| 1. |
- Ahlkrona, Josefin, et al.
(författare)
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A numerical study of the validity of Shallow Ice Approximations
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Improving numerical ice sheet models is a very active field of research. In part, this is because ice sheet modelling has gained societal relevance in the context of predictions of future sea level rise. Ice sheet modelling is however also a challenging mathematical and computational subject. Since the exact equations governing ice dynamics, the full Stokes equations, are computationally expensive to solve, approximations are crucially needed for many problems. Shallow ice approximations are a family of approximations derived by asymptotic expansion of the exact equations in terms of the aspect ratio, epsilon. Retaining only the zeroth order terms in this expansion yields the by far most frequently used approximation; the Shallow Ice Approximation (SIA). Including terms up to second order yields the Second Order Shallow Ice Approximation (SOSIA), which is a so-called higher order model. Here, we study the validity and accuracy of shallow ice approximations beyond previous analyses of the SIA. We perform a detailed analysis of the assumptions behind shallow ice approximations, i.e. of the order of magnitude of field variables. We do this by using a numerical solution of the exact equations for ice flow over a sloping, undulating bed. We also construct analytical solutions for the SIA and SOSIA and numerically compute the accuracy for varying epsilon by comparing to the exact solution. We find that the assumptions underlying shallow ice approximations are not entirely appropriate since they do not account for a high viscosity boundary layer developing near the ice surface as soon as small bumps are introduced at the ice base. This boundary layer is thick and has no distinct border. Other existing theories which do incorporate the boundary layer are in better, but not full, agreement with our numerical results. Our results reveal that neither the SIA nor the SOSIA is as accurate as suggested by the asymptotic expansion approach. Also, in SOSIA the ice rheology needs to be altered to avoid infinite viscosity, though both our analytical and numerical solutions show that, especially for high bump amplitudes, the accuracy of the SOSIA is highly sensitive to this alternation. However, by updating the SOSIA solution in an iterative manner, we obtain a model which utilises the advantages of shallow ice approximations, while reducing the disadvantages.
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| 2. |
- Bani-Hashemian, Hossein, et al.
(författare)
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Efficient sampling in event-driven algorithms for reaction-diffusion processes
- 2011
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In event-driven algorithms for simulation of diffusing, colliding, and reacting particles, new positions and events are sampled from the cumulative distribution function (CDF) of a probability distribution. The distribution is sampled frequently and it is important for the efficiency of the algorithm that the sampling is fast. The CDF is known analytically or computed numerically. Analytical formulas are sometimes rather complicated making them difficult to evaluate. The CDF may be stored in a table for interpolation or computed directly when it is needed. Different alternatives are compared for chemically reacting molecules moving by Brownian diffusion in two and three dimensions. The best strategy depends on the dimension of the problem, the length of the time interval, the density of the particles, and the number of different reactions.
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| 3. |
- Brüger, Arnim, et al.
(författare)
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High order accurate solution of the incompressible Navier-Stokes equations
- 2003
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- High order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation. With this goal in mind, the incompressible Navier-Stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid. The equations are integrated in time by a second order semi-implicit method. Stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions. In every time step, a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning. The convergence properties of the iterative method are analyzed. The order of accuracy of the method is demonstrated in numerical experiments. The method is used to compute the flow in a channel, the driven cavity and a constricted channel.
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| 4. |
- Edelvik, Fredrik, et al.
(författare)
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An Unconditionally Stable Subcell Model for Arbitrarily Oriented Thin Wires in the FETD Method
- 2002
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A computational subcell model for thin wires is developed for electromagnetic simulations. The Maxwell equations are discretized by a finite element approximation on a tetrahedral grid. The wires are described by a second-order equation for the current. The geometry of the wires can be chosen independent of the volume grid. A symmetric coupling between field and wires yields a stable semi-discrete field-wire system and an unconditionally stable fully discrete field-wire system. The system of equations is in each time step solved by a preconditioned conjugate gradient method. The accuracy of the subcell model is demonstrated for dipole and loop antenna with comparisons with the Method of Moments and experimental data.
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| 5. |
- Elf, Johan, et al.
(författare)
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Problems of High Dimension in Molecular Biology
- 2004
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The deterministic reaction rate equations are not an accurate description of many systems in molecular biology where the number of molecules of each species often is small. The master equation of chemical reactions is a more accurate stochastic description suitable for small molecular numbers. A computational difficulty is the high dimensionality of the equation. We describe how it can be solved by first approximating it by the Fokker-Planck equation. Then this equation is discretized in space and time by a finite difference method. The method is compared to a Monte Carlo method by Gillespie. The method is applied to a four-dimensional problem of interest in the regulation of cell processes.
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| 6. |
- Engblom, Stefan, et al.
(författare)
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Simulation of stochastic reaction-diffusion processes on unstructured meshes
- 2008
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Stochastic chemical systems with diffusion are modeled with a reaction-diffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with Brownian motion in space to obtain the reaction-diffusion master equation. The space is covered by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso- or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology.
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| 7. |
- Ferm, Lars, et al.
(författare)
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A hierarchy of approximations of the master equation scaled by a size parameter
- 2007
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Solutions of the master equation are approximated using a hierarchy of models based on the solution of ordinary differential equations: the macroscopic equations, the linear noise approximation and the moment equations. The advantage with the approximations is that the computational work with deterministic algorithms grows as a polynomial in the number of species instead of an exponential growth with conventional methods for the master equation. The relation between the approximations is investigated theoretically and in numerical examples. The solutions converge to the macroscopic equations when a parameter measuring the size of the system grows. A computational criterion is suggested for estimating the accuracy of the approximations. The numerical examples are models for the migration of people, in population dynamics and in molecular biology.
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| 8. |
- Ferm, Lars, et al.
(författare)
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Accurate and stable grid interfaces for finite volume methods
- 2002
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A convection-diffusion equation is discretized by a finite volume method in two space dimensions. The grid is partitioned into blocks with jumps in the grid size at the block interfaces. Interpolation in the cells adjacent to the interfaces is necessary to be able to apply the difference stencils. Second order accuracy is achieved and the stability of the discretizations is investigated. The interface treatment is tested in the solution of the compressible Navier-Stokes equations. The conclusions from the scalar equation are valid also for these equations.
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| 9. |
- Ferm, Lars, et al.
(författare)
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Adaptive, Conservative Solution of the Fokker-Planck Equation in Molecular Biology
- 2004
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The Fokker-Planck equation on conservation form is discretized by a finite volume method and advanced in time by a linear multistep method. The grid cells are refined and coarsened in blocks of the grid depending on an estimate of the spatial discretization error and the time step is chosen to satisfy a tolerance on the temporal discretization error. The solution is conserved across the block boundaries so that the total probability is constant. A similar effect is achieved by rescaling the solution. The steady state solution is determined as the eigenvector corresponding to the zero eigenvalue. The method is applied to the solution of a problem with two molecular species and the simulation of a circadian clock. Comparison is made with a stochastic method.
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| 10. |
- Ferm, Lars, et al.
(författare)
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Adaptive error control for steady state solutions of inviscid flow
- 2000
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The steady state solution of the Euler equations of inviscid flow is computed by an adaptive method. The grid is structured and is refined and coarsened in predefined blocks. The equations are discretized by a finite volume method. Error equations, satisfied by the solution errors, are derived with the discretization error as the driving right hand side. An algorithm based on the error equations is developed for errors propagated along streamlines. Numerical examples from two-dimensional compressible and incompressible flow illustrate the method.
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