# One-Dimensional Advection Equation Estimates the advection term in a one-dimensional model of a liquid (volume fraction constant and equal to one) or in a porous medium (volume fraction variable and lower than one). The interfaces between grid cells can have a variable cross-sectional area, e.g. when modelling spherical or cylindrical geometries (see example). TVD (total variation diminishing) slope limiters ensure monotonic and positive schemes in the presence of strong gradients. `advection.1-D`: uses finite differences. This implies the use of velocity (length per time) and fluxes (mass per unit of area per unit of time). `advection.volume.1D` Estimates the volumetric advection term in a one-dimensional model of an aquatic system (river, estuary). This routine is particularly suited for modelling channels (like rivers, estuaries) where the cross-sectional area changes, and hence the velocity changes. Volumetric transport implies the use of flows (mass per unit of time). When solved dynamically, the euler method should be used, unless the first-order upstream method is used. ```r advection.1D(C, C.up = NULL, C.down = NULL, flux.up = NULL, flux.down = NULL, v, VF = 1, A = 1, dx, dt.default = 1, adv.method = c("muscl", "super", "quick", "p3", "up"), full.check = FALSE) advection.volume.1D(C, C.up = C[1], C.down = C[length(C)], F.up = NULL, F.down = NULL, flow, V, dt.default = 1, adv.method = c("muscl", "super", "quick", "p3", "up"), full.check = FALSE) ``` ## Arguments - `C`: concentration, expressed per unit of phase volume, defined at the centre of each grid cell. A vector of length N [M/L3]. - `C.up`: concentration at upstream boundary. One value [M/L3]. If `NULL`, and `flux.up` is also `NULL`, then a zero-gradient boundary is assumed, i.e. `C.up = C[1]`. - `C.down`: concentration at downstream boundary. One value [M/L3]. If `NULL`, and `flux.down` is also `NULL`, then a zero-gradient boundary is assumed, i.e. `C.down = C[length(C)]`. - `flux.up`: flux across the upstream boundary, positive = INTO model domain. One value, expressed per unit of total surface [M/L2/T]. If `NULL`, the boundary is prescribed as a concentration boundary. - `flux.down`: flux across the downstream boundary, positive = OUT of model domain. One value, expressed per unit of total surface [M/L2/T]. If `NULL`, the boundary is prescribed as a concentration boundary. - `F.up`: total input across the upstream boundary, positive = INTO model domain; used with `advection.volume.1D`. One value, expressed in [M/T]. If `NULL`, the boundary is prescribed as a concentration boundary. - `F.down`: total input across the downstream boundary, positive = OUT of model domain; used with `advection.volume.1D`. One value, expressed in [M/T]. If `NULL`, the boundary is prescribed as a concentration boundary. - `v`: advective velocity, defined on the grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). One value, a vector of length N+1 [L/T], or a `1D property` list; the list contains at least the element `int` (see `setup.prop.1D`) [L/T]. Used with `advection.1D`. - `flow`: water flow rate, defined on grid cell interfaces. One value, a vector of length N+1, or a list as defined by `setup.prop.1D` [L^3/T]. Used with `advection.volume.1D`. - `VF`: Volume fraction defined at the grid cell interfaces. One value, a vector of length N+1, or a `1D property` list; the list contains at least the elements `int` and `mid` (see `setup.prop.1D`) [-]. - `A`: Interface area defined at the grid cell interfaces. One value, a vector of length N+1, or a `1D grid property` list; the list contains at least the elements `int` and `mid` (see `setup.prop.1D`) [L^2]. - `dx`: distance between adjacent cell interfaces (thickness of grid cells). One value, a vector of length N, or a `1D grid` list containing at least the elements `dx` and `dx.aux` (see `setup.grid.1D`) [L]. - `dt.default`: timestep to be