Energy decay of vortices in viscous fluids: an applied mathematics view

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: The Energy Methodsmentioning

confidence: 99%

A Roadmap to Well Posed and Stable Problems in Computational Physics

2016

J Sci Comput

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“…f ( x, ξ) and g( x, t, ξ) are the stochastic initial and boundary data to the problem. With a proper choice of the matrices involved, the problem (1) represents the linearized and symmetrized compressible Navier-Stokes equations [38].…”

Section: The Stochastic Formulationmentioning

confidence: 99%

Robust boundary conditions for stochastic incompletely parabolic systems of equations

Wahlsten

Journal of Computational Physics

2018

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We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier-Stokes equations.

“…is positive semi-definite under the standard assumption 2µ + 3λ ≥ 0 which is valid in ideal fluids [15,10,35]. Using the boundary conditions in (5), we can cancel the energy contribution from the viscous terms and get the final energy estimate as…”

Section: The Dual Navier-stokes Equationsmentioning

confidence: 99%

Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations

Berg

Journal of Computational Physics

2014

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In this paper we derive new farfield boundary conditions for the time-dependent Navier-Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedess of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier-Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations.We perform computations with a high-order finite difference scheme on summationby-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.