)==1) || (length(flux.y.down)==(Nx))))
stop("error: flux.y.down should be a vector of length 1 or nrow(C)")
}
## check input of grid
if (is.null(dx) && is.null(dy) && is.null(grid))
stop("error: dx, dy, and grid cannot be NULL at the same time")
gn <- names(grid)
if (! "dx" %in% gn)
stop("error: grid should be a list that contains 'dx' ")
if (! "dx.aux" %in% gn)
stop("error: grid should be a list that contains 'dx.aux' ")
if (! "dy" %in% gn)
stop("error: grid should be a list that contains 'dy' ")
if (! "dy.aux" %in% gn)
stop("error: grid should be a list that contains 'dy.aux' ")
if (is.null(grid$dx) || is.null(grid$dx.aux))
stop("error: the grid should be a list with (numeric) values for 'dx' and 'dx.aux' ")
if (is.null(grid$dy) || is.null(grid$dy.aux))
stop("error: the grid should be a list with (numeric) values for 'dy' and 'dy.aux' ")
if (any(grid$dx <= 0) || any(grid$dx.aux <= 0) )
stop("error: the grid distances dx and dx.aux should always be positive")
if (any(grid$dy <= 0) || any(grid$dy.aux <= 0) )
stop("error: the grid distances dy and dy.aux should always be positive")
## check input of AFDW.grid
gn <- names(AFDW.grid)
if (! "x.int" %in% gn)
stop("error: AFDW.grid should be a list that contains 'x.int', the AFDW values at the interfaces of the grid cells in x-direction")
if (! "y.int" %in% gn)
stop("error: AFDW.grid should be a list that contains 'y.int', the AFDW values at the interfaces of the grid cells in y-direction")
if (is.null(AFDW.grid$x.int))
stop("error: AFDW.grid$x.int should be a list with (numeric) values")
if (is.null(AFDW.grid$y.int))
stop("error: AFDW.grid$y.int should be a list with (numeric) values")
if (any (AFDW.grid$x.int < 0)||any (AFDW.grid$x.int > 1))
stop("error: the AFDW should range between 0 and 1")
if (any (AFDW.grid$y.int < 0)||any (AFDW.grid$y.int > 1))
stop("error: the AFDW should range between 0 and 1")
## check input of D.grid
gn <- names(D.grid)
if (! "x.int" %in% gn)
stop("error: D.grid should be a list that contains 'x.int', the D values at the interfaces of the grid cells in x-direction")
if (! "y.int" %in% gn)
stop("error: D.grid should be a list that contains 'y.int', the D values at the interfaces of the grid cells in y-direction")
if (is.null(D.grid$x.int))
stop("error: D.grid$x.int should be a list with (numeric) values")
if (is.null(D.grid$y.int))
stop("error: D.grid$y.int should be a list with (numeric) values")
if (any (D.grid$x.int < 0)||any (D.grid$y.int < 0))
stop("error: the diffusion coefficient should always be positive")
## check input of v.grid
gn <- names(v.grid)
if (! "x.int" %in% gn)
stop("error: v.grid should be a list that contains 'x.int', the velocity values at the interfaces of the grid cells in x-direction")
if (! "y.int" %in% gn)
