- Email: [email protected]
An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation
Journal Pre-proof An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation
Yayun Fu, Wenjun Cai, Yushun Wang
PII: DOI: Reference:
S0893-9659(19)30447-1 https://doi.org/10.1016/j.aml.2019.106123 AML 106123
To appear in:
Applied Mathematics Letters
Received date : 1 October 2019 Revised date : 1 November 2019 Accepted date : 1 November 2019 Please cite this article as: Y. Fu, W. Cai and Y. Wang, An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106123. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
Journal Pre-proof
*Manuscript Click here to view linked References
An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation Yayun Fu,
Wenjun Cai,
Yushun Wang∗
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Abstract
na lP repr oo f
In this paper, we propose a new approach to construct an explicit structure-preserving scheme for the nonlinear fractional wave equation. First, we reformulate the equation as a canonical Hamiltonian system. Then, we utilize the fourth-order fractional centered difference formula to discretize the equation in space direction, and obtain a conservative semi-discrete system. Subsequently, we develop an explicit fully-discrete energy-preserving scheme for the equation by using the proposed approach. Numerical examples are provided to confirm our theoretical analysis in long time computations at last. AMS subject classification: 35R11, 65M06, 65M12
Keywords: Nonlinear fractional wave equation; Hamiltonian system; Explicit energy-preserving scheme; Fourth-order difference formula
1
Introduction
From the physical perspective, the advances in science and engineering have shown that the use of fractional models may result in better descriptions of various complex phenomena with long-range temporal cumulative memory [11, 12]. The nonlinear fractional wave equation is one of the most important models in describing a physical system of interacting particles [9]. It can be regarded as fractional extension of the well known sine-Gordon and Klein-Gordon equations from relativistic quantum mechanics. In recent years, discussions regarding this equation have received much attention. For example, Alfimov [10] obtained the fractional sine-Gordon equation via replacing the Laplacian operator by the Riesz fractional differential operator, and proposed the energy functional of the equation. It is generally difficult to give the explicit forms of the analytical solutions of the nonlinear fractional wave equation, thus many numerical schemes have been discovered to solve this equation, the readers can refer to Refs. [1, 7, 8] and references therein for more details. In this paper, we consider the following initial and boundary value problems of the space nonlinear fractional wave equation α
0
(x, t) ∈ R × (0, T ],
utt + (−∆) 2 u + F (u) = 0,
x ∈ R,
u(x, 0) = ψ(x), ut (x, 0) = φ(x),
x ∈ R\Ω,
ur
u(x, t) = 0,
Ω = (xa , xb ),
(1.1) (1.2) t ∈ (0, T ],
(1.3)
0
Jo
where 1 < α ≤ 2, u(x, t) represents the wave displacement at the position x and time t, and F = ∇F (u) is the derivative of a smooth potential energy F (u) with respect to u. When α = 2, the equation reduces to the classical nonlinear wave equation. The fractional Laplacian is defined as a pseudo-differential operator with the symbol |ξ|α in the Fourier space [12] α
ˆ(ξ, t)), −(−∆) 2 u(x, t) = −F −1 (|ξ|α u
(1.4)
where F is the Fourier transform and u ˆ(ξ, t) = F[u(x, t)]. Yang [11] proposed that the fractional Laplacian is equivalent to the Riesz fractional derivative in one dimension, namely Z Z +∞ i α −1 ∂2 h x 1−α −(−∆) 2 u(x, t) = (x − s) u(s, t)ds + (s − x)1−α u(s, t)ds . (1.5) απ 2 2Γ(2 − α)cos( 2 ) ∂x −∞ x ∗ Correspondence
author. Email:[email protected].
