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| 2. |
- Abbas, Qaisar, et al.
(författare)
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Energy Stability of the MUSCL Scheme
- 2010
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Ingår i: Proc. 7th South African Conference on Computational and Applied Mechanics. - Berlin : South African Association for Theoretical and Applied Mechanics. ; , s. 65:1-8, s. 61-68
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)
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| 3. |
- Abbas, Qaisar, 1975-
(författare)
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Weak Boundary and Interface Procedures for Wave and Flow Problems
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems.The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case.The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach.Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks.Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability.Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.
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| 4. |
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| 5. |
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| 6. |
- Abgrall, Remi, et al.
(författare)
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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems
- 2020
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 85:2
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Tidskriftsartikel (refereegranskat)abstract
- In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.
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| 7. |
- Abgrall, Remi, et al.
(författare)
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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability
- 2023
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Ingår i: Communications on Applied Mathematics and Computation. - : Springer. - 2096-6385 .- 2661-8893. ; 5:2, s. 573-595
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Tidskriftsartikel (refereegranskat)abstract
- In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.
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| 8. |
- Amsallem, David, et al.
(författare)
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Energy Stable Model Reduction of Neurons by Non-negative Discrete Empirical Interpolation
- 2016
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM publishing. - 1064-8275 .- 1095-7197. ; 38:2, s. B297-B326
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Tidskriftsartikel (refereegranskat)abstract
- The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reduced-order basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups.
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| 9. |
- Amsallem, David, et al.
(författare)
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High-order accurate difference schemes for the Hodgkin-Huxley equations
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate dierence schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the rst demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial dierential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.
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| 10. |
- Amsallem, David, et al.
(författare)
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High-order accurate difference schemes for the Hodgkin-Huxley equations
- 2013
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 252, s. 573-590
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Tidskriftsartikel (refereegranskat)abstract
- A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin–Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.
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