Minor edits to the positivity-preserving section of step-25

diff --git a/examples/step-26/doc/results.dox b/examples/step-26/doc/results.dox
index da1c4dccc8e7befeffe58ed8c26302f262db433b..23790cc7114296183af3dcf17728ea827393f65f 100644 (file)
--- a/examples/step-26/doc/results.dox
+++ b/examples/step-26/doc/results.dox
@@ -140,7 +140,9 @@ The general form of the $i$th equation then reads:
   a_{ii} u^{n}_i &= b_{ii} u^{n-1}_i +
   \sum\limits_{j \in S_i} \left( b_{ij} u^{n-1}_j - a_{ij} u^{n}_j \right),
 @f}
-where $S_i$ is the set of nearest neighbors to node $i$. If all coefficients
+where $S_i$ is the set of degrees of freedom that DoF $i$ couples with (i.e.,
+for which either the matrix $A$ or matrix $B$ has a nonzero entry at position
+$(i,j)$). If all coefficients
 fulfill the following conditions:
 @f{align*}
   a_{ii} &> 0, & b_{ii} &\leq 0, & a_{ij} &\leq 0, & b_{ij} &\geq 0,
@@ -158,9 +160,9 @@ the Crank-Nicolson scheme,
 <a href="https://doi.org/10.2478/cmam-2010-0025">Schatz et. al.</a> have
 translated it to the following ones:
 @f{align*}
-  (1 - \theta) k a_{ii} &\leq m_{ii},~ \forall i,
+  (1 - \theta) k a_{ii} &\leq m_{ii},\qquad \forall i,
   &
-  \theta k \left| a_{ij} \right| &\geq m_{ij},~ j \neq i,
+  \theta k \left| a_{ij} \right| &\geq m_{ij},\qquad j \neq i,
 @f}
 where $M = m_{ij}$ denotes the mass matrix and $A = a_{ij}$ the stiffness
 matrix with $a_{ij} \leq 0$ for $j \neq i$, respectively. With
@@ -173,30 +175,34 @@ follows:
   k_{\text{min}} &= \frac{ 1 }{ \theta  }
   \max\left( \frac{ m_{ij} }{ \left|a_{ij}\right| } \right),~ j \neq i.
 @f}
-
-Here, it is worth mentioning that the time step is constrained by both a lower
-and upper bound in case of a Crank-Nicolson scheme. These bounds should be
+In other words, the time step is constrained by <i>both a lower
+and upper bound</i> in case of a Crank-Nicolson scheme. These bounds should be
 considered along with the CFL condition to ensure significance of the performed
 simulations.
 
-To ensure positivity preservation in this particular tutorial, we can use
+Being unable to make the time step as small as we want to get more
+accuracy without losing the positivity property is annoying. It raises
+the question of whether we can at least <i>compute</i> the minimal time step
+we can choose  to ensure positivity preservation in this particular tutorial.
+Indeed, we can use
 the SparseMatrix objects for both mass and stiffness that are created via
 the MatrixCreator functions. Iterating through each entry via SparseMatrixIterators
 lets us check for diagonal and off-diagonal entries to set a proper time step
 dynamically. For quadratic matrices, the diagonal element is stored as the
-first member of a row (see SparseMatrix documentation). A exemplary code
+first member of a row (see SparseMatrix documentation). An exemplary code
 snippet on how to grab the entries of interest from the <code>mass_matrix</code>
 is shown below.
 
 @code
 Assert (mass_matrix.m() == mass_matrix.n(), ExcNotQuadratic());
-const unsigned int& num_rows = mass_matrix.m();
-double mass_matrix_min_diag    = 1e100,
+const unsigned int num_rows = mass_matrix.m();
+double mass_matrix_min_diag    = std::numeric_limits<double>::max(),
        mass_matrix_max_offdiag = 0.;
 
 SparseMatrixIterators::Iterator<double,true> row_it (&mass_matrix, 0);
 
-for(unsigned int m = 0; m<num_rows; ++m) {
+for(unsigned int m = 0; m<num_rows; ++m) 
+{
   // check the diagonal element
   row_it = mass_matrix.begin(m);
   mass_matrix_min_diag = std::min(row_it->value(), mass_matrix_min_diag);
@@ -207,3 +213,6 @@ for(unsigned int m = 0; m<num_rows; ++m) {
     mass_matrix_max_offdiag = std::max(row_it->value(), mass_matrix_max_offdiag);
 }
 @endcode
+
+Using the information so computed, we can bound the time step via the formulas
+above.