A fractional time derivative is introduced into Burgers equation to model losses of nonlinear waves arising in acoustics. A diffusive representation of the fractional derivative replaces the non-local operator by a continuum of memory variables that satisfy local ordinary differential equations. A quadrature formula yields a system of local partial differential equations. The quadrature coefficients are computed by optimization with a positivity constraint. One resolves the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Extensive details can be found in [3]

Lombard, Bruno

Matignon, Denis

Open Archive Toulouse Archive Ouverte

											 	   		 	  	     	 			 									 an author's https://oatao.univ-toulouse.fr/19972Lombard, Bruno and Matignon, Denis Numerical modeling of a time-fractional Burgers equation. (2017) In: 13thInternational Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2017), 15 May 2017- 19 May 2017 (Minneapolis, United States).WAVES 2017, MinneapolisNumerical modeling of a time-fractional Burgers equationBruno Lombard1,∗, Denis Matignon21LMA, CNRS, UPR 7051, Aix-Marseille Univ., Centrale Marseille, 13453 Marseille, France2ISAE-Supae´ro, University of Toulouse, BP 54032, 31055 Toulouse, France∗Email: lombard@lma.cnrs-mrs.frAbstractA fractional time derivative is introduced intoBurgers equation to model losses of nonlinearwaves arising in acoustics. A diffusive represen-tation of the fractional derivative replaces thenonlocal operator by a continuum of memoryvariables that satisfy local ordinary differentialequations. A quadrature formula yields a sys-tem of local partial differential equations. Thequadrature coefficients are computed by opti-mization with a positivity constraint. One re-solves the hyperbolic part by a shock-capturingscheme, and the diffusive part exactly. Exten-sive details can be found in [3].Keywords: fractional derivatives, diffusive rep-resentation, nonlinear acoustics, Strang split-ting1 IntroductionWe investigate Burgers equation with a frac-tional time derivative Dαt (ε ≥ 0, 0 < α < 1):∂u∂t+∂∂x(au+ bu22)= −εDαt u, (1)For a causal function u(t), Dαt u refers to theCaputo fractional derivative in time of order α:Dαt u =1Γ(1− α)∫ t0(t− τ)−α∂u∂τ(x, τ) dτ, (2)where Γ is the Gamma function. The l.h.s. of(1) is a standard transport equation, with lin-ear advection at constant speed a and a nonlin-ear quadratic term with coefficient b. The r.h.s.of (1) models linear losses and memory effectsalong the propagation. Since α < 1, the hyper-bolic nature of Burgers equation is preserved.Various physical configurations are describedby (1). Particular values of ε and α enable torecover Chester’s equation describing propaga-tion of finite-amplitude sound waves in tubes,up to O(ε2) terms. This equation is widely usedto model brass instruments (trombones, trum-pets): the transport terms describe the steep-ening of waves, yielding the typical ”brassy”effect, and the fractional term models the vis-cothermal losses at the wall of the duct. More-over, the linear Lokshin equation can be seenas the superposition of two one-way fractionaltransport equations of this type [1]. Other ap-plications of (1) concern viscoelasticity, propa-gation in elastic-walled tubes, or more generallywave propagation in media with memory andcomplex rheological properties.2 Diffusive approximationThe convolution product (2) can be recast asDαt u =∫ +∞0φ(x, t, θ) dθ, (3)with the diffusive variable φ given byφ(x, t, θ) = γα θ2α−1∫ t0∂u∂τ(x, τ) e−(t−τ) θ2dτ,(4)with γα =2 sin(piα)pi> 0. From equation (4), φsatisfies the following first-order ordinary differ-ential equation (ODE):∂φ∂t= −θ2 φ+ γα θ2α−1∂u∂t. (5)The integral in (3) is approximated by a quadra-ture formula on L points, where the diffusivevariables φ` satisfy an ODE deduced from (5):Dαt u(x, t) ≈L∑`=1µ` φ(x, t, θ`) ≡L∑`=1µ` φ`(x, t),∂φ`∂t= −θ2` φ` + γα θ2α−1`∂u∂t, ` = 1, · · · , L.