<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><br class=""><div><blockquote type="cite" class=""><div class="">On Sep 13, 2016, at 9:47 PM, Gideon Simpson <<a href="mailto:gideon.simpson@gmail.com" class="">gideon.simpson@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div class="">I was looking around to see if there was any built in routine for implicit midpoint time stepping in petsc. By this I mean equation (1i) of <a href="https://en.wikipedia.org/wiki/Midpoint_method" class="">https://en.wikipedia.org/wiki/Midpoint_method</a>. <br class=""><br class="">I noticed the TSTHETA solver, and I see that the documentation says:<br class=""><br class="">-ts_type theta -ts_theta_theta 0.5 corresponds to the implicit midpoint rule<br class=""><br class="">But I wanted to verify that it agrees with what I interpret to be implicit midpoint. <br class=""><br class=""></div></div></blockquote><div>Yes, it does.</div><br class=""><blockquote type="cite" class=""><div class=""><div class="">Also, is the distinction with the ts_theta_endpoint flag as to whether it is the weighting of the argument to the f(t,y) function or the weighting of the f(t,y) evaluated at the different points?<br class=""><br class=""></div></div></blockquote><div><br class=""></div><div>The ts_theta_endpoint flag allows one to switch between two different types of theta methods; for ODEs they can be represented in the general forms</div><div><br class=""></div><div>y_{i+1} = y_i + h ( \theta f(t_i,y_i) + (1-\theta) f(t_{i+1},y_{i+1}) ) (w the endpoint flag)</div><div><br class=""></div><div>and </div><div><br class=""></div><div>y_s = y_i + h \theta f(t_i+\theta h, y_s)</div><div>y_{i+1} = y_i + (y_s-y_i)/\theta (w/o the endpoint flag)</div><div><br class=""></div><div>Hong</div><div><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo>+</mo><mi>h</mi><mo stretchy="false">[</mo><mi>θ</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>t</mi><mi>j</mi></msub><mo>,</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>" role="presentation" style="margin: 0px; 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padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; position: absolute; vertical-align: 0px; line-height: normal; clip: rect(3.426em, 1000.42em, 4.387em, -999.997em); top: -4.021em; left: 0em;" class=""><span class="mi" id="MathJax-Span-54" style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal; font-family: STIXGeneral-Italic;">y</span><span style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; display: inline-block; position: static; vertical-align: 0px; line-height: normal; width: 0px; height: 4.027em;" class=""></span></span><span style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; position: absolute; vertical-align: 0px; line-height: normal; top: -3.901em; left: 0.423em;" class=""><span class="texatom" id="MathJax-Span-55" style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-56" style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-57" style="margin: 0px; padding: 0px; border: 0px; font-size: 11.7715px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal; font-family: STIXGeneral-Italic;">j<span style="margin: 0px; padding: 0px; border: 0px; font-size: 11.7715px; transition: none; -webkit-transition: none; display: inline-block; position: static; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;" class=""></span></span><span class="mo" id="MathJax-Span-58" style="margin: 0px; padding: 0px; border: 0px; font-size: 11.7715px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal; font-family: STIXGeneral-Regular;">+</span><span class="mn" id="MathJax-Span-59" style="margin: 0px; padding: 0px; border: 0px; font-size: 11.7715px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal; font-family: STIXGeneral-Regular;">1</span></span></span><span style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; display: inline-block; position: static; vertical-align: 0px; line-height: normal; width: 0px; height: 4.027em;" class=""></span></span></span></span><span class="mo" id="MathJax-Span-60" style="margin: 0px; padding: 0px; border: 0px; font-size: 16.65px; transition: none; -webkit-transition: none; display: inline; position: static; vertical-align: 0px; line-height: normal; font-family: STIXGeneral-Regular;">)</span></span></span></span></span></nobr></span></div><div><br class=""></div><br class=""><blockquote type="cite" class=""><div class=""><div class="">-gideon<br class=""><br class=""></div></div></blockquote></div><br class=""></body></html>