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<div class="">Yes. I am talking about a general usage. To be accurate, direct application of these methods does not always work. There are examples of high-index DAEs for which backward Euler and all multi-step and RK methods fail. You can find one such example
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<div class="">This note also mentions the difficulties in solving the nonlinear system and calculating error estimate for index-2 DAEs.</div>
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<div class="">Thus we do not recommend solving index-2 DAEs directly with these methods.</div>
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<div class="">Hong<br class="">
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<div class="">On Feb 15, 2022, at 10:27 AM, Tang, Qi <<a href="mailto:tangqi@msu.edu" class="">tangqi@msu.edu</a>> wrote:</div>
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<div class="">Thanks a lot, Hong. </div>
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<div class="">I would think if one use BDF or backward Euler for incompressible Naiver Stokes, it should work with the index 2 equation. Why do you think it will not work? Or maybe you were talking about a general usage.</div>
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<div class="">INS is not my usage but I am still curious. We have some unconventional constraint equation.</div>
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<div class="">Qi<br class="">
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<div class="">On Feb 15, 2022, at 8:51 AM, Zhang, Hong <<a href="mailto:hongzhang@anl.gov" class="">hongzhang@anl.gov</a>> wrote:</div>
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<div class=""><span class="">Hi Qi,</span></div>
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The index-2 DAE cannot be solved directly with ARK or implicit methods such as backward Euler and Crank-Nicolson. You need to convert the system to an index-1 DAE as illustrated in the documentation.</span>
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<div class="">Hong (Mr.)</div>
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<div class="">On Feb 15, 2022, at 9:43 AM, Tang, Qi <<a href="mailto:tangqi@msu.edu" class="">tangqi@msu.edu</a>> wrote:</div>
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Hi,
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<div style="orphans: 2; widows: 2;" class="">Does PETSc’s ARK directly apply to <span style="color: rgb(51, 51, 51); font-family: -apple-system, system-ui, "Segoe UI", "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: var(--pst-font-size-h3); orphans: 2; widows: 2; background-color: rgb(255, 255, 255);" class="">Hessenberg
Index-2 DAE? Do we need to </span><span style="orphans: 2; widows: 2; background-color: rgb(255, 255, 255); caret-color: rgb(51, 51, 51);" class=""><font color="#333333" face="-apple-system, system-ui, Segoe UI, Helvetica Neue, Arial, sans-serif, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol" class="">perform</font></span><span style="color: rgb(51, 51, 51); font-family: -apple-system, system-ui, "Segoe UI", "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: var(--pst-font-size-h3); orphans: 2; widows: 2; background-color: rgb(255, 255, 255);" class=""> a
time derivative of the constraint equation by ourselves first? </span></div>
<div style="orphans: 2; widows: 2;" class=""><span style="orphans: 2; widows: 2; background-color: rgb(255, 255, 255); caret-color: rgb(51, 51, 51);" class=""><font color="#333333" face="-apple-system, system-ui, Segoe UI, Helvetica Neue, Arial, sans-serif, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol" class=""><a href="https://urldefense.com/v3/__https://petsc.org/main/docs/manual/ts/*hessenberg-index-2-dae__;Iw!!HXCxUKc!kM_qeX09WwLuE991Vn6u85M0vDlNnqMA0Diuy84sqZawkGLr-MNDkYVisStQBA$" class="">https://petsc.org/main/docs/manual/ts/#hessenberg-index-2-dae</a></font></span></div>
<div style="orphans: 2; widows: 2;" class=""><span style="orphans: 2; widows: 2; background-color: rgb(255, 255, 255); caret-color: rgb(51, 51, 51);" class=""><font color="#333333" face="-apple-system, system-ui, Segoe UI, Helvetica Neue, Arial, sans-serif, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol" class="">If
we do not have to, do we expect to get high order in time?</font></span></div>
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<div style="orphans: 2; widows: 2;" class=""><span style="color: rgb(51, 51, 51); font-family: -apple-system, system-ui, "Segoe UI", "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: var(--pst-font-size-h3); orphans: 2; widows: 2; background-color: rgb(255, 255, 255);" class="">Thanks,</span></div>
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<div class="">Qi</div>
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