(6)An adequate choice of the weights µ` and nodesθ` is crucial for the efficiency and accuracy ofthe diffusive approximation, see e.g. [3] and ref-erences therein. Injecting (6) into (1) yields∂∂tU+∂∂xF(U) = SU, (7)with U(x, t) = (u, φ1, . . . , φL)T . The energy ofU decreases if µ` > 0 and θ` > 0. A StrangWAVES 2017, Minneapolissplitting is used to solve (7). The propagativepart is solved by a finite-volume scheme withflux-limiters [2], whereas the diffusive part issolved exactly. The CFL condition of stabilityis the same as for the inviscid Burgers equation.3 Numerical experimentsα = 1/3 α = 1/38 9 10 11 12−1 −0.5 0 0.5 1 x (m)u (m/s)simulationanalytical0 0.01 0.02 0.03 0.04 0.05 0.06 0.070 5 10 15 20 t (s)Offset (m)α = 1/2 α = 1/28 9 10 11 12−1 −0.5 0 0.5 1 x (m)u (m/s)simulationanalytical0 0.01 0.02 0.03 0.04 0.05 0.06 0.070 5 10 15 20 t (s)Offset (m)Figure 1: linear fractional advection, for vari-ous fractional order α. Left row: snapshots ofthe numerical and exact solutions. Right row:seismograms.First, we consider linear advection (b = 0).A smooth truncated sinusoid is injected at theleft boundary of the domain. Closed-form an-alytical solutions are known for α = 1/3 andα = 1/2. These cases are illustrated in fig-ure 1. In the left row, one compares snapshotsof the numerical and exact solutions. Greatervalues of α yield a greater attenuation and aslower propagation, as predicted by the disper-sion analysis [3]. The right row illustates thetime and space evolution of the waves.Second, we consider both nonlinear prop-agation and fractional attenuation (figure 2).The initial data is a rectangular pulse. Withoutattenuation, classical phenomena are observed:the pulse splits into a left rarefaction wave anda right shock (a), which collide (b). With atten-uation, the right-going shock smears and evendisappears for sufficiently large ε.(a) ε = 0, t1 ε = 0, (b) t2 > t10 2 4 6 8 10 12 14 16 18 200 20 40 60 80 100 x (m)u (m/s)simulationanalytical0 2 4 6 8 10 12 14 16 18 200 20 40 60 80 100 x (m)u (m/s)simulationanalytical(c) α = 1/2, t1 α = 1/2, (d) t2 > t10 2 4 6 8 10 12 14 16 18 200 20 40 60 80 100 x (m)u (m/s)inviscideps = 0.5eps = 2.0eps = 5.00 2 4 6 8 10 12 14 16 18 200 20 40 60 80 100 x (m)u (m/s)inviscideps=0.5eps=2.0eps=5.0Figure 2: nonlinear advection and fractional at-tenuation. (a-b): numerical and exact solutionswithout attenuation. (c-d): numerical solutionsfor α = 1/2 and various values of ε.4 ConclusionThis article is an attempt for better understand-ing the competition between nonlinear effectsand nonlocal relaxation. Many theoretical ques-tions remain to be addressed. In particular,it seems that the emergence of shocks is con-ditional (unlike the inviscid Burgers equation).This question requires a deeper analysis to con-firm / infirm the numerical observations.References[1] T. He´lie and D. Matignon, Diffusive repre-sentations for the analysis and simulationof flared acoustic pipes with visco-thermallosses, Math. Models Meth. Appl. Sci. 16(2006), pp. 503–536.[2] R. J. LeVeque, Finite Volume Methods forHyperbolic Problems, Cambridge Univer-sity Press, 2002.[3] B. Lombard and D. Matignon, Diffusiveapproximation of a time-fractional Burg-ers equation in nonlinear acoustics, SIAMJ. Appl. Math. 76-5 (2016), pp. 1765–1791.

Numerical modeling of a time-fractional Burgers equation

http://oatao.univ-toulouse.fr/19972/1/Lombard_19972..pdf

Numerical modeling of a time-fractional Burgers equation

Abstract

Similar works

Full text

Available Versions

Open Archive Toulouse Archive Ouverte