15:45
MS23: Recent Advances in Numerical Simulation of Incompressible Flows (Part 4)
Chair: Christoph Lehrenfeld
15:45
25 mins
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Multipreconditioning with application to two-phase incompressible Navier–Stokes flow
Niall Bootland, Andy Wathen
Abstract: We consider the use of multipreconditioning to solve linear systems when more than one preconditioner is available but the optimal choice is not known. In particular, we consider a selective multipreconditioned GMRES algorithm where we incorporate a weighting that allows us to prefer one preconditioner over another. Our target application lies in the simulation of incompressible two-phase flow. Since it is not always known if a preconditioner will perform well within all regimes found in a simulation, we also consider robustness of the multipreconditioning to a poorly performing preconditioner. Overall, we obtain promising results with the approach.
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16:10
25 mins
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Stability Analysis of Time Stepping Schemes for Incompressible Flows from a DAE Perspective
Robert Altmann, Jan Heiland
Abstract: We consider the time discretization of the semi-discrete incompressible Navier-Stokes equations (NSE)
\begin{align*}
M \dot v(t) &= N(v(t)) + J^Tp(t) + f(t), \\
0 &= Jv(t),
\end{align*}
formulated in the velocity $v(t) \in \mathbb R^{n_v}$ and pressure $p(t) \in \mathbb R^{n_p}$, with $M\in \mathbb R^{n_v, n_v}$ being the mass matrix from the spatial discretization, $N\colon \mathbb R^{n_v} \to \mathbb R^{n_v}$ modelling the discretized convection and diffusion, and with $J\in \mathbb R^{n_p, n_v}$ and $J^T$ representing the discrete divergence and gradient operators.
It is commonly known that the semi-discrete incompressible Navier-Stokes equations can be classified as a differential-algebraic equation (DAE) of *differentiation index* $\nu=2$ so that a straight-forward time discretization, e.g. by the *implicit-Euler* method, will likely suffer from instabilities. To overcome these numerical difficulties, a large number of time stepping schemes, with *Chorin*'s projection or the *SIMPLE* scheme as notable examples, have been developed.
In this talk, we trace down and illustrate the instability that origins from the index-2 structure and show that commonly and succesfully applied time-stepping schemes correspond to a reformulation of the semi-discrete NSE as an equivalent system of index-1. Also we show how a finite-element discretization by *Taylor-Hood* or *Crouzeix-Raviart* schemes can be modified such that the resulting semi-discrete NSE already is of index-1 so that standard time-integration schemes lead to stable pressure approximations.
**References**
Altmann, R. & Heiland, J.: *Finite Element Decomposition and Minimal Extension for Flow Equations*, ESAIM: M2AN, **2015**, 49, 1489--1509
Altmann, R. & Heiland, J.: *Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations*, DAE-Forum, Issue: *Applications of Differential-Algebraic Equations: Examples and Benchmarks*, Springer, **2018**
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16:35
25 mins
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A proof that Anderson acceleration increases the convergence rate in linearly converging fixed point methods (but not in quadratically converging ones), and application to NSE
Leo Rebholz
Abstract: This paper provides the first proof that Anderson acceleration (AA) improves the convergence rate of general fixed point iterations. AA has been used for decades to speed up nonlinear solvers in many applications, however a rigorous mathematical justification of the improved convergence rate has remained lacking. The key ideas of the analysis presented here are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is that AA improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step. In addition to improving the convergence rate, our results indicate that AA increases the radius of convergence. Lastly, our estimate shows that while the linear convergence rate is improved, additional quadratic terms arise in the estimate, which shows why AA does not typically improve convergence in quadratically converging fixed point iterations. Results of several numerical tests for fluid applications are given which illustrate the theory.
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17:00
25 mins
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Comparison Between Algebraic and Matrix-free Geometric Multigrid for a Stokes Problem
Conrad Clevenger, Timo Heister
Abstract: We present numerical results of a geometric multigrid preconditioner applied to a Stokes problem with variable viscosity that is implemented for adaptively refined finite element meshes.
The Stokes problem is motivated by computation of flow in the Geosciences. The problem has varying viscosity and localized features, requiring adaptive mesh refinement to resolve thiese features. Scalability is shown to 65k cores.
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