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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/TSROSW.html "><small>Report Typos and Errors</small></a></div>
<A NAME="TSROSW"><H1>TSROSW</H1></A>
ODE solver using Rosenbrock-W schemes These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly
nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part
of the equation using <A HREF="../TS/TSSetIFunction.html#TSSetIFunction">TSSetIFunction</A>() and the non-stiff part with <A HREF="../TS/TSSetRHSFunction.html#TSSetRHSFunction">TSSetRHSFunction</A>().
<P>
<H3><FONT COLOR="#CC3333">Notes</FONT></H3>
This method currently only works with autonomous ODE and DAE.
<P>
Consider trying <A HREF="../TS/TSARKIMEX.html#TSARKIMEX">TSARKIMEX</A> if the stiff part is strongly nonlinear.
<P>
Since this uses a single linear solve per time-step if you wish to lag the jacobian or preconditioner computation you must use also -snes_lag_jacobian_persists true or -snes_lag_jacobian_preconditioner true
<P>
<H3><FONT COLOR="#CC3333">Developer Notes</FONT></H3>
Rosenbrock-W methods are typically specified for autonomous ODE
<P>
<pre>
udot = f(u)
</pre>
<P>
by the stage equations
<P>
<pre>
k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j
</pre>
<P>
and step completion formula
<P>
<pre>
u_1 = u_0 + sum_j b_j k_j
</pre>
<P>
with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u)
and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner,
we define new variables for the stage equations
<P>
<pre>
y_i = gamma_ij k_j
</pre>
<P>
The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define
<P>
<pre>
A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}
</pre>
<P>
to rewrite the method as
<P>
<pre>
[M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j
</pre>
<pre>
u_1 = u_0 + sum_j bt_j y_j
</pre>
<P>
where we have introduced the mass matrix M. Continue by defining
<P>
<pre>
ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j
</pre>
<P>
or, more compactly in tensor notation
<P>
<pre>
Ydot = 1/h (Gamma^{-1} \otimes I) Y .
</pre>
<P>
Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current
stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the
equation
<P>
<pre>
g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0
</pre>
<P>
with initial guess y_i = 0.
<P>
<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/TSRosWSetType.html#TSRosWSetType">TSRosWSetType</A>(), <A HREF="../TS/TSRosWRegister.html#TSRosWRegister">TSRosWRegister</A>(), <A HREF="../TS/TSROSWTHETA1.html#TSROSWTHETA1">TSROSWTHETA1</A>, <A HREF="../TS/TSROSWTHETA2.html#TSROSWTHETA2">TSROSWTHETA2</A>, <A HREF="../TS/TSROSW2M.html#TSROSW2M">TSROSW2M</A>, <A HREF="../TS/TSROSW2P.html#TSROSW2P">TSROSW2P</A>, <A HREF="../TS/TSROSWRA3PW.html#TSROSWRA3PW">TSROSWRA3PW</A>, <A HREF="../TS/TSROSWRA34PW2.html#TSROSWRA34PW2">TSROSWRA34PW2</A>, <A HREF="../TS/TSROSWRODAS3.html#TSROSWRODAS3">TSROSWRODAS3</A>,
<BR><A HREF="../TS/TSROSWSANDU3.html#TSROSWSANDU3">TSROSWSANDU3</A>, <A HREF="../TS/TSROSWASSP3P3S1C.html#TSROSWASSP3P3S1C">TSROSWASSP3P3S1C</A>, <A HREF="../TS/TSROSWLASSP3P4S2C.html#TSROSWLASSP3P4S2C">TSROSWLASSP3P4S2C</A>, <A HREF="../TS/TSROSWLLSSP3P4S2C.html#TSROSWLLSSP3P4S2C">TSROSWLLSSP3P4S2C</A>, <A HREF="../TS/TSROSWGRK4T.html#TSROSWGRK4T">TSROSWGRK4T</A>, <A HREF="../TS/TSROSWSHAMP4.html#TSROSWSHAMP4">TSROSWSHAMP4</A>, <A HREF="../TS/TSROSWVELDD4.html#TSROSWVELDD4">TSROSWVELDD4</A>, <A HREF="../TS/TSROSW4L.html#TSROSW4L">TSROSW4L</A>
<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/rosw/rosw.c.html#TSROSW">src/ts/impls/rosw/rosw.c</A>
<P><H3><FONT COLOR="#CC3333">Examples</FONT></H3>
<A HREF="../../../src/ts/tutorials/ex8.c.html">src/ts/tutorials/ex8.c.html</A><BR>
<A HREF="../../../src/ts/tutorials/ex40.c.html">src/ts/tutorials/ex40.c.html</A><BR>
<A HREF="../../../src/ts/tutorials/ex41.c.html">src/ts/tutorials/ex41.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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