used, if it cannot be estimated (e.g. when calculating steady-state conditions. - `V`: volume of cells. One value, or a vector of length N [L^3]. - `adv.method`: the advection method, slope limiter used to reduce the numerical dispersion. One of "quick","muscl","super","p3","up" - see details. - `full.check`: logical flag enabling a full check of the consistency of the arguments (default = `FALSE`; `TRUE` slows down execution by 50 percent). ## Returns - **dC**: the rate of change of the concentration C due to advective transport, defined in the centre of each grid cell. The rate of change is expressed per unit of (phase) volume [M/L^3/T]. - **adv.flux**: advective flux across at the interface of each grid cell. A vector of length N+1 [M/L2/T] - only for `advection.1D`. - **flux.up**: flux across the upstream boundary, positive = INTO model domain. One value [M/L2/T] - only for `advection.1D`. - **flux.down**: flux across the downstream boundary, positive = OUT of model domain. One value [M/L2/T] - only for `advection.1D`. - **adv.F**: advective mass flow across at the interface of each grid cell. A vector of length N+1 [M/T] - only for `advection.volume.1D`. - **F.up**: mass flow across the upstream boundary, positive = INTO model domain. One value [M/T] - only for `advection.volume.1D`. - **F.down**: flux across the downstream boundary, positive = OUT of model domain. One value [M/T] - only for `advection.volume.1D`. - **it**: number of split time iterations that were necessary. ## Author(s) Karline Soetaert ## Examples ```r ## ============================================================================= ## EXAMPLE 1: Testing the various methods - moving a square pulse ## use of advection.1D ## The tests as in Pietrzak ## ============================================================================= #--------------------# # Model formulation # #--------------------# model <- function (t, y, parms,...) { adv <- advection.1D(y, v = v, dx = dx, C.up = y[n], C.down = y[1], ...) # out on one side -> in at other return(list(adv$dC)) } #--------------------# # Parameters # #--------------------# n <- 100 dx <- 100/n y <- c(rep(1, 5), rep(2, 20), rep(1, n-25)) v <- 2 times <- 0:300 # 3 times out and in #--------------------# # model solution # #--------------------# ## a plotting function plotfun <- function (Out, ...) { plot(Out[1, -1], type = "l", col = "red", ylab = "y", xlab = "x", ...) lines(Out[nrow(Out), 2:(1+n)]) } # courant number = 2 pm <- par(mfrow = c(2, 2)) ## third order TVD, quickest limiter out <- ode.1D(y = y, times = times, func = model, parms = 0, hini = 1, method = "euler", nspec = 1, adv.method = "quick") plotfun(out, main = "quickest, euler") ## third-order ustream-biased polynomial out2 <- ode.1D(y = y, times = times, func = model, parms = 0, hini = 1, method = "euler", nspec = 1, adv.method = "p3") plotfun(out2, main = "p3, euler") ## third order TVD, superbee limiter out3 <- ode.1D(y = y, times = times, func = model, parms = 0, hini = 1, method = "euler", nspec = 1, adv.method = "super") plotfun(out3, main = "superbee, euler") ## third order TVD, muscl limiter out4 <- ode.1D(y = y, times = times, func = model, parms = 0, hini = 1, method = "euler", nspec = 1, adv.method = "muscl") plotfun(out4, main = "muscl, euler") ## ============================================================================= ## upstream, different time-steps , i.e. different courant number ## ============================================================================= out <- ode.1D(y = y, times = times, func = model, parms = 0, method = "euler", nspec = 1, adv.method = "up") plotfun(out, main = "upstream, courant number = 2") out2 <- ode.1D(y = y, times = times, func = model, parms = 0, hini = 0.5, method = "euler", nspec = 1, adv.method = "up") plotfun(out2, main = "upstream, courant number = 1") ## Now muscl scheme, velocity against x-axis y <- rev(c(rep(0, 