Numerical methods for a fractional diffusion/anti-diffusion equation

Numerical methods for a fractional diffusion/anti-diffusion equation Afaf Bouharguane Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux 1, France Berlin, November 2012 Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 1 / 30

Outline 1 Introduction 2 Finite difference schemes - Collaboration : Pascal Azerad (Univ. Montp2 ) 3 Splitting methods - Collaboration : Rémi Carles (CNRS & Univ. Montp2) Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 2 / 30

Model for dune morphodynamics A conservative nonlinear model For all t (0,T) et x R, { ( ) u t (t,x)+ u 2 2 (t,x) u xx(t,x)+i[u(t, )](x) = 0, x u(0,x) = u 0 (x). Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 4 / 30

Model for dune morphodynamics A conservative nonlinear and nonlocal model (A.C Fowler, Oxford) For all t (0,T) et x R, { ) u t (t,x)+ ( u 2 2 u(0,x) = u 0 (x). x (t,x) u xx(t,x)+ + 0 ξ 1/3 u xx (t,x ξ)dξ = 0, Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 4 / 30

Model for dune morphodynamics A nonlinear and nonlocal conservative model (A.C Fowler, Oxford) For all t (0,T) and x R, { ) u t (t,x)+ ( u 2 2 u(0,x) = u 0 (x). x (t,x) u xx(t,x)+ + 0 ξ 1/3 u xx (t,x ξ)dξ = 0, Remark : This nonlocal term also appears in the work of P.-Y. Lagrée (Paris VI). References : P-Y Lagrée, Asymptotic Methods in Fluid Mechanics : Survey and Recent Advances, lecture notes 523, CISM International Centre for Mechanical Sciences Udine, H. STEINRÜCK Ed., Springer, (2010). A.C. Fowler, Mathematics and environment, lecture note, 2006. Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 4 / 30

Some expressions for I Integral formula : For all ϕ S(R) and all x R I[ϕ](x) = 4 9 0 ϕ(x +z) ϕ(x) ϕ (x)z z 7/3 dz Fractional derivative : For causal functions (i.e. ϕ(x) = 0 for x < 0) 1 + Γ(2/3) 0 ϕ (x ξ) ξ 1/3 dξ = d 2/3 dx 2/3ϕ (x) = d4/3 dx 4/3ϕ(x) Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 5 / 30

Anti-diffusive effect Pseudo-differential formula For all ϕ S(R) and all ξ R F (I[ϕ])(ξ) = (a I ±b I i) ξ 4/3 Fϕ(ξ), Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 6 / 30

Anti-diffusive effect Pseudo-differential formula For all ϕ S(R) and all ξ R F (I[ϕ])(ξ) = (a I ± b I i) ξ 4/3 Fϕ(ξ), = I ( ) λ 2 with λ = 4 3. anti-diffusive effect Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 6 / 30

Some results Existence, uniqueness and continuous dependence of the solution u w.r.t. initial datum in L 2 (R), Failure of maximum principle, Existence of travelling waves φ C 1 b (R), Global existence of L 2 perturbations of travelling waves, Instability of constant solutions. References : N. Alibaud, P. Azerad, D. Isèbe, A non-monotone conservation law for dune morphodynamics, Differential Integral Equations, 2010. B. Alvarez-Samaniego, P. Azerad, Travelling wave solutions of the Fowler equation, Discrete and Continuous Dynamical Systems, B, 2009. A.B. On the instability of a nonlocal scalar conservation law, Disc. Cont. Dyn. Syst., Ser. S Vol. 5, no 3 (2012). A.B. Global existence of solutions to the Fowler equation in a neighbourhood of travelling-waves, Int. J. Diff. Eq. (2011). Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 7 / 30

Linearized problem { vt (t,x) v xx (t,x)+i[v(t, )](x) = 0, v(0,x) = v 0 (x). = v(t,x) = K(t, ) v 0, Kernel of I 2 xx K(t, ) = F 1( e tψ I) with ψ I (ξ) = 4π 2 ξ 2 a I ξ 4/3 +b I iξ ξ 1/3 Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 8 / 30

