| 1. |
- 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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| 2. |
- Erickson, Brittany. A., et al.
(författare)
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Stable, High Order Accurate Adaptive Schemes for Long Time, Highly Intermittent Geophysics Problems
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.
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| 3. |
- 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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| 4. |
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| 5. |
- Frenander, Hannes, et al.
(författare)
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A stable and accurate data assmimilation technique using multiple penalty terms in space and time
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A new method for data assimilation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.
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| 6. |
- Frenander, Hannes, et al.
(författare)
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Constructing non-reflecting boundary conditions using summation-by-parts in time
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments.
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| 7. |
- Frenander, Hannes, et al.
(författare)
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Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative.
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A one dimensional steady-state advection-diffusion problem using summation-by-parts operators has been investigated. For approximating the second derivative, a wide stencil has been used, which has spurious, oscillating, modes for all mesh-sizes. We show that the size of the spurious modes are equal to the size of the truncation error for a stable approximation. The theoretical results are veried with numerical experiments.
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| 8. |
- Ghader, Sarmad, et al.
(författare)
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High-order compact finite difference schemes for the spherical shallow water equations
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- This work is devoted to the application of the super compact finite difference (SCFDM) and the combined compact finite difference (CCFDM) methods for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence and height. Five high-order schemes including the fourth-order compact, the sixth-order and eighth-order SCFDM and the sixth-order and eighth-order CCFDM schemes are used for spatial differencing of the spherical shallow water equations. To advance the solution in time, a semi-implicit Runge-Kutta method is used. In addition, to control the nonlinear instability and avoiding the polar problem a high-order spatial filter is proposed. An unstable barotropic mid-latitude zonal jet is employed as an initial condition. For the numerical solution of the elliptic equations in the problem, a direct hybrid method which consists of using a high-order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate is utilized. The convergence rate for all methods is studied and veried. Qualitative and quantitative assessment of the results, such as measures of maximum vorticity gradient, power spectrum of total energy, relative change in potential enstrophy and potential palinstrophy, reveal that the sixth-order and eighth-order CCFDM and the sixth-order and eighth-order SCFDM methods lead to a remarkable improvement of the solution over the fourth-order compact method. It is also shown that the performance of the sixth-order and eighth-order CCFDM methods are superior to the sixth-order and eighth-order SCFDM methods. Copyright c 2013 John Wiley & Sons, Ltd.
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| 9. |
- Ghader, Sarmad, et al.
(författare)
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Well-posed boundary conditions for the shallow water equations
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We derive well-posed boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. Then, as an example, the boundary conditions are discussed for a rectangular domain.
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| 10. |
- Ghasemi, Fatemeh, et al.
(författare)
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Coupling requirements for multi-physics problems
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider two hyperbolic systems onfirst order form of different size posed on two domains. Our ambition is to derive general conditions for when the two systems can and cannot be coupled.The adjoint equations are derived and well-posedness of the primal and dual problems are discussed. By applying the energy method, interface conditions for the primal and dual problems are derived such that the continuous problems are well posed.The equations are discretized using a high order finite difference method on summation-by-parts form and the interface conditions are imposed weakly in a stable way, using penalty formulations. It is shown that one specific choice of penalty matrices leads to a dual consistent scheme and superconverging functionals.By considering an example, it is shown that the correct physical coupling conditions are contained in the set of well posed coupling conditions. It is also shown that the correct convergence rates are obtained for both the solutions and functionals.
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