5), rep(1, 20), rep(0, n-25))) v <- -2.0 out6 <- ode.1D(y = y, times = times, func = model, parms = 0, hini = 1, method = "euler", nspec = 1, adv.method = "muscl") plotfun(out6, main = "muscl, reversed velocity, , courant number = 1") image(out6, mfrow = NULL) par(mfrow = pm) ## ============================================================================= ## EXAMPLE 2: moving a square pulse in a widening river ## use of advection.volume.1D ## ============================================================================= #--------------------# # Model formulation # #--------------------# river.model <- function (t=0, C, pars = NULL, ...) { tran <- advection.volume.1D(C = C, C.up = 0, flow = flow, V = Volume,...) return(list(dCdt = tran$dC, F.down = tran$F.down, F.up = tran$F.up)) } #--------------------# # Parameters # #--------------------# # Initialising morphology river: nbox <- 100 # number of grid cells lengthRiver <- 100000 # [m] BoxLength <- lengthRiver / nbox # [m] Distance <- seq(BoxLength/2, by = BoxLength, len = nbox) # [m] # Cross sectional area: sigmoid function of distance [m2] CrossArea <- 4000 + 72000 * Distance^5 /(Distance^5+50000^5) # Volume of boxes (m3) Volume <- CrossArea*BoxLength # Transport coefficients flow <- 1000*24*3600 # m3/d, main river upstream inflow #--------------------# # Model solution # #--------------------# pm <- par(mfrow=c(2,2)) # a square pulse yini <- c(rep(10, 10), rep(0, nbox-10)) ## third order TVD, muscl limiter Conc <- ode.1D(y = yini, fun = river.model, method = "euler", hini = 1, parms = NULL, nspec = 1, times = 0:40, adv.method = "muscl") image(Conc, main = "muscl", mfrow = NULL) plot(Conc[30, 2:(1+nbox)], type = "l", lwd = 2, xlab = "x", ylab = "C", main = "muscl after 30 days") ## simple upstream differencing Conc2<- ode.1D(y = yini, fun = river.model, method = "euler", hini = 1, parms = NULL, nspec = 1, times = 0:40, adv.method = "up") image(Conc2, main = "upstream", mfrow = NULL) plot(Conc2[30, 2:(1+nbox)], type = "l", lwd = 2, xlab = "x", ylab = "C", main = "upstream after 30 days") par(mfrow = pm) # Note: the more sophisticated the scheme, the more mass lost/created # increase tolerances to improve this. CC <- Conc[ , 2:(1+nbox)] MASS <- t(CC)*Volume colSums(MASS) ## ============================================================================= ## EXAMPLE 3: A steady-state solution ## use of advection.volume.1D ## ============================================================================= Sink <- function (t, y, parms, ...) { C1 <- y[1:N] C2 <- y[(N+1):(2*N)] C3 <- y[(2*N+1):(3*N)] # Rate of change= Flux gradient and first-order consumption # upstream can be implemented in two ways: dC1 <- advection.1D(C1, v = sink, dx = dx, C.up = 100, adv.method = "up", ...)$dC - decay*C1 # same, using tran.1D # dC1 <- tran.1D(C1, v = sink, dx = dx, # C.up = 100, D = 0)$dC - # decay*C1 dC2 <- advection.1D(C2, v = sink, dx = dx, C.up = 100, adv.method = "p3", ...)$dC - decay*C2 dC3 <- advection.1D(C3, v = sink, dx = dx, C.up = 100, adv.method = "muscl", ...)$dC - decay*C3 list(c(dC1, dC2, dC3)) } N <- 10 L <- 1000 dx <- L/N # thickness of boxes sink <- 10 decay <- 0.1 out <- steady.1D(runif(3*N), func = Sink, names = c("C1", "C2", "C3"), parms = NULL, nspec = 3, bandwidth = 2) matplot(out$y, 1:N, type = "l", ylim = c(10, 0), lwd = 2, main = "Steady-state") legend("bottomright", col = 1:3, lty = 1:3, c("upstream", "p3", "muscl")) ``` ## References Pietrzak J (1998) The use of TVD limiters for forward-in-time upstream-biased advection schemes in ocean modeling. Monthly Weather Review 126: 812 .. 