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Abstract We investigate the temporal accuracy of two generalized‐ schemes for the incompressible Navier‐Stokes equations. In a widely‐adopted approach, the pressure is collocated at the time steptn + 1while the remainder of the Navier‐Stokes equations is discretized following the generalized‐ scheme. That scheme has been claimed to besecond‐order accurate in time. We developed a suite of numerical code using inf‐sup stable higher‐order non‐uniform rational B‐spline (NURBS) elements for spatial discretization. In doing so, we are able to achieve high spatial accuracy and to investigate asymptotic temporal convergence behavior. Numerical evidence suggests that onlyfirst‐order accuracyis achieved, at least for the pressure, in this aforesaid temporal discretization approach. On the other hand, evaluating the pressure at the intermediate time step recovers second‐order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized‐ scheme of choice when integrating the incompressible Navier‐Stokes equations.
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Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations *
Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations.
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- Award ID(s):
- 2309748
- PAR ID:
- 10585550
- Publisher / Repository:
- IOP Publishing, London Mathematical Society
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 37
- Issue:
- 11
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 115014
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6544/ad7f18
