Time-discrete higher order ALE formulations: a priori error analysis

Bonito, Andrea; Kyza, Irene; Nochetto, Ricardo H.

doi:10.1007/s00211-013-0539-3

Cited by 15 publications

(11 citation statements)

References 20 publications

(46 reference statements)

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“…We point out that the time step restriction is not a CFL condition because the space is continuous, and that RKR methods are much cheaper than Reynolds' methods for the same accuracy since the number of nodes q + 1 compare favorably with p. A detailed discussion of the convergence rates for both methods is given in [6].…”

Section: Runge-kutta-radau Methods In the Ale Framework: Conditional mentioning

confidence: 99%

“…However, stability results (as well as error bounds; cf. [6]) for the conservative RKR method can be derived similarly.…”

Section: Runge-kutta-radau Methods In the Ale Framework: Conditional mentioning

confidence: 99%

“…Invoking a perturbation argument, we prove in [6] that there is no loss of accuracy by replacing the exact map with its L 2 -projection.…”

Section: Reynolds' Quadrature and Geometric Conservation Lawmentioning

confidence: 98%

“…The latter is computational less intensive that Reynolds' quadrature and leads to dG schemes with the same accuracy; cf. [6]. However, Radau quadrature requires a restriction on the time steps for stability.…”

Section: 1mentioning

confidence: 99%

“…This paper is the first in a series devoted to the analysis of dG for ALE formulations. We perform an a priori error analysis in [6] and an a posteriori error analysis in [5], both based on the stability notions developed here.…”

Section: Introductionmentioning

confidence: 99%

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Time-Discrete Higher-Order ALE Formulations: Stability

Bonito¹,

Kyza²,

Nochetto³

2013

SIAM J. Numer. Anal.

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Abstract. Arbitrary Lagrangian Eulerian (ALE) formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. This arbitrary extension has no effect on the stability of the PDE but may influence that of a discrete scheme. We propose time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion model problem in moving domains, and study their stability properties. The analysis hinges on the validity of the Reynolds' identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and non-conservative dG schemes are equivalent and unconditionally stable. The same results remain true for piecewise polynomial ALE maps of any degree and suitable quadrature that guarantees the validity of the Reynolds' identity. This approach generalizes the so-called geometric conservation law (GCL) to higher order methods. We also prove that simpler Runge-Kutta-Radau (RKR) methods of any order are conditionally stable, that is subject to a mild ALE constraint on the time steps. Numerical experiments corroborate and complement our theoretical results.

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Section: Runge-kutta-radau Methods In the Ale Framework: Conditional mentioning

confidence: 99%

“…However, stability results (as well as error bounds; cf. [6]) for the conservative RKR method can be derived similarly.…”

Section: Runge-kutta-radau Methods In the Ale Framework: Conditional mentioning

confidence: 99%

“…Invoking a perturbation argument, we prove in [6] that there is no loss of accuracy by replacing the exact map with its L 2 -projection.…”

Section: Reynolds' Quadrature and Geometric Conservation Lawmentioning

confidence: 98%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time-Discrete Higher-Order ALE Formulations: Stability

Bonito¹,

Kyza²,

Nochetto³

2013

SIAM J. Numer. Anal.

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High‐order methods for low Reynolds number flows around moving obstacles based on universal meshes

Gawlik

Kabaria

Lew

2015

Numerical Meth Engineering

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We propose a family of methods for simulating two-dimensional incompressible, low Reynolds number flow around a moving obstacle whose motion is prescribed. The methods make use of a universal mesh: a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle's boundary. The resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off-the-shelf time integrators. We demonstrate the approach by using Taylor-Hood elements to approximate the fluid velocity and pressure. To integrate in time, we consider implicit Runge-Kutta schemes as well as a fractional step scheme. We illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations.Organization. This paper is organized as follows. In Section 2, we recall the governing equations for incompressible, viscous flow around a moving obstacle with prescribed evolution, and we recast the equations in weak form. In Section 3, we propose a discretization of the aforementioned equations using a universal mesh in conjunction with Taylor-Hood finite elements [61]. To integrate in time, we propose the use of implicit Runge-Kutta schemes as well as a fractional step scheme. In Section 4, we apply the proposed methods to simulate flow around various obstacles with prescribed evolution: The setup we have described thus far offers the freedom to employ a time integrator of one's choosing to numerically integrate (17), a system of DAEs of index 2, from t D t n 1 to t D t n . We present two examples of integration schemes: a singly diagonally implicit Runge-Kutta (SDIRK) scheme [69,70] and a fractional step scheme [71][72][73]. In accordance with common guidelines for numerically solving DAEs, the SDIRK schemes we consider are stiffly accurate (and hence L-stable) methods [70,74]. The same schemes are considered by, for instance, [75,76] in their studies of high-order methods for the Navier-Stokes equations on fixed domains.We immersed the oscillating disk in a domain D D OE 6; 6 OE 3; 3 and prescribed boundary conditions ‡ In the case of the fractional step scheme, the error in p was measured at t D T t=2 rather than at t D T .

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Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

Kovács

2018

Numerical Methods Partial

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The good mesh quality of a discretized closed evolving surface is often compromised during time evolution. In recent years this phenomenon has been theoretically addressed in a few ways, one of them uses arbitrary Lagrangian Eulerian (ALE) maps. However, the numerical computation of such maps still remained an unsolved problem in the literature. An approach, using differential algebraic problems, is proposed here to numerically compute an arbitrary Lagrangian Eulerian map, which preserves the mesh properties over time. The ALE velocity is obtained by finding an equilibrium of a simple spring system, based on the connectivity of the nodes in the mesh. We also consider the algorithmic question of constructing acute surface meshes. We present various numerical experiments illustrating the good properties of the obtained meshes and the low computational cost of the proposed approach.

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Time-discrete higher order ALE formulations: a priori error analysis

Cited by 15 publications

References 20 publications

Time-Discrete Higher-Order ALE Formulations: Stability

Time-Discrete Higher-Order ALE Formulations: Stability

High‐order methods for low Reynolds number flows around moving obstacles based on universal meshes

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

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