Description Usage Arguments Details Value Note Author(s) References See Also Examples
Estimates the transport term (i.e. the rate of change of a concentration due to diffusion and advection) in a twodimensional model domain.
1 2 3 4 5 6 7 8 9 10 11 12 13  tran.2D (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, a.bl.x.down = NULL,
a.bl.y.up = NULL, a.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)

C 
concentration, expressed per unit volume, defined at the centre of each grid cell; Nx*Ny matrix [M/L3]. 
C.x.up 
concentration at upstream boundary in xdirection; vector of length Ny [M/L3]. 
C.x.down 
concentration at downstream boundary in xdirection; vector of length Ny [M/L3]. 
C.y.up 
concentration at upstream boundary in ydirection; vector of length Nx [M/L3]. 
C.y.down 
concentration at downstream boundary in ydirection; vector of length Nx [M/L3]. 
flux.x.up 
flux across the upstream boundary in xdirection, positive = INTO model domain; vector of length Ny [M/L2/T]. 
flux.x.down 
flux across the downstream boundary in xdirection, positive = OUT of model domain; vector of length Ny [M/L2/T]. 
flux.y.up 
flux across the upstream boundary in ydirection, positive = INTO model domain; vector of length Nx [M/L2/T]. 
flux.y.down 
flux across the downstream boundary in ydirection, positive = OUT of model domain; vector of length Nx [M/L2/T]. 
a.bl.x.up 
transfer coefficient across the upstream boundary layer. in xdirection;

a.bl.x.down 
transfer coefficient across the downstream boundary layer in xdirection;

a.bl.y.up 
transfer coefficient across the upstream boundary layer. in ydirection;

a.bl.y.down 
transfer coefficient across the downstream boundary layer in ydirection;

D.grid 
diffusion coefficient defined on all grid cell
interfaces. A 
D.x 
diffusion coefficient in xdirection, defined on grid cell
interfaces. One value, a vector of length (Nx+1),
a 
D.y 
diffusion coefficient in ydirection, defined on grid cell
interfaces. One value, a vector of length (Ny+1),
a 
v.grid 
advective velocity defined on all grid cell
interfaces. Can be positive (downstream flow) or negative (upstream flow).
A 
v.x 
advective velocity in the xdirection, defined on grid cell
interfaces. Can be positive (downstream flow) or negative (upstream flow).
One value, a vector of length (Nx+1),
a 
v.y 
advective velocity in the ydirection, defined on grid cell
interfaces. Can be positive (downstream flow) or negative (upstream flow).
One value, a vector of length (Ny+1),
a 
AFDW.grid 
weight used in the finite difference scheme for advection
in the x and y direction, defined on grid cell interfaces; backward = 1,
centred = 0.5, forward = 0; default is backward.
A 
AFDW.x 
weight used in the finite difference scheme for advection
in the xdirection, defined on grid cell interfaces; backward = 1,
centred = 0.5, forward = 0; default is backward.
One value, a vector of length (Nx+1),
a 
AFDW.y 
weight used in the finite difference scheme for advection
in the ydirection, defined on grid cell interfaces; backward = 1,
centred = 0.5, forward = 0; default is backward.
One value, a vector of length (Ny+1),
a 
VF.grid 
Volume fraction. A 
VF.x 
Volume fraction at the grid cell interfaces in the xdirection.
One value, a vector of length (Nx+1),
a 
VF.y 
Volume fraction at the grid cell interfaces in the ydirection.
One value, a vector of length (Ny+1),
a 
A.grid 
Interface area. A 
A.x 
Interface area defined at the grid cell interfaces in
the xdirection. One value, a vector of length (Nx+1),
a 
A.y 
Interface area defined at the grid cell interfaces in
the ydirection. One value, a vector of length (Ny+1),
a 
dx 
distance between adjacent cell interfaces in the xdirection (thickness of grid cells). One value or vector of length Nx [L]. 
dy 
distance between adjacent cell interfaces in the ydirection (thickness of grid cells). One value or vector of length Ny [L]. 
grid 
discretization grid, a list containing at least elements

