| 1. |
- Eriksson, Sofia, et al.
(författare)
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Finite difference schemes with transferable interfaces for parabolic problems
- 2018
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Ingår i: Journal of Computational Physics. - Linköping : Elsevier. - 0021-9991 .- 1090-2716. ; 375, s. 935-949
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Tidskriftsartikel (refereegranskat)abstract
- We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent. (C) 2018 Elsevier Inc. All rights reserved.
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| 2. |
- Nordström, Jan, et al.
(författare)
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Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations
- 2012
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 231:14, s. 4867-4884
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Tidskriftsartikel (refereegranskat)abstract
- We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.
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| 3. |
- Nordström, Jan, et al.
(författare)
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Weak and Strong Wall Boundary Procedures and Convergence to Steady-State of the Navier-Stokes Equations
- 2011
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Rapport (refereegranskat)abstract
- We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.
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| 4. |
- Eriksson, Sofia, et al.
(författare)
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Exact Non-reflecting Boundary Conditions Revisited : Well-Posedness and Stability
- 2017
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Ingår i: Foundations of Computational Mathematics. - : Springer. - 1615-3375 .- 1615-3383. ; 17:4, s. 957-986
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Tidskriftsartikel (refereegranskat)abstract
- Exact non-reflecting boundary conditions for a linear incompletely parabolic system in one dimension have been studied. The system is a model for the linearized compressible Navier-Stokes equations, but is less complicated which allows for a detailed analysis without approximations. It is shown that well-posedness is a fundamental property of the exact non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, it is also shown that energy stability follows automatically.
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| 5. |
- Eriksson, Sofia, et al.
(författare)
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Summation-by-Parts Approximations of the Second Derivative : Pseudoinverse and Revisitation of a High Order Accurate Operator
- 2021
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM Publications. - 0036-1429 .- 1095-7170. ; 59:5, s. 2669-2697
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Tidskriftsartikel (refereegranskat)abstract
- We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation, and the wave equation. The finite difference operators satisfy a summation-by-parts (SBP) property, which mimics the integration-by-parts principle. Since the operators approximate the second derivative, they are singular by construction. When imposing boundary conditions weakly, these operators are modified using simultaneous approximation terms. The modification makes the discretization matrix nonsingular for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, for problems with only Neumann boundary conditions, the modified matrices are still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose inverse, which can be used for solving elliptic problems and some time-dependent problems. For this explicit expression to be valid, it is required that the modified matrix does not have more than one zero eigenvalue. This condition holds for the SBP operators with second and fourth order accurate interior stencil. For the sixth order accurate case, we have reconstructed the operator with a free parameter and show that there can be more than one zero eigenvalue. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative SBP operator. We complement the derivations by numerical experiments to demonstrate the improvements.
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| 6. |
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| 7. |
- Eliasson, Peter, et al.
(författare)
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The influence of weak and strong solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations
- 2009
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Ingår i: 19th AIAA Computational Fluid Dynamics Conference 2009. - Reston, Virigina : American Institute of Aeronautics and Astronautics. - 9781563479755
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Konferensbidrag (refereegranskat)abstract
- In the present paper we study the influence of weak and strong no-slip solid wall boundary conditions on the convergence to steady-state. Our Navier-Stokes solver is edge based and operates on unstructured grids. The two types of boundary conditions are applied to no-slip adiabatic walls. The two approaches are analyzed for a simplified model problem and the reason for the different convergence rates are discussed in terms of the theoretical findings for the model problem. Numerical results for a 2D viscous steady state low Reynolds number problem show that the weak boundary conditions often provide faster convergence. It is shown that strong boundary conditions can prevent the steady state convergence. It is also demonstrated that the two approaches converge to the same solution. Similar results are obtained for high Reynolds number flow in two and three dimensions.
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| 8. |
- Eriksson, Sofia
(författare)
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A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators
- 2018
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 75:2, s. 906-940
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Tidskriftsartikel (refereegranskat)abstract
- We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP-SAT). Recently it was shown that SBP-SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.
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| 9. |
- Eriksson, Sofia, et al.
(författare)
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A stable and conservative method for locally adapting the design order of finite difference schemes
- 2011
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 230:11, s. 4216-4231
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Tidskriftsartikel (refereegranskat)abstract
- A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable. Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.
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| 10. |
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