\name{fiadeiro}
\alias{fiadeiro}
\title{
Advective Finite Difference Weights
}
\description{
Weighing coefficients used in the finite difference scheme for advection
calculated according to Fiadeiro and Veronis (1977).
This particular AFDW (advective finite difference weights) scheme switches
from backward differencing (in advection dominated conditions; large Peclet
numbers) to central differencing (under diffusion dominated conditions;
small Peclet numbers).
This way it forms a compromise between stability, accuracy and reduced
numerical dispersion.
}
\usage{
fiadeiro(v, D, dx.aux = NULL, grid = list(dx.aux = dx.aux))
}
\arguments{
\item{v }{advective velocity; either one value or a vector of length N+1,
with N the number of grid cells [L/T]
}
\item{D }{diffusion coefficient; either one value or a vector of length N+1
[L2/T]
}
\item{dx.aux }{auxiliary vector containing the distances between the
locations where the concentration is defined (i.e. the grid cell centers
and the two outer interfaces);
either one value or a vector of length N+1 [L]
}
\item{grid }{discretization grid as calculated by \code{\link{setup.grid.1D}}
}
}
\value{
the Advective Finite Difference Weighing (AFDW) coefficients as used in
the transport routines \code{\link{tran.1D}} and \code{\link{tran.volume.1D}};
either one value or a vector of length N+1 [-]
}
\author{
Filip Meysman ,
Karline Soetaert
}
\examples{
#============================================
# Model formulation (differential equations)
#============================================
# This is a test model to evaluate the different finite difference schemes
# and evaluate their effect on munerical diffusion. The model describes the
# decay of organic carbon (OC) as it settles through the ocean water column.
model <- function (time, OC, pars, AFDW = 1) {
dOC <- tran.1D(OC, flux.up = F_OC, D = D.eddy,
v = v.sink, AFDW = AFDW, dx = dx)$dC - k*OC
return(list(dOC))
}
#============================================
# Parameter set
#============================================
L <- 1000 # water depth model domain [m]
x.att <- 200 # attenuation depth of the sinking velocity [m]
v.sink.0 <- 10 # sinking velocity at the surface [m d-1]
D.eddy <- 10 # eddy diffusion coefficient [m2 d-1]
F_OC <- 10 # particle flux [mol m-2 d-1]
k <- 0.1 # decay coefficient [d-1]
## =============================================================================
## Model solution for a coarse grid (10 grid cells)
## =============================================================================
# Setting up the grid
N <- 10 # number of grid layers
dx <- L/N # thickness of boxes [m]
dx.aux <- rep(dx, N+1) # auxilliary grid vector
x.int <- seq(from = 0, to = L, by = dx) # water depth at box interfaces [m]
x.mid <- seq(from = dx/2, to = L, by = dx) # water depth at box centres [m]
# Exponentially declining sink velocity
v.sink <- v.sink.0 * exp(-x.int/x.att) # sink velocity [m d-1]
Pe <- v.sink * dx/D.eddy # Peclet number
# Calculate the weighing coefficients
AFDW <- fiadeiro(v = v.sink, D = D.eddy, dx.aux = dx.aux)
par(mfrow = c(2, 1), cex.main = 1.2, cex.lab = 1.2)
# Plot the Peclet number over the grid
plot(Pe, x.int, log = "x", pch = 19, ylim = c(L,0), xlim = c(0.1, 1000),
xlab = "", ylab = "depth [m]",
main = "Peclet number", axes = FALSE)
abline(h = 0)
axis(pos = NA, side = 2)
axis(pos = 0, side = 3)
# Plot the AFDW coefficients over the grid
plot(AFDW, x.int, pch = 19, ylim = c(L, 0), xlim = c(0.5, 1),
xlab = "", ylab = "depth [m]", main = "AFDW coefficient", axes = FALSE)
abline(h = 0)
axis(pos = NA, side = 2)
axis(pos = 0, side = 3)
# Three steady-state solutions for a coarse grid based on:
# (1) backward differences (BD)
# (2) central differences (CD)
# (3) Fiadeiro & Veronis scheme (FV)
BD <- steady.1D(y = runif(N), func = model, AFDW = 1.0, nspec = 1)
CD <- steady.1D(y = runif(N), func = model, AFDW = 0.5, nspec = 1)
FV <- steady.1D(y = runif(N), func = model, AFDW = AFDW, nspec = 1)
# Plotting output - use rootSolve's plot method
plot(BD, CD, FV, grid = x.mid, xyswap = TRUE, mfrow = c(1,2),
xlab = "", ylab = "depth [m]", main = "conc (Low resolution grid)")
legend("bottomright", col = 1:3, lty = 1:3,