stop("error: v.grid should be a list that contains 'y.int', the velocity values at the interfaces of the grid cells in y-direction")
if (is.null(v.grid$x.int))
stop("error: the advective velocity v.grid$x.int should be a list with (numeric) values")
if (is.null(v.grid$y.int))
stop("error: the advective velocity v.grid$y.int should be a list with (numeric) values")
## check input of VF.grid
gn <- names(VF.grid)
if (! "x.int" %in% gn)
stop("error: VF.grid should be a list that contains 'x.int'")
if (! "y.int" %in% gn)
stop("error: VF.grid should be a list that contains 'y.int'")
if (! "x.mid" %in% gn)
stop("error: VF.grid should be a list that contains 'x.mid'")
if (! "y.mid" %in% gn)
stop("error: VF.grid should be a list that contains 'y.mid'")
if (is.null(VF.grid$x.int) || is.null(VF.grid$y.int) || is.null(VF.grid$x.mid) || is.null(VF.grid$y.mid))
stop("error: VF should contain (numeric) values")
if (any (VF.grid$x.int < 0) || any (VF.grid$y.int < 0) || any (VF.grid$x.mid < 0) || any (VF.grid$y.mid < 0))
stop("error: the VF values should always be positive")
## check input of A.grid
gn <- names(A.grid)
if (! "x.int" %in% gn)
stop("error: A.grid should be a list that contains 'x.int'")
if (! "y.int" %in% gn)
stop("error: A.grid should be a list that contains 'y.int'")
if (! "x.mid" %in% gn)
stop("error: A.grid should be a list that contains 'x.mid'")
if (! "y.mid" %in% gn)
stop("error: A.grid should be a list that contains 'y.mid'")
if (is.null(A.grid$x.int) || is.null(A.grid$y.int) || is.null(A.grid$x.mid) || is.null(A.grid$y.mid))
stop("error: the VF should contain (numeric) values")
if (any (A.grid$x.int < 0) || any (A.grid$y.int < 0) || any (A.grid$x.mid < 0) || any (A.grid$y.mid < 0))
stop("error: the A values should always be positive")
}
## FUNCTION BODY: CALCULATIONS
## Impose boundary flux at upstream x-boundary when needed
## Default boundary condition is no gradient
if (! is.null (flux.x.up[1])) {
nom <- flux.x.up + VF.grid$x.int[1,]*(D.grid$x.int[1,]/grid$dx.aux[1] +
(1-AFDW.grid$x.int[1,])*v.grid$x.int[1,])*C[1,]
denom <- VF.grid$x.int[1,]*(D.grid$x.int[1,]/grid$dx.aux[1]+
AFDW.grid$x.int[1,]*v.grid$x.int[1,])
C.x.up <- nom/denom
}
## Impose boundary flux at downstream x-boundary when needed
## Default boundary condition is no gradient
if (! is.null (flux.x.down[1])) {
nom <- flux.x.down - VF.grid$x.int[(Nx+1),]*(D.grid$x.int[(Nx+1),]/
grid$dx.aux[Nx+1] + AFDW.grid$x.int[(Nx+1),]*v.grid$x.int[(Nx+1),])*C[Nx,]
denom <- -VF.grid$x.int[(Nx+1),]*(D.grid$x.int[(Nx+1),]/grid$dx.aux[Nx+1]+
(1-AFDW.grid$x.int[(Nx+1),])*v.grid$x.int[(Nx+1),])
C.x.down <- nom/denom
}
# Impose boundary flux at upstream y-boundary when needed
# Default boundary condition is no gradient
if (! is.null (flux.y.up[1])) {
nom <- flux.y.up + VF.grid$y.int[,1]*(D.grid$y.int[,1]/grid$dy.aux[1] +
