The long-time L2 and H1 stability of linearly extrapolated second-order time-stepping schemes for the 2D incompressible Navier–Stokes equations

2019 ◽  
Vol 342 ◽  
pp. 263-279
Author(s):  
Sean Breckling ◽  
Sidney Shields
Keyword(s):  
Stokes Equations ◽  
Second Order ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Time Stepping ◽  
Long Time

scholarly journals A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

2019 ◽  
Vol 19 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Max Gunzburger ◽  
Nan Jiang ◽  
Zhu Wang
Keyword(s):  
Stokes Equations ◽  
Second Order ◽  
Navier Stokes ◽  
Conditional Stability ◽  
Initial Condition ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Simulations ◽  
Flow Problems ◽  
Time Stepping

AbstractWe consider settings for which one needs to perform multiple flow simulations based on the Navier–Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

scholarly journals Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Milani ◽  
Jérôme Bonelle ◽  
Alexandre Ern
Keyword(s):  
Stokes Equations ◽  
Time Discretization ◽  
Second Order ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Time Stepping ◽  
Monolithic Approach ◽  
Polytopal Meshes

Abstract We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier–Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier–Stokes equations. The AC time-stepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.

scholarly journals RELATIVISTIC VISCOELASTIC FLUID MECHANICS

2013 ◽  
Vol 21 ◽  
pp. 189-190
Author(s):  
MASAFUMI FUKUMA ◽  
YUHO SAKATANI
Keyword(s):  
Fluid Mechanics ◽  
Viscoelastic Fluid ◽  
Stokes Equations ◽  
Second Order ◽  
Time Limit ◽  
Navier Stokes ◽  
Viscoelastic System ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Short Time

We explain the relativistic theory of viscoelasticity which we have recently constructed on the basis of Onsager's linear nonequilibrium thermodynamics. This theory universally reduces to the standard relativistic Navier-Stokes fluid mechanics in the long time limit. Since effects of elasticity are taken into account, the dynamics at short time scales is modified from that given by the Navier-Stokes equations, so that acausal problems intrinsic to relativistic Navier-Stokes fluids are significantly remedied. We then present conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We particularly show that a conformal second-order fluid with all possible parameters in the constitutive equations can be obtained without breaking the hypothesis of local thermodynamic equilibrium, if the conformal fluid is defined as the long time limit of a conformal second-order viscoelastic system.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 127-150
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Minimum Principle ◽  
Immiscible Fluids ◽  
Second Order ◽  
Navier Stokes ◽  
Distributed Optimal Control ◽  
Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

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