diff --git a/doc/ADI_scheme.org b/doc/ADI_scheme.org
index 9408089..575249f 100644
--- a/doc/ADI_scheme.org
+++ b/doc/ADI_scheme.org
@@ -1,6 +1,7 @@
 #+TITLE: Finite Difference Schemes for the numerical solution of heterogeneous diffusion equation in 2D
 #+LaTeX_CLASS_OPTIONS: [a4paper,10pt]
 #+LATEX_HEADER: \usepackage{fullpage}
+#+LATEX_HEADER: \usepackage{charter}
 #+LATEX_HEADER: \usepackage{amsmath, systeme, cancel, xcolor}
 #+OPTIONS: toc:nil
 
diff --git a/doc/ADI_scheme.pdf b/doc/ADI_scheme.pdf
index 62f2f9f..1e389d7 100644
Binary files a/doc/ADI_scheme.pdf and b/doc/ADI_scheme.pdf differ
diff --git a/doc/ValidationHetDiff.org b/doc/ValidationHetDiff.org
index c9b5613..bb46e3a 100644
--- a/doc/ValidationHetDiff.org
+++ b/doc/ValidationHetDiff.org
@@ -1,12 +1,12 @@
-#+TITLE: 2D Validation Examples
+#+TITLE: Validation Examples for 2D Heterogeneous Diffusion
 #+AUTHOR: MDL <delucia@gfz-potsdam.de>
-#+DATE: 2023-07-31
+#+DATE: 2023-08-26
 #+STARTUP: inlineimages
 #+LATEX_CLASS_OPTIONS: [a4paper,9pt]
 #+LATEX_HEADER: \usepackage{fullpage}
 #+LATEX_HEADER: \usepackage{amsmath, systeme}
 #+LATEX_HEADER: \usepackage{graphicx}
-#+LATEX_HEADER: \usepackage{}
+#+LATEX_HEADER: \usepackage{charter}
 #+OPTIONS: toc:nil
 
 
@@ -38,7 +38,7 @@ constant in 4 quadrants:
 The relevant part of the R script used to produce these results is
 presented in listing 1; the whole script is at [[file:scripts/HetDiff.R]].
 A visualization of the output of the reference simulation is given in
-figure [[#fig:1][1]].
+figure [[fig:1][1]].
 
 Note: all results from this script are stored in the =outc= matrix by
 the =deSolve= function. I stored a different version into
@@ -47,68 +47,69 @@ for each time step including initial conditions) and 121 rows, one for
 each domain element, with no headers.
 
 #+caption: Result of ReacTran/deSolve solution of the above problem at 4
+#+name: fig:1
 [[./images/deSolve_AlphaHet1.png]]
 
 
 #+name: lst:1
 #+begin_src R :language R :frame single :caption Listing 1, generate reference simulation using R packages deSolve/ReacTran :captionpos b :label lst:1
-library(ReacTran)
-library(deSolve)
+  library(ReacTran)
+  library(deSolve)
 
-## harmonic mean
-harm <- function(x,y) {
-  if (length(x) != 1 || length(y) != 1)
-  stop("x & y have different lengths")
-  2/(1/x+1/y)
-}
-
-N     <- 11          # number of grid cells
-ini   <- 1           # initial value at x=0
-N2    <- ceiling(N/2)
-L     <- 10          # domain side
-
-## Define diff.coeff per cell, in  4 quadrants
-alphas <- matrix(0, N, N) 
-alphas[1:N2, 1:N2] <- 1
-alphas[1:N2, seq(N2+1,N)] <- 0.1
-alphas[seq(N2+1,N), 1:N2] <- 0.01
-alphas[seq(N2+1,N), seq(N2+1,N)] <- 0.001
-
-cmpharm <- function(x) {
-  y <- c(0, x, 0)
-  ret <- numeric(length(x)+1)
-  for (i in seq(2, length(y))) {
-    ret[i-1] <- harm(y[i], y[i-1])
+  ## harmonic mean
+  harm <- function(x,y) {
+    if (length(x) != 1 || length(y) != 1)
+    stop("x & y have different lengths")
+    2/(1/x+1/y)
   }
-  ret
-}
 