full.check 
logical flag enabling a full check of the consistency
of the arguments (default = 
full.output 
logical flag enabling a full return of the output
(default = 
The boundary conditions are either
(1) zerogradient
(2) fixed concentration
(3) convective boundary layer
(4) fixed flux
This is also the order of priority. The zero gradient is the default, the fixed flux overrules all other.
a list containing:
dC 
the rate of change of the concentration C due to transport, defined in the centre of each grid cell, a Nx*Ny matrix. [M/L3/T]. 
C.x.up 
concentration at the upstream interface in xdirection.
A vector of length Ny [M/L3]. Only when 
C.x.down 
concentration at the downstream interface in xdirection.
A vector of length Ny [M/L3]. Only when 
C.y.up 
concentration at the the upstream interface in ydirection.
A vector of length Nx [M/L3]. Only when 
C.y.down 
concentration at the downstream interface in ydirection.
A vector of length Nx [M/L3]. Only when 
x.flux 
flux across the interfaces in xdirection of the grid cells.
A (Nx+1)*Ny matrix [M/L2/T]. Only when 
y.flux 
flux across the interfaces in ydirection of the grid cells.
A Nx*(Ny+1) matrix [M/L2/T]. Only when 
flux.x.up 
flux across the upstream boundary in xdirection, positive = INTO model domain. A vector of length Ny [M/L2/T]. 
flux.x.down 
flux across the downstream boundary in xdirection, positive = OUT of model domain. A vector of length Ny [M/L2/T]. 
flux.y.up 
flux across the upstream boundary in ydirection, positive = INTO model domain. A vector of length Nx [M/L2/T]. 
flux.y.down 
flux across the downstream boundary in ydirection, positive = OUT of model domain. A vector of length Nx [M/L2/T]. 
It is much more efficient to use the grid input rather than vectors or single numbers.
Thus: to optimise the code, use setup.grid.2D to create the
grid
, and use setup.prop.2D to create D.grid
,
v.grid
, AFDW.grid
, VF.grid
, and A.grid
,
even if the values are 1 or remain constant.
There is no provision (yet) to deal with crossdiffusion.
Set D.x
and D.y
different only if crossdiffusion effects
are unimportant.
Filip Meysman <[email protected]>, Karline Soetaert <[email protected]>
Soetaert and Herman, 2009. a practical guide to ecological modelling  using R as a simulation platform. Springer
tran.polar
for a discretisation of 2D transport equations
in polar coordinates
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## Testing the functions
## =============================================================================
# Parameters
F < 100 # input flux [micromol cm2 yr1]
por < 0.8 # constant porosity
D < 400 # mixing coefficient [cm2 yr1]
v < 1 # advective velocity [cm yr1]
# Grid definition
x.N < 4 # number of cells in xdirection
y.N < 6 # number of cells in ydirection
x.L < 8 # domain size xdirection [cm]
y.L < 24 # domain size ydirection [cm]
dx < x.L/x.N # cell size xdirection [cm]
dy < y.L/y.N # cell size ydirection [cm]
# Intial conditions
C < matrix(nrow = x.N, ncol = y.N, data = 0, byrow = FALSE)
# Boundary conditions: fixed concentration
C.x.up < rep(1, times = y.N)
C.x.down < rep(0, times = y.N)
C.y.up < rep(1, times = x.N)
C.y.down < rep(0, times = x.N)
# Only diffusion
tran.2D(C = C, D.x = D, D.y = D, v.x = 0, v.y = 0,
VF.x = por, VF.y = por, dx = dx, dy = dy,
C.x.up = C.x.up, C.x.down = C.x.down,
C.y.up = C.y.up, C.y.down = C.y.down, full.output = TRUE)
# Strong advection, backward (default), central and forward
#finite difference schemes
tran.2D(C = C, D.x = D, v.x = 100*v,
VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, C.x.down = C.x.down,
C.y.up = C.y.up, C.y.down = C.y.down)
tran.2D(AFDW.x = 0.5, C = C, D.x = D, v.x = 100*v,
VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, C.x.down = C.x.down,
C.y.up = C.y.up, C.y.down = C.y.down)
tran.2D(AFDW.x = 0, C = C, D.x = D, v.x = 100*v,
VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, C.x.down = C.x.down,
C.y.up = C.y.up, C.y.down = C.y.down)
# Boundary conditions: fixed fluxes
flux.x.up < rep(200, times = y.N)
flux.x.down < rep(200, times = y.N)
flux.y.up < rep(200, times = x.N)
flux.y.down < rep(200, times = x.N)
tran.2D(C = C, D.x = D, v.x = 0,
VF.x = por, dx = dx, dy = dy,
flux.x.up = flux.x.up, flux.x.down = flux.x.down,
flux.y.up = flux.y.up, flux.y.down = flux.y.down)
# Boundary conditions: convective boundary layer on all sides
a.bl < 800 # transfer coefficient
C.x.up < rep(1, times = (y.N)) # fixed conc at boundary layer
C.y.up < rep(1, times = (x.N)) # fixed conc at boundary layer
tran.2D(full.output = TRUE, C = C, D.x = D, v.x = 0,
VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, a.bl.x.up = a.bl,
C.x.down = C.x.up, a.bl.x.down = a.bl,
C.y.up = C.y.up, a.bl.y.up = a.bl,
C.y.down = C.y.up, a.bl.y.down = a.bl)
# Runtime test with and without argument checking
n.iterate <500
test1 < function() {
for (i in 1:n.iterate )
