In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain - decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term is the same as for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.
Accepté le :
DOI : 10.1051/m2an/2017029
Mots-clés : Augmented Burgers equation, numerical approximation, large-time behavior
@article{M2AN_2017__51_6_2367_0, author = {Ignat, Liviu I. and Pozo, Alejandro}, title = {A semi-discrete large-time behavior preserving scheme for the augmented {Burgers} equation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {2367--2398}, publisher = {EDP-Sciences}, volume = {51}, number = {6}, year = {2017}, doi = {10.1051/m2an/2017029}, mrnumber = {3745175}, zbl = {1394.65089}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2017029/} }
TY - JOUR AU - Ignat, Liviu I. AU - Pozo, Alejandro TI - A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 2367 EP - 2398 VL - 51 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017029/ DO - 10.1051/m2an/2017029 LA - en ID - M2AN_2017__51_6_2367_0 ER -
%0 Journal Article %A Ignat, Liviu I. %A Pozo, Alejandro %T A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 2367-2398 %V 51 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017029/ %R 10.1051/m2an/2017029 %G en %F M2AN_2017__51_6_2367_0
Ignat, Liviu I.; Pozo, Alejandro. A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2367-2398. doi : 10.1051/m2an/2017029. http://archive.numdam.org/articles/10.1051/m2an/2017029/
Multidisciplinary Optimization with Applications to Sonic-boom Minimization. Ann. Rev. Fluid Mech. 44 (2012) 505–526. | DOI | MR | Zbl
and ,Splitting methods for the nonlocal fowler equation. Math. Comput. 83 (2013) 1121–1141. | DOI | MR | Zbl
and ,Ph. Brenner, V. Thomée and L.B. Wahlbin, Besov spaces and applications to difference methods for initial value problems. In vol. 434 of Lect. Notes Math. Springer Verlag (1975). | MR | Zbl
Application of a model system to illustrate some points of the statistical theory of free turbulence. Proc. R. Netherlands Acad. Sci. 43 (1940) 2–12. | JFM | MR | Zbl
,T.W. Carlton and D.T. Blackstock, Propagation of plane waves of finite amplitude in inhomogeneous media with applications to vertical propagation in the ocean. Tech. Report ARL-TR-74-31, Applied Research laboratories, The University of Texas at Austin (1974).
C. Castro, F. Palacios and E. Zuazua, Optimal control and vanishing viscosity for the burgers equation. Integral Methods in Science and Engineering, edited by C. Costanda and M.E. Pérez, vol. 2. Birkhäuser Verlag (2010) 65–90. | MR | Zbl
R.O. Cleveland, Propagation of sonic booms through a real, stratified atmosphere. Ph.D. thesis, University of Texas at Austin (1995).
Sonic boom theory: its status in prediction and minimization. J. Aircraft 14 (1977) 569–576. | DOI
,Moments, masses de dirac et décomposition de fonctions. C. R. Acad. Sci. 315 (1992) 693–698. | MR | Zbl
and ,Large time behavior for convection-diffusion equations in . J. Functional Anal. 100 (1991) 119–161. | DOI | MR | Zbl
and ,Propagation of a strong sound wave in a plane layered medium. Soviet Phys. Acoustics 22 (1976) 349–350.
,E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Math. Appl. Ellipses (1991). | MR | Zbl
W.D. Hayes, Linearized supersonic flow. Ph.D. thesis, California Institute of Technology, Pasadena, California (1947).
Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws, Math. Comput. 84 (2015) 1633–1662. | DOI | MR | Zbl
, and ,A nonlocal convection-diffusion equation. J. Functional Anal. 251 (2007) 399–437. | DOI | MR | Zbl
and ,A class of singular integrals. Amer. J. Math. 86 (1964) 441–462. | DOI | MR | Zbl
,Lower bounds for sonic bangs. J. Royal Aeronautical Soc. 65 (1961) 433–436. | DOI
,Spikes and diffusion waves in one-dimensional model of chemotaxis. Nonlinearity 23 (2010) 3119–3137. | DOI | MR | Zbl
and ,Diffusive n-waves and metastability in the burgers equation. SIAM J. Math. Anal. 33 (2001) 607–633. | DOI | MR | Zbl
and ,M.J. Lighthill, Viscosity effects in sound waves of finite amplitude. Surveys in Mechanics, edited by Ge.K. Batchelor and R.M. Davies. Cambridge University Press (1956) 250–351. | MR
Source-solutions and asymptotic behavior in conservation laws. J. Differ. Equ. 51 (1984) 419–441. | DOI | MR | Zbl
and ,Effects of meteorological variability on sonic boom propagation from hypersonic aircraft. AIAA J. 47 (2009) 2632–2641. | DOI
and ,A.D. Pierce, Acoustics: an introduction to its physical principles and applications. Acoustical Soc. Amer. (1989).
A. Pozo, Large-time behavior of some numerical schemes: application to the sonic-boom phenomenon, Ph.D. thesis, Universidad del País Vasco (2014).
Advanced sonic boom prediction using augmented burger’s equation. J. Aircraft 48 (2011) 1245–1253. | DOI
,S.K. Rallabhandi, Sonic boom adjoint methodology and its applications, 29th AIAA Applied Aerodynamics Conference. American Institute of Aeronautics and Astronautics (2011).
Minimum sonic boom shock strengths and overpressures. Nature 221 (1969) 651–653. | DOI
,Sonic boom theory. J. Aircraft 6 (1969) 177–184. | DOI
,Sonic-boom Minimization. J. Acoustical Soc. Amer. 51 (1972) 686–694. | DOI
and ,Compact sets in the space . Ann. Mat. pura ed Applicata 146 (1987) 65–96. | DOI | MR | Zbl
,N.Th. Varopoulos, L. Saloff−Coste and Th. Coulhon, Analysis and geometry on groups. In vol. 100. Cambridge Tracts in Mathematics. Cambridge University Press (1992). | MR | Zbl
Effets régularisants de semi-groupes non linéaires dans des espaces de banach. Ann. fac. Sci. Toulouse 5e Série 1 (1979) 171–200. | DOI | Numdam | MR | Zbl
,The flow pattern of a supersonic projectile. Commun. Pure Appl. Math. 5 (1952) 301–348. | DOI | MR | Zbl
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