-## Construction of the 2D grid
-x.grid <- setup.grid.1D(x.up = 0, L = L, N = N)
-y.grid <- setup.grid.1D(x.up = 0, L = L, N = N)
-grid2D <- setup.grid.2D(x.grid, y.grid)
-dx <- dy <- L/N
+  N     <- 11          # number of grid cells
+  ini   <- 1           # initial value at x=0
+  N2    <- ceiling(N/2)
+  L     <- 10          # domain side
 
-D.grid <- list()
-## Diffusion coefs on x-interfaces
-D.grid$x.int <- apply(alphas, 1, cmpharm)
-## Diffusion coefs on y-interfaces
-D.grid$y.int <- t(apply(alphas, 2, cmpharm))
+  ## Define diff.coeff per cell, in  4 quadrants
+  alphas <- matrix(0, N, N) 
+  alphas[1:N2, 1:N2] <- 1
+  alphas[1:N2, seq(N2+1,N)] <- 0.1
+  alphas[seq(N2+1,N), 1:N2] <- 0.01
+  alphas[seq(N2+1,N), seq(N2+1,N)] <- 0.001
 
-# The model
-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
-  return(list(dCONC))
-}
+  cmpharm <- function(x) {
+    y <- c(0, x, 0)
+    ret <- numeric(length(x)+1)
+    for (i in seq(2, length(y))) {
+      ret[i-1] <- harm(y[i], y[i-1])
+    }
+    ret
+  }
 
-## 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...
+  ## Construction of the 2D grid
+  x.grid <- setup.grid.1D(x.up = 0, L = L, N = N)
+  y.grid <- setup.grid.1D(x.up = 0, L = L, N = N)
+  grid2D <- setup.grid.2D(x.grid, y.grid)
+  dx <- dy <- L/N
 
-## solve for 10 time units
-times <- 0:10
-outc <- ode.2D(y = y, func = Diff2Dc, t = times, parms = NULL,
-               dim = c(N, N), lrw = 1860000)
+  D.grid <- list()
+  ## Diffusion coefs on x-interfaces
+  D.grid$x.int <- apply(alphas, 1, cmpharm)
+  ## Diffusion coefs on y-interfaces
+  D.grid$y.int <- t(apply(alphas, 2, cmpharm))
+
+  # The model
+  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
+    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
+  outc <- ode.2D(y = y, func = Diff2Dc, t = times, parms = NULL,
+		 dim = c(N, N), lrw = 1860000)
 #+end_src
 
diff --git a/docs_sphinx/index.rst b/docs_sphinx/index.rst
index 740fbbc..1301f7e 100644
--- a/docs_sphinx/index.rst
+++ b/docs_sphinx/index.rst
@@ -5,12 +5,25 @@
 
 Welcome to Tug's documentation!
 ===============================
-Welcome to the documentation of the TUG project, a simulation program 
-for solving one- and two-dimensional diffusion problems with heterogeneous diffusion coefficients, more 
-generally, for solving the following differential equation
+
+Welcome to the documentation of the TUG project, a simulation program
+for solving transport equations in one- and two-dimensional uniform
+grids using cell centered finite differences.
+
+Diffusion
+-----------
+
+TUG can solve diffusion problems with heterogeneous and anisotropic
+diffusion coefficients. The partial differential equation expressing
+diffusion reads:
 
 .. math::
-   \frac{\partial C}{\partial t} = \alpha_x \frac{\partial^2 C}{\partial x^2} + \alpha_y \frac{\partial^2 C}{\partial y^2}.
+   \frac{\partial C}{\partial t} =  \nabla \cdot \left[ \mathbf{\alpha} \nabla C \right]
+
+In 2D, the equation reads:
+   
+.. math::
+   \frac{\partial C}{\partial t} =  \frac{\partial}{\partial x}\left[ \alpha_x \frac{\partial C}{\partial x}\right] + \frac{\partial}{\partial y}\left[ \alpha_y \frac{\partial C}{\partial y}\right]
 
 .. toctree::
    :maxdepth: 2