(1-AFDW.grid$y.int[,1])*v.grid$y.int[,1])*C[,1]
denom <- VF.grid$y.int[,1]*(D.grid$y.int[,1]/grid$dy.aux[1]+
AFDW.grid$y.int[,1]*v.grid$y.int[,1])
C.y.up <- nom/denom
}
# Impose boundary flux at downstream y-boundary when needed
# Default boundary condition is no gradient
if (! is.null (flux.y.down[1])) {
nom <- flux.y.down - VF.grid$y.int[,(Ny+1)]*(D.grid$y.int[,(Ny+1)]/
grid$dy.aux[Ny+1] + AFDW.grid$y.int[,(Ny+1)]*v.grid$y.int[,(Ny+1)])*C[,Ny]
denom <- -VF.grid$y.int[,(Ny+1)]*(D.grid$y.int[,(Ny+1)]/grid$dy.aux[Ny+1]+
(1-AFDW.grid$y.int[,(Ny+1)])*v.grid$y.int[,(Ny+1)])
C.y.down <- nom/denom
}
## when upper boundary layer is present, calculate new C.x.up
if (!is.null(a.bl.x.up) & !is.null(C.bl.x.up[1])) {
nom <- a.bl.x.up*C.bl.x.up + VF.grid$x.int[1,]*(D.grid$x.int[1,]/
grid$dx.aux[1] + (1-AFDW.grid$x.int[1,])*v.grid$x.int[1,])*C[1,]
denom <- a.bl.x.up + VF.grid$x.int[1,]*(D.grid$x.int[1,]/grid$dx.aux[1]+
AFDW.grid$x.int[1,]*v.grid$x.int[1,])
C.x.up <- nom/denom
}
## when lower boundary layer is present, calculate new C.x.down
if (!is.null(a.bl.x.down) & !is.null(C.bl.x.down[1])) {
nom <- a.bl.x.down*C.bl.x.down + VF.grid$x.int[(Nx+1),]*(D.grid$x.int[(Nx+1),]/
grid$dx.aux[(Nx+1)] + (1-AFDW.grid$x.int[(Nx+1),])*
v.grid$x.int[(Nx+1),])*C[Nx,]
denom <- a.bl.x.down + VF.grid$x.int[(Nx+1),]*(D.grid$x.int[(Nx+1),]/
grid$dx.aux[(Nx+1)]+ AFDW.grid$x.int[(Nx+1),]*v.grid$x.int[(Nx+1),])
C.x.down <- nom/denom
}
## when upper y boundary layer is present, calculate new C.y.up
if (!is.null(a.bl.y.up) & !is.null(C.bl.y.up[1])) {
nom <- a.bl.y.up*C.bl.y.up + VF.grid$y.int[,1]*(D.grid$y.int[,1]/
grid$dy.aux[1] + (1-AFDW.grid$y.int[,1])*v.grid$y.int[,1])*C[,1]
denom <- a.bl.y.up + VF.grid$y.int[,1]*(D.grid$y.int[,1]/grid$dy.aux[1]+
AFDW.grid$y.int[,1]*v.grid$y.int[,1])
C.y.up <- nom/denom
}
## when lower y boundary layer is present, calculate new C.y.down
if (!is.null(a.bl.y.down) & !is.null(C.bl.y.down[1])) {
nom <- a.bl.y.down*C.bl.y.down + VF.grid$y.int[,(Ny+1)]*
(D.grid$y.int[,(Ny+1)]/grid$dy.aux[(Ny+1)] +
(1-AFDW.grid$y.int[,(Ny+1)])*v.grid$y.int[,(Ny+1)])*C[,Ny]
denom <- a.bl.y.down + VF.grid$y.int[,(Ny+1)]*(D.grid$y.int[,(Ny+1)]/
grid$dy.aux[(Ny+1)]+ AFDW.grid$y.int[,(Ny+1)]*v.grid$y.int[,(Ny+1)])
C.y.down <- nom/denom
}
## Calculate diffusive part of the flux
x.Dif.flux <- as.matrix(-VF.grid$x.int * D.grid$x.int *
diff(rbind(C.x.up, C, C.x.down, deparse.level = 0))/
matrix(data=grid$dx.aux,nrow=(Nx+1),ncol=Ny,byrow=FALSE))
y.Dif.flux <- as.matrix(-VF.grid$y.int * D.grid$y.int *
t(diff(t(cbind(C.y.up,C,C.y.down,deparse.level = 0))))/
matrix(data=grid$dy.aux,nrow=Nx,ncol=(Ny+1),byrow=TRUE))
## Calculate advective part of the flux
x.Adv.flux <- 0
if (any(v.grid$x.int >0) ) {
vv <- v.grid$x.int
