| 1. |
- Kreiss, Gunilla, et al.
(författare)
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Stability of viscous shocks on finite intervals
- 2008
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 187:1, s. 157-183
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Tidskriftsartikel (refereegranskat)abstract
- Consider the Cauchy problem for a system of viscous conservation laws with a solution consisting of a thin, viscous shock layer connecting smooth regions. We expect the time-dependent behavior of such a solution to involve two processes. One process consists of the large-scale evolution of the solution. This process is well modeled by the corresponding inviscid equations. The other process is the adjustment in shape and position of the shock layer to the large-scale solution. The time scale of the second process is much faster than the first, 1/nu compared to 1. The second process can be divided into two parts, adjustment of the shape and of the position. During this adjustment the end states are essentially constant. In order to answer the question of stability we have developed a technique where the two processes can be separated. To isolate the fast process, we consider the region in the vicinity of the shock layer. The equations are augmented with special boundary conditions that reflect the slow change of the end states. We show that, for the isolated fast process, the perturbations decay exponentially in time.
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| 2. |
- Babiuc, M. C., et al.
(författare)
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Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations
- 2007
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Ingår i: PHYSICAL REVIEW D. - 1550-7998. ; 75:4, s. 044002-
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Tidskriftsartikel (refereegranskat)abstract
- The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A new formulation of constraint-preserving boundary conditions of the Sommerfeld-type for such systems has recently been proposed. We implement these boundary conditions in a nonlinear 3D evolution code and test their accuracy.
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| 3. |
- Kozameh, Carlos, et al.
(författare)
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On the well posedness of Robinson-Trautman-Maxwell solutions
- 2008
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 25:2, s. 025004-
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Tidskriftsartikel (refereegranskat)abstract
- We show that the so-called Robinson - Trautman - Maxwell equations do not constitute a well-posed initial-value problem. That is, the dependence of the solution on the initial data is not continuous in any norm built out from the initial data and a finite number of its derivatives. Thus, they cannot be used to solve for solutions outside the analytic domain.
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| 4. |
- Kreiss, Heinz-Otto, et al.
(författare)
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A note on viscous conservation laws with complex characteristics
- 2006
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 46, s. S55-S59
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Tidskriftsartikel (refereegranskat)abstract
- There are several physical set-ups involving multi-phase fluids that result in highly unstable behavior already at rather low flow rates. Mathematical models of these flow problems consist typically of conservation laws like conservation of mass and momentum for each phase together with coupling terms connecting the phases. For multi-phase flow the characteristics are often complex and without the dissipative terms the problem is ill-posed and not computable. We will discuss why the nonlinearity of the system can prevent blow-up.
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| 5. |
- Kreiss, Heinz -Otto, et al.
(författare)
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Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems
- 2009
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 289:3, s. 1099-1129
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Tidskriftsartikel (refereegranskat)abstract
- We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T] x I pound, where I pound is a compact manifold with smooth boundaries a,I pound. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on a,I pound. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell's equations in the Lorentz gauge and Einstein's gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.
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| 6. |
- Kreiss, Heinz-Otto, et al.
(författare)
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Problems which are well posed in a generalized sense with applications to the Einstein equations
- 2006
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Ingår i: Classical and quantum gravity. - 0264-9381 .- 1361-6382. ; 23:16, s. S405-S420
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Tidskriftsartikel (refereegranskat)abstract
- In the harmonic description of general relativity, the principal part of the Einstein equations reduces to a constrained system of ten curved space wave equations for the components of the spacetime metric. We use the pseudo-differential theory of systems which are strongly well posed in the generalized sense to establish the well posedness of constraint-preserving boundary conditions for this system when treated in a second-order differential form. The boundary conditions are of a generalized Sommerfeld type that is benevolent for numerical calculation.
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| 7. |
- Kreiss, Heinz-Otto, et al.
(författare)
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Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates
- 2007
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 24:23, s. 5973-5984
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Tidskriftsartikel (refereegranskat)abstract
- In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second-order systems of wave equations be strongly well-posed in a generalized sense. The applications included the harmonic version of the Einstein equations. Here we show that these results can also be obtained via standard energy estimates, thus establishing strong well-posedness of the harmonic Einstein problem in the classical sense.
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| 8. |
- Motamed, Mohammad, et al.
(författare)
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Finite Difference Schemes for Second Order Systems Describing Black Holes
- 2006
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Ingår i: Physical Review D. Particles and fields. - 0556-2821 .- 1089-4918. ; 73:12, s. 124008-1-124008-14
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Tidskriftsartikel (refereegranskat)abstract
- In the harmonic description of general relativity, the principal part of Einstein's equations reduces to 10 curved space wave equations for the components of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.
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| 9. |
- Szilagyi, B., et al.
(författare)
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Modeling the black hole excision problem
- 2005
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Ingår i: Physical Review D. - 1550-7998 .- 1550-2368. ; 71:10, s. 104035-
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Tidskriftsartikel (refereegranskat)abstract
- We analyze the excision strategy for simulating black holes. The problem is modeled by the propagation of quasilinear waves in a 1-dimensional spatial region with timelike outer boundary, spacelike inner boundary and a horizon in between. Proofs of well-posed evolution and boundary algorithms for a second differential order treatment of the system are given for the separate pieces underlying the finite-difference problem. These are implemented in a numerical code which gives accurate long term simulations of the quasilinear excision problem. Excitation of long wavelength exponential modes, which are latent in the problem, are suppressed using conservation laws for the discretized system. The techniques are designed to apply directly to recent codes for the Einstein equations based upon the harmonic formulation.
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