Linearized problem { vt (t,x) v xx (t,x)+i[v(t, )](x) = 0, v(0,x) = v 0 (x). = v(t,x) = K(t, ) v 0, Properties of K(t, ),t > 0 C 0 -semi-group : Kernel of I 2 xx K(t, ) = F 1( e tψ I) with ψ I (ξ) = 4π 2 ξ 2 a I ξ 4/3 +b I iξ ξ 1/3 K(t) K(s) = K(t +s) u 0 L 2 (R), lim t 0 K(t) u 0 = u 0 Regularity K(t,x) C ((0,+ ) R) Estimates for the gradient : x K(t) L 2 Ct 3/4 x K(t) L 1 Ct 1/2 Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 8 / 30

Linearized problem { vt (t,x) v xx (t,x)+i[v(t, )](x) = 0, v(0,x) = v 0 (x). = v(t,x) = K(t, ) v 0, Kernel of I 2 xx K(t, ) = F 1( e tψ I) with ψ I (ξ) = 4π 2 ξ 2 a I ξ 4/3 +b I iξ ξ 1/3 Evolution of K for t = 0.1 and t = 0.5. = K(t, ) is not positive! Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 8 / 30

Explicit finite differences u n+1 j t u n j +F(u n j 1,un j,un j+1 ) un j+1 2un j +u n j 1 x 2 +I x [u n ] j = 0 Two discretizations for the nonlocal term : I x 1 [ϕ] + j = x 4/3 l 1/3 (ϕ j l+1 2ϕ j l +ϕ j l 1 ) l=1 I x 2 [ϕ] j = 4 + ( 9 x 4/3 l 7/3 ϕ j l ϕ j + ϕ ) j+1 ϕ j 1 l 2 l=1 Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 10 / 30

Linearization For v > 0, we have u n+1 j t u n j +v un j u n j 1 x ǫ un j+1 2un j +u n j 1 x 2 +ηi x [u n ] j = 0 Numerical scheme reads for I 1 x u n+1 j = ǫ t + ( x 2 un j+1 + 1 v t x 2ǫ t x 2 η t ( v t x + ǫ t x 2 +(2 2 1/3 ) η t η t x 4/3 + l=2 x 4/3 x 4/3 ) u n j 1 [ (l +1) 1/3 2l 1/3 +(l 1) 1/3] u n j l. ) u n j Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 11 / 30

Linearization For v > 0, we have u n+1 j t u n j +v un j u n j 1 x ǫ un j+1 2un j +u n j 1 x 2 +ηi x [u n ] j = 0 Numerical scheme reads for I 1 x u n+1 j Note : u n+1 j = ǫ t + ( x 2 un j+1 + 1 v t x 2ǫ t x 2 η t ( v t x + ǫ t x 2 +(2 2 1/3 ) η t η t x 4/3 + l=2 x 4/3 x 4/3 ) u n j 1 [ (l +1) 1/3 2l 1/3 +(l 1) 1/3] u n j l. is not a convex combination of (u n j ) j N ) u n j Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 11 / 30

Amplification factor u(t,x) = e ikx+σt is solution of iff dispersion relationship u t +vu x ǫu xx +ηi[u] = 0, σ +iµk +ǫk 2 ηk 4/3 Γ Exact amplification factor ( ) ( 2 1 3 2 i ) 3 = 0 2 ( G cont = e t( ǫk2 +ηk 4/3 1 2 Γ(2 i t µk+ηk 3)) e 4/3 3 2 Γ(2 3) ) Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 12 / 30

Strategy High frequencies are responsible of numerical instabilities High frequencies are not amplified in Fowler s model Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 13 / 30

Strategy High frequencies are responsible of numerical instabilities High frequencies are not amplified in Fowler s model + Definition (High Frequency stability) We say that a numerical scheme is HF-stable if the high frequencies are strongly stable that is to say : 0 < θ 0 < π such that (θ 0,π], g( x, t,θ) < 1, where g is the discrete amplification factor. = Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 13 / 30