legend = c("backward diff", "centred diff", "Fiadeiro&Veronis"))
## =============================================================================
## Model solution for a fine grid (1000 grid cells)
## =============================================================================
# Setting up the grid
N <- 1000 # number of grid layers
dx <- L/N # thickness of boxes[m]
dx.aux <- rep(dx, N+1) # auxilliary grid vector
x.int <- seq(from = 0, to = L, by = dx) # water depth at box interfaces [m]
x.mid <- seq(from = dx/2, to = L, by = dx) # water depth at box centres [m]
# Exponetially declining sink velocity
v.sink <- v.sink.0 * exp(-x.int/x.att) # sink velocity [m d-1]
Pe <- v.sink * dx/D.eddy # Peclet number
# Calculate the weighing coefficients
AFDW <- fiadeiro(v = v.sink, D = D.eddy, dx.aux = dx.aux)
# Three steady-state solutions for a coarse grid based on:
# (1) backward differences (BD)
# (2) centered differences (CD)
# (3) Fiadeiro & Veronis scheme (FV)
BD <- steady.1D(y = runif(N), func = model, AFDW = 1.0, nspec = 1)
CD <- steady.1D(y = runif(N), func = model, AFDW = 0.5, nspec = 1)
FV <- steady.1D(y = runif(N), func = model, AFDW = AFDW, nspec = 1)
# Plotting output
plot(BD, CD, FV, grid = x.mid, xyswap = TRUE, mfrow = NULL,
xlab = "", ylab = "depth [m]", main = "conc (High resolution grid)")
legend("bottomright", col = 1:3, lty = 1:3,
legend = c("backward diff", "centred diff", "Fiadeiro&Veronis"))
# Results and conclusions:
# - For the fine grid, all three solutions are identical
# - For the coarse grid, the BD and FV solutions show numerical dispersion
# while the CD provides more accurate results
}
\references{
\itemize{
\item Fiadeiro ME and Veronis G (1977) Weighted-mean schemes for
finite-difference approximation to advection-diffusion equation.
Tellus 29, 512-522.
\item Boudreau (1997) Diagnetic models and their implementation.
Chapter 8: Numerical Methods. Springer.
}
}
\details{
The Fiadeiro and Veronis (1977) scheme adapts the differencing method to
the local situation (checks for advection or diffusion dominance).
Finite difference schemes are based on following rationale:
\itemize{
\item When using forward differences (AFDW = 0), the scheme is first
order accurate, creates a low level of (artificial) numerical dispersion,
but is highly unstable (state variables may become negative).
\item When using backward differences (AFDW = 1), the scheme is first
order accurate, is universally stable (state variables always remain
positive), but the scheme creates high levels of numerical dispersion.
\item When using central differences (AFDW = 0.5), the scheme is second
order accurate, is not universally stable, and has a moderate level of
numerical dispersion, but state variables may become negative.
}
Because of the instability issue, forward schemes should be avoided.
Because of the higher accuracy, the central scheme is preferred over the
backward scheme.
The central scheme is stable when sufficient physical dispersion is present,
it may become unstable when advection is the only transport process.
The Fiadeiro and Veronis (1977) scheme takes this into account: it uses
central differencing when possible (when physical dispersion is high enough),
and switches to backward differing when needed (when advection dominates).
The switching is determined by the Peclet number
\code{Pe = abs(v)*dx.aux/D}
\itemize{
\item the higher the diffusion \code{D} (\code{Pe > 1}), the closer the
AFDW coefficients are to 0.5 (central differencing)
\item the higher the advection \code{v} (\code{Pe < 1}), the closer the
AFDW coefficients are to 1 (backward differencing)
}
}
\note{
\itemize{
\item If the state variables (concentrations) decline in the direction
of the 1D axis, then the central difference scheme will be stable.
If this is known a prioiri, then central differencing is
preferred over the fiadeiro scheme.
\item Each scheme will always create some numerical diffusion. This
principally depends on the resolution of the grid (i.e. larger
\code{dx.aux} values create higher numerical diffusion). In order to reduce numerical dispersion, one should
increase the grid resolution (i.e. decrease \code{dx.aux}).
}
}
\keyword{utilities}