From 97303e05b2b840334e01a64bfed98dec8776fc1a Mon Sep 17 00:00:00 2001
From: Elias Pipping <elias.pipping@fu-berlin.de>
Date: Tue, 30 Oct 2012 12:24:01 +0100
Subject: [PATCH] Whitespace
---
src/timestepping.org | 19 ++++++++++++-------
1 file changed, 12 insertions(+), 7 deletions(-)
diff --git a/src/timestepping.org b/src/timestepping.org
index 074f2ad0..ed04802b 100644
--- a/src/timestepping.org
+++ b/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
--
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