High frequency stability Definition (High Frequency stability) We say that a numerical scheme is HF-stable if the high frequencies are strongly stable that is to say : 0 < θ 0 < π such that (θ 0,π], g( x, t,θ) < 1, where g is the discrete amplification factor. Von-Neumann method : Input : one single Fourier mode u n j = û n k eikx j Output : û n+1 k = g( x, t,k)û n k Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 14 / 30

Stability criteria Proposition A scheme is HF-stable if x and t satisfy the following conditions : For I x 1. v t x + 2ǫ t x 2 +(2 2 1/3 )η t < 1, x4/3 (1 2 1/3 ) η t 2ǫ t x4/3 x 2. For I 2 x. v t x 4 9 where ζ is the Riemann zeta function. t + 2ǫ x 2 + 4 ( ζ( 4 ) η t 9 3 ) 1 < 1, x4/3 ( ζ( 7 ) η t 3 ) 1+ζ(4 3 ) 2ǫ t x4/3 x 2. Azerad P., B. A., Finite difference approximations for a diffusion/anti-diffusion equation, submitted Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 15 / 30

Numerical experiments Modified CFL µ t x +2ǫ t x 2+(2 2 1/3 )η t x 4/3 < 1 θ = kδx, CFL mod 0.94. 4 3.5 max θ g(θ,δ t) 3 2.5 2 1.5 1 0.5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 δ t δt max = 0.042,CFL mod 0.99 CFL mod 1.22 Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 16 / 30

Numerical experiments Modified CFL For I x 1 : v t x +2ǫ t +(2 2 1/3 )η t < 1 x 2 x 4/3 VS. For I 2 x : v t x +2ǫ t x 2 + 4 9 ( ζ( 4 9 ) 1) η t x 4/3 < 1 Amplification factors for I x 1 (blue line) and I2 x (dashed line) Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 17 / 30

Splitting method { ( ) u u t (t,x)+ 2 2 (t,x) u xx(t,x)+i[u(t, )](x) = 0, x u(0,x) = u 0 (x). Notation : u(t, ) = S t u 0 { ( v t + v 2 2 ւ ) ǫv xx = 0 x v(0, ) = v 0, Notation : v(t, ) = Y t v 0 { ց wt +I[w] ηw xx = 0 w(0, ) = w 0, Notation : w(t, ) = X t w 0 Lie method : Z t L = Xt Y t Finite difference method FFT Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 19 / 30

Splitting method { ( ) u u t (t,x)+ 2 2 (t,x) u xx(t,x)+i[u(t, )](x) = 0, x u(0,x) = u 0 (x). Notation : u(t, ) = S t u 0 { ( v t + v 2 2 ւ ) ǫv xx = 0 x v(0, ) = v 0, Notation : v(t, ) = Y t v 0 { ց wt +I[w] ηw xx = 0 w(0, ) = w 0, Notation : w(t, ) = X t w 0 Lie method : Z t L = Xt Y t Note : H. Holden, C. Lubich, N.-H. Risebro; Operator splitting for partial differential equations with Burgers nonlinearity, to appear, Math. Comp (2012). Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 19 / 30

Expressions for the flows Linear flow X t v 0 = D(t, ) v 0, where D(t, ) = F 1( ) e t φi with φi (ξ) = 4π 2 ηξ 2 a I ξ 4/3 +b I ξ ξ 1/3 Nonlinear flow (viscous Burgers equation) Y t w 0 = G(t, ) w 0 1 2 where G is the heat kernel. Exact flow S t u 0 = K(t, ) u 0 1 2 Splitting operator Z t Lu 0 = K(t) u 0 1 2 t 0 t 0 t 0 x G(t s, ) (Y s w 0 ) 2 ds, x K(t s, ) (S s u 0 ) 2 ds D(t) G(t s) x (Y s u 0 ) 2 ds Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 20 / 30

L 2 local error estimate Proposition Let u 0 H 3 (R). There exists C ( u 0 L (R)) 2 such that for all t [0,1], ZL t u 0 S t u 0 L 2 (R) C ( u 0 L (R)) 2 t 2 u 0 2 H 3 (R). Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 21 / 30