ST < tran.2D(full.check = TRUE, C = C, D.x = D,
v.x = 0, VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, a.bl.x.up = a.bl, C.x.down = C.x.down)
}
system.time(test1())
test2 < function() {
for (i in 1:n.iterate )
ST < tran.2D(full.output = TRUE, C = C, D.x = D,
v.x = 0, VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, a.bl.x.up = a.bl, C.x.down = C.x.down)
}
system.time(test2())
test3 < function() {
for (i in 1:n.iterate )
ST < tran.2D(full.output = TRUE, full.check = TRUE, C = C,
D.x = D, v.x = 0, VF.x = por, dx = dx, dy = dy,
C.x.up = C.x.up, a.bl.x.up = a.bl, C.x.down = C.x.down)
}
system.time(test3())
## =============================================================================
## A 2D model with diffusion in x and y direction and firstorder
## consumption  unefficient implementation
## =============================================================================
N < 51 # number of grid cells
XX < 10 # total size
dy < dx < XX/N # grid size
Dy < Dx < 0.1 # diffusion coeff, X and Ydirection
r < 0.005 # consumption rate
ini < 1 # initial value at x=0
N2 < ceiling(N/2)
X < seq (dx, by = dx, len = (N21))
X < c(rev(X), 0, X)
# The model equations
Diff2D < function (t, y, parms) {
CONC < matrix(nrow = N, ncol = N, y)
dCONC < tran.2D(CONC, D.x = Dx, D.y = Dy, dx = dx, dy = dy)$dC + r * CONC
return (list(dCONC))
}
# initial condition: 0 everywhere, except in central point
y < matrix(nrow = N, ncol = N, data = 0)
y[N2, N2] < ini # initial concentration in the central point...
# solve for 10 time units
times < 0:10
out < ode.2D (y = y, func = Diff2D, t = times, parms = NULL,
dim = c(N,N), lrw = 160000)
pm < par (mfrow = c(2, 2))
# Compare solution with analytical solution...
for (i in seq(2, 11, by = 3)) {
tt < times[i]
mat < matrix(nrow = N, ncol = N,
data = subset(out, time == tt))
plot(X, mat[N2,], type = "l", main = paste("time=", times[i]),
ylab = "Conc", col = "red")
ana < ini*dx^2/(4*pi*Dx*tt)*exp(r*ttX^2/(4*Dx*tt))
points(X, ana, pch = "+")
}
legend ("bottom", col = c("red","black"), lty = c(1, NA),
pch = c(NA, "+"), c("tran.2D", "exact"))
par("mfrow" = pm )
## =============================================================================
## A 2D model with diffusion in x and y direction and firstorder
## consumption  more efficient implementation, specifying ALL 2D grids
## =============================================================================
N < 51 # number of grid cells
Dy < Dx < 0.1 # diffusion coeff, X and Ydirection
r < 0.005 # consumption rate
ini < 1 # initial value at x=0
x.grid < setup.grid.1D(x.up = 5, x.down = 5, N = N)
y.grid < setup.grid.1D(x.up = 5, x.down = 5, N = N)
grid2D < setup.grid.2D(x.grid, y.grid)
D.grid < setup.prop.2D(value = Dx, y.value = Dy, grid = grid2D)
v.grid < setup.prop.2D(value = 0, grid = grid2D)
A.grid < setup.prop.2D(value = 1, grid = grid2D)
AFDW.grid < setup.prop.2D(value = 1, grid = grid2D)
VF.grid < setup.prop.2D(value = 1, grid = grid2D)
# The model equations  using the grids
Diff2Db < function (t, y, parms) {
CONC < matrix(nrow = N, ncol = N, data = y)
dCONC < tran.2D(CONC, grid = grid2D, D.grid = D.grid,
A.grid = A.grid, VF.grid = VF.grid, AFDW.grid = AFDW.grid,
v.grid = v.grid)$dC + r * CONC
return (list(dCONC))
}
# initial condition: 0 everywhere, except in central point
y < matrix(nrow = N, ncol = N, data = 0)
y[N2,N2] < ini # initial concentration in the central point...
# solve for 8 time units
times < 0:8
outb < ode.2D (y = y, func = Diff2Db, t = times, parms = NULL,
dim = c(N, N), lrw = 160000)
image(outb, ask = FALSE, mfrow = c(3, 3), main = paste("time", times))
## =============================================================================
## Same 2D model, but now with spatiallyvariable diffusion coefficients
## =============================================================================
N < 51 # number of grid cells
r < 0.005 # consumption rate
ini < 1 # initial value at x=0
N2 < ceiling(N/2)
D.grid < list()
# Diffusion on xinterfaces
D.grid$x.int < matrix(nrow = N+1, ncol = N, data = runif(N*(N+1)))
# Diffusion on yinterfaces
D.grid$y.int < matrix(nrow = N, ncol = N+1, data = runif(N*(N+1)))
dx < 10/N
dy < 10/N
# The model equations
Diff2Dc < function (t, y, parms) {
CONC < matrix(nrow = N, ncol = N, data = y)
dCONC < tran.2D(CONC, dx = dx, dy = dy, D.grid = D.grid)$dC + r * CONC
return (list(dCONC))
}
# initial condition: 0 everywhere, except in central point
y < matrix(nrow = N, ncol = N, data = 0)
y[N2, N2] < ini # initial concentration in the central point...
# solve for 8 time units
times < 0:8
outc < ode.2D (y = y, func = Diff2Dc, t = times, parms = NULL,
dim = c(N, N), lrw = 160000)
outtimes < c(1, 3, 5, 7)
image(outc, ask = FALSE, mfrow = c(2, 2), main = paste("time", outtimes),
legend = TRUE, add.contour = TRUE, subset = time %in% outtimes)

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