| 1. |
- Nikkar, Samira, et al.
(författare)
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A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this article, well-posedness and dual consistency of the linearized incompressible Navier-Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem, the second order formulation is transformed to rst order form. Boundary conditions that simultaneously lead to well-posedness of the primal and dual problems are derived.We construct fully discrete nite di erence schemes on summation-byparts form, in combination with the simultaneous approximation technique. We prove energy stability and discrete dual consistency. Moreover, we show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain, and as a result, stability and discrete dual consistency follow simultaneously.The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.
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| 2. |
- Nikkar, Samira, et al.
(författare)
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A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
- 2019
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 376, s. 322-338
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Tidskriftsartikel (refereegranskat)abstract
- In this article, well-posedness and dual consistency of the linearized constant coefficient incompressible Navier–Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem and improve the accuracy of gradients, the second order formulation is transformed to first order form. Boundary conditions that simultaneously lead to boundedness of the primal and dual problems are derived.Fully discrete finite difference schemes on summation-by-parts form, in combination with the simultaneous approximation technique, are constructed. We prove energy stability and discrete dual consistency and show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain. As a result of the aforementioned formulations, stability and discrete dual consistency follow simultaneously.The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.
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| 3. |
- Nikkar, Samira, et al.
(författare)
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A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We introduce an interface/coupling procedure for hyperbolic problems posedon time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions.Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.
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| 4. |
- Nikkar, Samira, et al.
(författare)
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A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
- 2017
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Ingår i: Journal of Computational Physics. - : Academic Press. - 0021-9991 .- 1090-2716. ; 339, s. 500-524
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Tidskriftsartikel (refereegranskat)abstract
- We introduce an interface/coupling procedure for hyperbolic problems posed on time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions. Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected. The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.
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| 5. |
- Nikkar, Samira, et al.
(författare)
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Energy Stable High Order Finite Difference Methods for Hyperbolic Equations in Moving Coordinate Systems
- 2013
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Ingår i: AIAA Aerospace Sciences - Fluid Sciences Event. - Reston, Virginia : American Institute of Aeronautics and Astronautics. ; , s. 1-14
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- A time-dependent coordinate transformation of a constant coeffcient hyperbolic equation which results in a variable coeffcient problem is considered. By using the energy method, we derive well-posed boundary conditions for the continuous problem. It is shown that the number of boundary conditions depend on the coordinate transformation. By using Summation-by-Parts (SBP) operators for the space discretization and weak boundary conditions, an energy stable finite dieffrence scheme is obtained. We also show how to construct a time-dependent penalty formulation that automatically imposes the right number of boundary conditions. Numerical calculations corroborate the stability and accuracy of the approximations.
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| 6. |
- Nikkar, Samira, et al.
(författare)
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Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains : An initial investigation
- 2015
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Ingår i: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. - Cham : Springer. - 9783319197999 - 9783319198002 ; , s. 385-395
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Bokkapitel (refereegranskat)abstract
- A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summation-by-Parts (SBP) operators together with a weak imposition of the boundary and initial conditions using Simultaneously Approximation Terms (SATs) guarantee energy-stability of the fully discrete scheme. We construct a time-dependent SAT formulation that automatically imposes the boundary conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stability and accuracy of the approximations. As an application we study the sound propagation in a deforming domain using the linearized Euler equations.
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| 7. |
- Nikkar, Samira, et al.
(författare)
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Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coecient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the numerical Geometric Conservation Law holds automatically by using SBP-SAT in time. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
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| 8. |
- Nikkar, Samira, et al.
(författare)
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Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains
- 2015
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 291, s. 82-98
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Tidskriftsartikel (refereegranskat)abstract
- A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
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| 9. |
- Nikkar, Samira
(författare)
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Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We construct stable, accurate and efficient numerical schemes for wave propagation and flow problems posed on spatial geometries that are moving, deforming, erroneously described or non-simply connected. The schemes are on Summation-by-Parts (SBP) form, combined with the Simultaneous Approximation Term (SAT) technique for imposing initial and boundary conditions. The main analytical tool is the energy method, by which well-posedness, stability and conservation are investigated. To handle the deforming domains, time-dependent coordinate transformations are used to map the problem to fixed geometries.The discretization is performed in such a way that the Numerical Geometric Conservation Law (NGCL) is satisfied. Additionally, even though the schemes are constructed on fixed domains, time-dependent penalty formulations are necessary, due to the originally moving boundaries. We show how to satisfy the NGCL and present an automatic formulation for the penalty operators, such that the correct number of boundary conditions are imposed, when and where required.For problems posed on erroneously described geometries, we investigate how the accuracy of the solution is affected. It is shown that the inaccurate geometry descriptions may lead to wrong wave speeds, a misplacement of the boundary condition, the wrong boundary operator or a mismatch of data. Next, the SBP-SAT technique is extended to time-dependent coupling procedures for deforming interfaces in hyperbolic problems. We prove conservation and stability and show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the variations of the interface location while the NGCL is preserved.Moreover, dual consistent SBP-SAT schemes for the linearized incompressible Navier-Stokes equations posed on deforming domains are investigated. To simplify the derivations of the dual problem and incorporate the motions of the boundaries, the second order formulation is reduced to first order and the problem is transformed to a fixed domain. We prove energy stability and dual consistency. It is shown that the solution as well as the divergence of the solution converge with the design order of accuracy, and that functionals of the solution are superconverging.Finally, initial boundary value problems posed on non-simply connected spatial domains are investigated. The new formulation increases the accuracy of the scheme by minimizing the use of multi-block couplings. In order to show stability, the spectrum of the semi-discrete SBP-SAT formulation is studied. We show that the eigenvalues have the correct sign, which implies stability, in combination with the SBP-SAT technique in time.
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| 10. |
- Nikkar, Samira, et al.
(författare)
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Summation-by-parts operators for non-simply connected domains
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique.In the theoretical part, we consider the two dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries.Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multi-block technique. Finally, an application using the linearized Euler equations for sound propagation is presented.
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