4,463 research outputs found
Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary conditions, and of the Cauchy problem for
the same system (we also prove computable dependence on the coefficients) in a
cube . Such systems describe a wide variety of physical
processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many
boundary-value problems for the wave equation also can be reduced to this case,
thus we partially answer a question raised in Weihrauch and Zhong (2002).
Compared with most of other existing methods of proving computability for PDEs,
this method does not require existence of explicit solution formulas and is
thus applicable to a broader class of (systems of) equations.Comment: 31 page
Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
We present a modification to the Berger and Oliger adaptive mesh refinement
algorithm designed to solve systems of coupled, non-linear, hyperbolic and
elliptic partial differential equations. Such systems typically arise during
constrained evolution of the field equations of general relativity. The novel
aspect of this algorithm is a technique of "extrapolation and delayed solution"
used to deal with the non-local nature of the solution of the elliptic
equations, driven by dynamical sources, within the usual Berger and Oliger
time-stepping framework. We show empirical results demonstrating the
effectiveness of this technique in axisymmetric gravitational collapse
simulations. We also describe several other details of the code, including
truncation error estimation using a self-shadow hierarchy, and the
refinement-boundary interpolation operators that are used to help suppress
spurious high-frequency solution components ("noise").Comment: 31 pages, 15 figures; replaced with published versio
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
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