| 1. |
- Persson, Ingemar, et al.
(författare)
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On the convergence of multigrid methods for flow problems
- 1999
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Ingår i: Electronic Transactions on Numerical Analysis. - 1068-9613. ; 8, s. 46-87
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Tidskriftsartikel (refereegranskat)abstract
- We prove two theorems on the residual damping in multigrid methods when solving convection dominated diffusion equations and shock wave problems, discretized by the streamline diffusion finite element method. The first theorem shows that a V-cycle, including sufficiently many pre and post smoothing steps, damps the residual in LIloc for a constant coefficient convection problem with small diffusion in two space dimensions, without the assumption that the coarse grid is sufficiently fine. The proof is based on discrete Green's functions for the smoothing and correction operators on a uniform unbounded mesh aligned with the characteristic. The second theorem proves a similar result for a certain continuous version of a two grid method, with Isotropic artificial diffusion, applied to a two dimensional Burgers shock wave problem. We also present numerical experiments that verify the residual damping dependence on the equation, the choice of artificial diffusion and the number of smoothing steps. In particular numerical experiments show improved convergence of the multigrid method, with damped Jacobi smoothing steps, for the compressible Navier-Stokes equations in two space dimensions by using the theoretically suggested exponential increase of the number of smoothing steps on coarser meshes, as compared to the same amount of work with constant number of smoothing steps on each level.
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| 2. |
- Johnson, Claes, et al.
(författare)
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Adaptive finite element methods for conservation laws based on a posteriori error estimates
- 1995
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Ingår i: Communications on Pure and Applied Mathematics. - NEW YORK : John Wiley & Sons. - 0010-3640 .- 1097-0312. ; 48:3, s. 199-234
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Tidskriftsartikel (refereegranskat)abstract
- We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way.
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| 3. |
- Szepessy, Anders, 1960-
(författare)
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Dynamics and stability of a weak detonation wave
- 1999
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Ingår i: Communications in Mathematical Physics. - NEW YORK : Springer-Verlag New York. - 0010-3616 .- 1432-0916. ; 202:3, s. 547-569
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Tidskriftsartikel (refereegranskat)abstract
- One dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable, i.e. initially perturbed waves tend asymptotically to translated weak detonation waves. This model system was derived as the low Math number limit of the one component reactive Navier-Stokes equations by Majda and Roytburd [SIAM J. Sci. Stat. Comput. 43, 1086-1118 (1983)], and its weak detonation waves have been numerically observed as stable. The analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave, The shock translation solves a nonlinear integral equation, obtained by Green's function techniques, and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder. The inverse operator of the convolution and detailed properties of the traveling wave reduce, by monotonicity, the remainder to a small L-1 perturbation.
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| 4. |
- Szepessy, Anders, 1960-, et al.
(författare)
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Stability of rarefaction waves in viscous media
- 1996
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Ingår i: Archive for Rational Mechanics and Analysis. - NEW YORK : Springer-Verlag New York. - 0003-9527 .- 1432-0673. ; 133:3, s. 249-298
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Tidskriftsartikel (refereegranskat)abstract
- We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, ''Burgers'' rarefaction wave, for initial perturbations w(o) with small mass and localized as w(o)(x)= O(\x\(-1)). The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error. This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass O(log(t)). These ''diffusion waves'' have amplitude O(t(-1/2) log t) in linear degenerate transversal fields and O(t(-1/2)) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof
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