Weak and strong wall boundary procedures and convergence to steady-state of the Navier–Stokes equations

2016

J Sci Comput

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All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier-Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

Section: Remark 51mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Roadmap to Well Posed and Stable Problems in Computational Physics

2016

J Sci Comput

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“…For application of this technique to finite difference methods, nodecentered finite volume methods, spectral domain methods and various hybrid methods see the references in the original article [1] In this section we will consider a new effect of using weak boundary procedures, namely that it in many cases (all that we tried) speeds up the convergence to steady-state.…”

Section: Weak Solid Wall Boundary Conditions and Steady Statementioning

confidence: 99%

“…The complete description of the weak boundary procedures for convergence to steady state is given in [1] while the development of dual consistence schemes is presented in [2], [3], [4]. The reader is referred to articles mentioned above for potentially missing details.…”

Section: Introductionmentioning

confidence: 99%

New Developments for Increased Performance of the SBP-SAT Finite Difference Technique

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

Eliasson

2015

Abstract. In this article, recent developments for increased performance of the high order and stable SBP-SAT finite difference technique is described. In particular we discuss the use of weak boundary conditions and dual consistent formulations. The use of weak boundary conditions focus on increased convergence to steady state, and hence efficiency. Dual consistent schemes produces superconvergent functionals and increases accuracy.

“…In this article it is extended to the analysis of the time-dependent advection-diffusion equation. The continuous problem is analyzed using the energy method, and strong well-posedness is proved [14,15].…”

Section: Introductionmentioning

confidence: 99%

The effect of uncertain geometries on advection–diffusion of scalar quantities

Wahlsten

2017

Bit Numer Math

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The two dimensional advection-diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.