L 2 local error estimate Proposition Let u 0 H 3 (R). There exists C ( u 0 L (R)) 2 such that for all t [0,1], ZL t u 0 S t u 0 L 2 (R) C ( u 0 L (R)) 2 t 2 u 0 2 H 3 (R). Ingredients of the proof t ZL tu 0 S t u 0 = 1 2 0 xk(t s) ( (S s u 0 ) 2 (ZL su 0) 2) ds +R(t), The remainder R(t) is written as R(t) = 1 t 2 0 R 1(s)ds, with R 1 (s) = x K(t s) (ZL s u 0) 2 D(t) x G(t s, ) (Y s u 0 ) 2, and satisfies : R(t) L 2 (R) C ( u 0 L (R)) 2 t 2 u 0 2 H 3 (R). Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 21 / 30

Modified fractional Gronwall Lemma Lemma Let φ : [0,T] R + be a bounded measurable function and P be a polynomial with positive coefficients and no constant term. We assume there exists two positive constants C and θ ]0,1[ such that for all t [0,T], 0 φ(t) φ(0)+p(t)+c d θ dt θφ(t). Then there exists C T (θ) such that for all t [0,T], Riemann-Liouville operator : φ(t) C T (θ)φ(0)+c T (θ)p(t). d θ dt θφ(t) = 1 Γ(θ) t 0 (t s) θ 1 φ(s)ds. Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 22 / 30

We will also need the fact that the flow map S t is uniformly Lipschitzean on balls of H 2 (R). Proposition Let T,R > 0. There exists K = K(R,T) < such that if u 0 H 2 (R) R, v 0 H 2 (R) R, then S t u 0 S t v 0 L 2 (R) K u 0 v 0 L 2 (R), t [0,T]. Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 23 / 30

Convergence Theorem For all u 0 H 3 (R) and for all T > 0, there exist positive constants c 1,c 2 and t 0 such that for all t ]0, t 0 ] and for all n N such that 0 n t T, (Z t L )n u 0 S n t u 0 L 2 (R) c 1 t and (Z t L )n u 0 H 3 (R) c 2. Here, c 1,c 2 and t 0 depend only on T, ρ = max t [0,T] S t u 0 H 2 (R), and u 0 H 3 (R). B.A., Carles R., Splitting methods for the nonlocal Fowler equation, Math. Comp (2012), to appear. Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 24 / 30

Proof The proof follows the same idea as in [1,2] for instance. We prove by induction that there exists γ, t 0 such that if 0 < t t 0, for all n N with n t T, (Z t L )n u 0 L 2 (R) 2ρ, (Z t L )n u 0 S n t u 0 L 2 (R) γ t. The triangle inequality yields (Z t L ) n u 0 S n t u 0 L 2 with u k = (Z t L )k n 1 S (n k 1) t ( ZL t ) u k S (n k 1) t ( S t ) u k L, 2 k=0 References : [1] C. Besse, B. Bidegaray, S. Descombes; Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002). [2] H. Holden, C. Lubich, N.-H. Risebro; Operator splitting for partial differential equations with Burgers nonlinearity, to appear, Math. Comp (2012). Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 25 / 30

Proof Lipschitz property of S t yields S (n k 1) t ( ZL t ) u k S (n k 1) t ( S t ) u k L K 2 Z t L u k S t u L k 2 From L 2 local error estimate, we infer S (n k 1) t ( Z t L u k ) S (n k 1) t ( S t u k ) L 2 CK( t)2 u 0 2 H 3, for some constant C. Therefore, (ZL t ) n u 0 S n t u 0 L 2 nck( t) 2 u 0 2 H3 CTK t, which yields the two estimates of the induction, provided one takes γ = CTK, which is uniform in n and t. Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 26 / 30

Numerical experiments : initial data Figure: Initial data used for numerical experiments. Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 27 / 30

Numerical convergence for Lie operator Figure: Lie method Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 28 / 30

Strang method Strang operator Z t S = Xt/2 Y t X t/2 Figure: Strang method Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 29 / 30

Merci, Thanks, Danke! Afaf Bouharguane Numerical schemes for the Fowler equation Berlin, November 2012 30 / 30