vv[vv<0]<-0
x.Adv.flux <- x.Adv.flux + as.matrix(VF.grid$x.int * vv * (
(1-AFDW.grid$x.int) * rbind(C.x.up,C,deparse.level = 0)
+ AFDW.grid$x.int * rbind(C,C.x.down,deparse.level = 0)))
}
if (any (v.grid$x.int < 0)) {
vv <- v.grid$x.int
vv[vv>0]<-0
x.Adv.flux <- x.Adv.flux + as.matrix(VF.grid$x.int * vv * (
AFDW.grid$x.int * rbind(C.x.up,C,deparse.level = 0)
+ (1-AFDW.grid$x.int) * rbind(C,C.x.down,deparse.level = 0)))
}
y.Adv.flux <- 0
if (any(v.grid$y.int >0) ) {
vv <- v.grid$y.int
vv[vv<0]<-0
y.Adv.flux <- y.Adv.flux + as.matrix(VF.grid$y.int * vv * (
(1-AFDW.grid$y.int) * cbind(C.y.up,C,deparse.level = 0)
+ AFDW.grid$y.int * cbind(C,C.y.down,deparse.level = 0)))
}
if (any (v.grid$y.int < 0)) {
vv <- v.grid$y.int
vv[vv>0]<-0
y.Adv.flux <- y.Adv.flux + as.matrix(VF.grid$y.int * vv * (
AFDW.grid$y.int * cbind(C.y.up,C,deparse.level = 0)
+ (1-AFDW.grid$y.int) * cbind(C,C.y.down,deparse.level = 0)))
}
x.flux <- x.Dif.flux + x.Adv.flux
y.flux <- y.Dif.flux + y.Adv.flux
## Impose boundary fluxes when needed
## Default boundary condition is no gradient
if (! is.null (flux.x.up[1]))
x.flux[1,] <- flux.x.up
if (! is.null (flux.x.down[1]))
x.flux[nrow(x.flux),] <- flux.x.down
if (! is.null (flux.y.up[1]))
y.flux[,1] <- flux.y.up
if (! is.null (flux.y.down[1]))
y.flux[,ncol(y.flux)] <- flux.y.down
## Calculate rate of change = flux gradient
dFdx <- - (diff(A.grid$x.int*x.flux) / A.grid$x.mid/grid$dx ) / VF.grid$x.mid
dFdy <- -t(diff(t(A.grid$y.int*y.flux))/t(A.grid$y.mid)/grid$dy) / VF.grid$y.mid
if (!full.output) {
return (list (dC = dFdx + dFdy, # Rate of change due to advective-diffuisve transport in each grid cell
flux.x.up = x.flux[1,], # flux across lower boundary interface; positive = IN
flux.x.down = x.flux[nrow(x.flux),], # flux across lower boundary interface; positive = OUT
flux.y.up = y.flux[,1], # flux across lower boundary interface; positive = IN
flux.y.down = y.flux[,ncol(y.flux)])) # flux across lower boundary interface; positive = OUT
} else {
return (list (dC = dFdx + dFdy, # Rate of change in the centre of each grid cells
C.x.up = C.x.up, # concentration at upper interface
C.x.down = C.x.down, # concentration at upper interface
C.y.up = C.y.up, # concentration at upper interface
C.y.down = C.y.down, # concentration at upper interface
x.flux = x.flux, # flux across at the interface of each grid cell
y.flux = y.flux, # flux across at the interface of each grid cell
flux.x.up = x.flux[1,], # flux across lower boundary interface; positive = IN
flux.x.down = x.flux[nrow(x.flux),], # flux across lower boundary interface; positive = OUT
flux.y.up = y.flux[,1], # flux across lower boundary interface; positive = IN
flux.y.down = y.flux[,ncol(y.flux)])) # flux across lower boundary interface; positive = OUT
}
} # end tran.2D
