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src/timestepping.org

@@ -213,7 +213,7 @@
                     VectorType const &_ud_initial,
                     Dune::BitSetVector<dim> const &_dirichletNodes,
                     FunctionType const &_dirichletFunction);
-
+  
   <<virtual_function_declaration>>
   
   private:
@@ -363,7 +363,8 @@
     &= \frac 2\tau (\dot u_1 - \dot u_0) - \ddot u_0
     \intertext{so that it yields}
     \label{eq:4} u_1
-    &= u_0 + \tau \dot u_0 + \frac {\tau^2}4 \ddot u_0 + \frac {\tau^2}4 \left( \frac 2\tau (\dot u_1 - \dot u_0) - \ddot u_0\right)\\
+    &= u_0 + \tau \dot u_0 + \frac {\tau^2}4 \ddot u_0 + \frac {\tau^2}4 \left(
+      \frac 2\tau (\dot u_1 - \dot u_0) - \ddot u_0\right)\\
     &= u_0 + \tau \dot u_0 + \frac \tau 2 (\dot u_1 - \dot u_0)\nonumber\\
     &= u_0 + \frac \tau 2 (\dot u_1 + \dot u_0)\nonumber
   \end{align}
@@ -372,11 +373,15 @@
     b(\ddot u_1, v - \dot u_1) + a(u_1, v - \dot u_1) + j(v) - j(\dot u_1)
     &\ge \ell (v - \dot u_1)
     \intertext{with unknowns $u_1$, $\dot u_1$, and $\ddot u_1$, we first derive}
-    \frac 2\tau b(\dot u_1, v - \dot u_1) + a(u_1, v - \dot u_1) + j(v) - j(\dot u_1)
-    &\ge \ell (v - \dot u_1) + b\left(\frac 2\tau \dot u_0 + \ddot u_0, v - \dot u_1\right)
+    \frac 2\tau b(\dot u_1, v - \dot u_1)
+    + a(u_1, v - \dot u_1) + j(v) - j(\dot u_1)
+    &\ge \ell (v - \dot u_1) + b\left(
+      \frac 2\tau \dot u_0 + \ddot u_0, v - \dot u_1\right)
     \intertext{from \eqref{eq:3} and then}
-    \frac 2\tau b(\dot u_1, v - \dot u_1) + \frac \tau 2 a(\dot u_1, v - \dot u_1) + j(v) - j(\dot u_1)
-    &\ge \ell (v - \dot u_1) + b\left(\frac 2\tau \dot u_0 + \ddot u_0, v - \dot u_1\right)\\
+    \frac 2\tau b(\dot u_1, v - \dot u_1) + \frac \tau 2 a(\dot u_1, v - \dot u_1)
+    + j(v) - j(\dot u_1)
+    &\ge \ell (v - \dot u_1) + b\left(
+      \frac 2\tau \dot u_0 + \ddot u_0, v - \dot u_1\right)\\
     &\quad - a\left(u_0 + \frac \tau 2 \dot u_0, v - \dot u_1\right)
   \end{align*}
   from \eqref{eq:4}. The only unknown is now $\dot u_1$.
@@ -529,7 +534,7 @@
   <<ImplicitEuler.cc>>
   <<ImplicitTwoStep.cc>>
   <<Newmark.cc>>
-
+  
   #include "timestepping_tmpl.cc"
 #+end_src