830 Hundsdorfer W and Verwer JG (2003) Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 471 pages Burchard H, Bolding K, Villarreal MR (1999) GOTM, a general ocean turbulence model. Theory, applications and test cases. Tech Rep EUR 18745 EN. European Commission Leonard BP (1988) Simple high accuracy resolution program for convective modeling of discontinuities. Int. J. Numer. Meth.Fluids 8: 1291--1318. Roe PL (1985) Some contributions to the modeling of discontinuous flows. Lect. Notes Appl. Math. 22: 163-193. van Leer B. (1979) Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method. J. Comput. Phys. 32: 101-136 ## Details This implementation is based on the GOTM code The boundary conditions are either * zero-gradient. * fixed concentration. * fixed flux. The above order also shows the priority. The default condition is the zero gradient. The fixed concentration condition overrules the zero gradient. The fixed flux overrules the other specifications. Ensure that the boundary conditions are well defined: for instance, it does not make sense to specify an influx in a boundary cell with the advection velocity pointing outward. Transport properties: The **advective velocity** (`v`), the **volume fraction** (VF), and the **interface surface** (`A`), can either be specified as one value, a vector, or a 1D property list as generated by `setup.prop.1D`. When a vector, this vector must be of length N+1, defined at all grid cell interfaces, including the upper and lower boundary. The finite difference grid (`dx`) is specified either as one value, a vector or a 1D grid list, as generated by `setup.grid.1D`. Several slope limiters are implemented to obtain monotonic and positive schemes also in the presence of strong gradients, i.e. to reduce the effect of numerical dispersion. The methods are (Pietrzak, 1989, Hundsdorfer and Verwer, 2003): * "quick": third-order scheme (TVD) with ULTIMATE QUICKEST limiter (quadratic upstream interpolation for convective kinematics with estimated stream terms) (Leonard, 1988) * "muscl": third-order scheme (TVD) with MUSCL limiter (monotonic upstream centered schemes for conservation laws) (van Leer, 1979). * "super": third-order scheme (TVD) with Superbee limiter (method=Superbee) (Roe, 1985) * "p3": third-order upstream-biased polynomial scheme (method=P3) * "up": first-order upstream ( method=UPSTREAM) - this is the same method as implemented in tran.1D or tran.volume.1D where "TVD" means a total variation diminishing scheme Some schemes may produce artificial steepening. Scheme "p3" is not necessarily monotone (may produce negative concentrations!). 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. The maximal number of such iterations is set to 100. These limiters are supposed to work with explicit methods (euler ). However, they will also work with implicit methods, although less effectively. Integrate ode.1D only if the model is stiff (see first example). ## Note The advective equation is not checked for mass conservation. Sometimes, this is not an issue, for instance when `v` represents a sinking velocity of particles or a swimming velocity of organisms. In others cases however, mass conservation needs to be accounted for. To ensure mass conservation, the advective velocity must obey certain continuity constraints: in essence the product of the volume fraction (VF), interface surface area (A) and advective velocity (v) should be constant. In sediments, one can use `setup.compaction.1D` to ensure that the advective velocities for the pore water and solid phase meet these constraints. In terms of the units of concentrations and fluxes we follow the convention in the geosciences. The concentration `C`, `C.up`, `C.down` as well at the rate of change of the concentration `dC` are always expressed per unit of phase volume (i.e. per unit volume of solid or liquid). Total concentrations (e.g. per unit volume of bulk sediment) can be obtained by multiplication with the appropriate volume fraction. In contrast, fluxes are always expressed per unit of total interface area (so here the volume fraction is accounted for). ## See Also `tran.1D`, for a discretisation of the general transport equations

advection function