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<div id="version" align=right><b>petsc-3.14.5 2021-03-03</b></div>
<div id="bugreport" align=right><a href="mailto:petsc-maint@mcs.anl.gov?subject=Typo or Error in Documentation &body=Please describe the typo or error in the documentation: petsc-3.14.5 v3.14.5 docs/manualpages/TS/TSTHETA.html "><small>Report Typos and Errors</small></a></div>
<A NAME="TSTHETA"><H1>TSTHETA</H1></A>
DAE solver using the implicit Theta method
<P>
<H3><FONT COLOR="#CC3333">Options Database</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ts_theta_theta <Theta> </B></TD><TD>- Location of stage (0<Theta<=1)
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ts_theta_endpoint <flag> </B></TD><TD>- Use the endpoint (like Crank-Nicholson) instead of midpoint form of the Theta method
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>-ts_theta_initial_guess_extrapolate <flg> </B></TD><TD>- Extrapolate stage initial guess from previous solution (sometimes unstable)
</TD></TR></TABLE>
<P>
<H3><FONT COLOR="#CC3333">Notes</FONT></H3>
<pre>
-ts_type theta -ts_theta_theta 1.0 corresponds to backward Euler (<A HREF="../TS/TSBEULER.html#TSBEULER">TSBEULER</A>)
</pre>
<pre>
-ts_type theta -ts_theta_theta 0.5 corresponds to the implicit midpoint rule
</pre>
<pre>
-ts_type theta -ts_theta_theta 0.5 -ts_theta_endpoint corresponds to Crank-Nicholson (<A HREF="../TS/TSCN.html#TSCN">TSCN</A>)
</pre>
<P>
This method can be applied to DAE.
<P>
This method is cast as a 1-stage implicit Runge-Kutta method.
<P>
<PRE>
Theta | Theta
-------------
| 1
</PRE>
<P>
For the default Theta=0.5, this is also known as the implicit midpoint rule.
<P>
<H3><FONT COLOR="#CC3333">When the endpoint variant is chosen, the method becomes a 2-stage method with first stage explicit</FONT></H3>
<P>
<PRE>
0 | 0 0
1 | 1-Theta Theta
-------------------
| 1-Theta Theta
</PRE>
<P>
For the default Theta=0.5, this is the trapezoid rule (also known as Crank-Nicolson, see <A HREF="../TS/TSCN.html#TSCN">TSCN</A>).
<P>
To apply a diagonally implicit RK method to DAE, the stage formula
<P>
<pre>
Y_i = X + h sum_j a_ij Y'_j
</pre>
<P>
is interpreted as a formula for Y'_i in terms of Y_i and known values (Y'_j, j<i)
<P>
<H3><FONT COLOR="#CC3333">See Also</FONT></H3>
<A HREF="../TS/TSCreate.html#TSCreate">TSCreate</A>(), <A HREF="../TS/TS.html#TS">TS</A>, <A HREF="../TS/TSSetType.html#TSSetType">TSSetType</A>(), <A HREF="../TS/TSCN.html#TSCN">TSCN</A>, <A HREF="../TS/TSBEULER.html#TSBEULER">TSBEULER</A>, <A HREF="../TS/TSThetaSetTheta.html#TSThetaSetTheta">TSThetaSetTheta</A>(), <A HREF="../TS/TSThetaSetEndpoint.html#TSThetaSetEndpoint">TSThetaSetEndpoint</A>()
<BR>
<P>
<P><B></B><H3><FONT COLOR="#CC3333">Level</FONT></H3>beginner<BR>
<H3><FONT COLOR="#CC3333">Location</FONT></H3>
</B><A HREF="../../../src/ts/impls/implicit/theta/theta.c.html#TSTHETA">src/ts/impls/implicit/theta/theta.c</A>
<P><H3><FONT COLOR="#CC3333">Examples</FONT></H3>
<A HREF="../../../src/ts/tutorials/ex10.c.html">src/ts/tutorials/ex10.c.html</A><BR>
<A HREF="../../../src/ts/tutorials/ex14.c.html">src/ts/tutorials/ex14.c.html</A><BR>
<A HREF="../../../src/ts/tutorials/ex17.c.html">src/ts/tutorials/ex17.c.html</A><BR>
<A HREF="../../../src/ts/tutorials/ex31.c.html">src/ts/tutorials/ex31.c.html</A><BR>
<BR><A HREF="./index.html">Index of all TS routines</A>
<BR><A HREF="../../index.html">Table of Contents for all manual pages</A>
<BR><A HREF="../singleindex.html">Index of all manual pages</A>
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