A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes

We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity an...

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Bibliographic Details
Published in:Journal of computational physics Vol. 431; p. 110145
Main Authors: Beyer, Florian, LeFloch, Philippe G.
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 15.04.2021
Elsevier Science Ltd
Subjects:
Algorithms
Asymptotic behavior
Boundary value problems
Cauchy problems
Compressible fluids
Computational fluid dynamics
Computational physics
Error analysis
Evolution
Flow control
Fluid flow
Fluids on Kasner
Fuchsian equations
General relativity
Hyperspaces
Mathematical analysis
Method of lines
Numerical analysis
Numerical approximation
Runge-Kutta method
Singularities
Spacetime
Numerical approximation
Fluids on Kasner
Fuchsian equations
General relativity
Asymptotic behavior
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity and the evolution is performed from the singularity hypersurface. We approximate the singular Cauchy problem of Fuchsian type by a sequence of regular Cauchy problems, which we next discretize by pseudo-spectral and Runge-Kutta techniques. Our main contribution is a detailed analysis of the numerical error which has two distinct sources, and our main proposal here is to keep in balance the errors arising at the continuum and at the discrete levels of approximation. We present numerical experiments which strongly support our theoretical conclusions. This strategy is finally applied to compressible fluid flows evolving on a Kasner spacetime, and we numerically demonstrate the nonlinear stability of such flows, at least in the so-called sub-critical regime identified earlier by the authors.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2021.110145