| 1. |
- Babiuc, M. C., et al.
(författare)
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Testing the well-posedness of characteristic evolution of scalar waves
- 2014
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 31:2, s. 025022-
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Tidskriftsartikel (refereegranskat)abstract
- Recent results have revealed a critical way in which lower order terms affect the well-posedness of the characteristic initial value problem for the scalar wave equation. The proper choice of such terms can make the Cauchy problem for scalar waves well posed even on a background spacetime with closed lightlike curves. These results provide new guidance for developing stable characteristic evolution algorithms. In this regard, we present here the finite difference version of these recent results and implement them in a stable evolution code. We describe test results which validate the code and exhibit some of the interesting features due to the lower order terms.
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| 2. |
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| 3. |
- Kreiss, Heinz-Otto, et al.
(författare)
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Geometric boundary data for the gravitational field
- 2014
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 31:6, s. 065004-
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Tidskriftsartikel (refereegranskat)abstract
- An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic curvature of the initialCauchy hypersurface, subject to constraints. This Cauchy data determine a solution to Einstein's equations which is unique up to a diffeomorphism. Here, we show how three pieces of unconstrained boundary data, which are associated locally with the geometry of the boundary, likewise determine a solution of the initial-boundary value problem which is unique, up to a diffeomorphism. Two pieces of this data constitute a conformal class of rank-2 metrics, which represent the two gravitational degrees of freedom. The third piece, constructed from the extrinsic curvature of the boundary, determines the dynamical evolution of the boundary.
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| 4. |
- Kreiss, Heinz-Otto, et al.
(författare)
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Initial-Boundary Value Problems For Second Order Systems Of Partial Differential Equations
- 2012
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 46:3, s. 559-593
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Tidskriftsartikel (refereegranskat)abstract
- We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of n equations as a larger first order system. Unfortunately, the resulting first order system consists, in general, of more than 2n equations which leads to many complications, such as side conditions which must be satisfied by the solution of the larger first order system. Here we will use the theory of pseudo-differential operators combined with mode analysis. There are many desirable properties of this approach: (1) the reduction to first order systems of pseudo-differential equations poses no difficulty and always gives a system of 2n equations. (2) We can localize the problem, i.e., it is only necessary to study the Cauchy problem and halfplane problems with constant coefficients. (3) The class of problems we can treat is much larger than previous approaches based on "integration by parts". (4) The relation between boundary conditions and boundary phenomena becomes transparent.
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| 5. |
- Kreiss, Heinz-Otto, et al.
(författare)
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The well-posedness of the null-timelike boundary problem for quasilinear waves
- 2011
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 28:14, s. 145020-
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Tidskriftsartikel (refereegranskat)abstract
- The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the worldtube. We establish the well-posedness of this problem for the evolution of a quasilinear scalar wave by means of energy estimates. The treatment is given in characteristic coordinates and thus provides a guide for developing stable finite difference algorithms. A new technique underlying the approach has potential application to other characteristic initial-boundary value problems.
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