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Register a new userEulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains
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Added by
Stefan Frei
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld & Olshanskii [ESAIM: M2AN, 53(2):585-614, 2019], where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsches method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $L^2(L^2)$-norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.
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The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport with Faradaic reactions, and have applications in a number of fields. In this article, we develop an adaptive time-stepping scheme for the solution of the PNP-FBV equations based on two time-stepping methods: a fully implicit (BDF2) method, and an implicit-explicit (SBDF2) method. We present simulations under both current and voltage boundary conditions and demonstrate the ability to simulate a large range of parameters, including any value of the singular perturbation parameter $epsilon$. When the underlying dynamics is one that would have the solutions converge to a steady-state solution, we observe that the adaptive time-stepper based on the SBDF2 method produces solutions that ``nearly converge to the steady state and that, simultaneously, the time-step sizes stabilize to a limiting size $dt_infty$. In the companion to this article cite{YPD_Part2}, we linearize the SBDF2 scheme about the steady-state solution and demonstrate that the linearized scheme is conditionally stable. This conditional stability is the cause of the adaptive time-steppers behaviour. While the adaptive time-stepper based on the fully-implicit (BDF2) method is not subject to such time-step constraints, the required nonlinear solve yields run times that are significantly longer.
We present a projection-based framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work we extend the fully implicit method presented in Khanwale et al. [{it A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes.}, arXiv:2009.06628 (2020)], to a block iterative hybrid method. We use a projection-based semi-implicit time discretization for the Navier-Stokes and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.
In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter.
This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.
We present a second-order ensemble method based on a blended three-step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier-Stokes equations. Compared with the only existing second-order ensemble method that combines the two-step BDF timestepping scheme and a special explicit second-order Adams-Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method.
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