1
Journal Pre-proof In the last few years, some numerical schemes have been devoted to approximating the Riesz fractional derivative, see the details in [15] and references therein. As well known, many continuous systems possess some physical quantities that naturally arise from the physical context, such as energy, momentum and mass. Following a similar argument for the fractional sine-Gordon equation in Ref. [10], we can derive that system (1.1) with the boundary condition (1.3) has the following fractional energy conservation law [7] Z α 2 dE(t) 1 = 0, with E(t) = u2t + (−∆) 4 u + 2F (u) dx. (1.6) dt 2 Ω
na lP repr oo f
Structure-preserving algorithms are achieved by constructing numerical methods which can preserve some properties of continuous systems. These methods have long time computing stability and many advantages over other numerical methods [13], and have been successfully applied to fractional differential equations [5, 6]. The conservation of energy is a crucial property for the nonlinear fractional wave equation, thus constructing various energy-preserving schemes for this equation exerted a tremendous fascination on scholars. For instance, Mac´ıas-D´ıaz [7] studied the energy conservation law and proposed energy-preserving difference schemes for the problem. Xie et al. [8] developed dissipation-preserving schemes for the fractional nonlinear wave equation with damping. All mentioned energy-preserving schemes for the equation are fully-implicit, one has to use iterations to solve algebraic system at each time step, which brings a large number of calculations. To overcome the nonlinear difficulty of these methods [7, 8], scholars developed linearly-implicit energy-preserving scheme [1, 4] for the equation. Compared with the fully-implicit scheme, the linearly-implicit scheme can reduce the computational complexity in practical computation. However, this is still a computationally expensive procedure in long time numerical simulation. The explicit scheme is very efficient, but rarely preserves the structure of the equation. Therefore, it is important to develop explicit structure-preserving schemes for the equation. Up to now, there exist few works focusing on the above aspect. Our aim is to present and analyse an explicit energy-preserving scheme for the nonlinear fractional wave equation. To this end, we consider the following second-order differential equation q¨ = −∇G(q).
(1.7)
Introducing a conjugate momentum p = q, ˙ the system can be rewritten as q˙ = p, p˙ = −∇G(q).
(1.8) (1.9)
Multiplying p and q˙ on the above system respectively, and summing them together, we can derive the energy conservation law d H(p, q) = 0, dt
with H(p, q) =
1 T p p + G(q). 2
Employing the centered difference scheme and averaged vector field (AVF) method [14] for system (1.8)-(1.9), respectively, we obtain q n+1 − q n−1 = pn , 2τ Z 1 pn+1 − pn =− ∇G ηq n+1 + (1 − η)q n dη. τ 0
ur
Equivalent form with one single variable q is given by Z 1 q n+2 − q n+1 − q n + q n−1 = − ∇G ηq n+1 + (1 − η)q n dη. 2τ 2 0
(1.10) (1.11)
(1.12)
Jo
Multiplying both sides of the above scheme with q n+1 − q n , we have q n+2 − q n+1 T q n+1 − q n q n+1 − q n T q n − q n−1 − 2τ 2 2τ 2 Z 1 = − q n+1 − q n ∇G ηq n+1 + (1 − η)q n dη Z
0
1
d =− G ηq n+1 + (1 − η)q n dη dη 0 = − G q n+1 − G (q n ) .
(1.13)
2
Journal Pre-proof Consequently, H n+1 = H n ,
with
Hn =
q n+1 − q n
T
2τ 2
q n − q n−1
+ G (q n ) .
(1.14)
The above scheme is a four level explicit energy-preserving scheme, we can obtain q 1 , q 2 by using the conservative scheme for the first two steps The outline of this paper is as follows. In Section 2, we reformulate the nonlinear fractional wave equation as a canonical Hamiltonian system. In Section 3, we construct an explicit difference scheme for the equation and prove that the scheme can preserve the discrete energy. Numerical examples are presented in Section 4 to demonstrate the theoretical results. We draw some conclusions in Section 5.
2
Hamiltonian formulation and conservation law
The Hamiltonian structure is important to analyse the conservative systems and further to construct numerical schemes for them. In this section, we will derive the Hamiltonian formulation of system (1.1) with homogeneous boundary conditions. Let v = ut , system (1.1) can be rewritten as a first-order system ut = v,
(2.1) α 2
0
vt = −(−∆) u − F (u).
(2.2)
na lP repr oo f
Taking the inner products of (2.1)-(2.2) with vt , v, respectively, we derive the energy conservation law as follows Z α 2 d 1 2 H = 0, with H = v + (−∆) 4 u + 2F (u) dx. dt Ω 2 Lemma 2.1. [1, 6] For a functional G[ρ] with the following form Z α G[ρ] = g ρ(η), (−∆) 4 ρ(η) dη, Ω
(2.3)
(2.4)
where g is a smooth function on the Ω, the variational derivative of G[ρ] is given as follows α ∂g δG ∂g = + (−∆) 4 α . δρ ∂ρ ∂ (−∆) 4 ρ
(2.5)
Based on the fractional variational derivative formula in Lemma 2.1, we obtain the following result straightforwardly. Theorem 2.1. The system (2.1) is an infinite-dimensional canonical Hamiltonian system ut 0 1 δH/δu = , vt −1 0 δH/δv where the energy functional H is given in (2.3).
3
Construction of the explicit energy-preserving scheme
3.1
ur
In this section, we construct an explicit energy-preserving difference scheme for system (2.1).
Structure-preserving spatial discretization
Jo
−xa T and τ := N be the space size and time step, respectively. We choose two positive integers M and N , let h := xbM Denoting Ωh = {xj | xj = xa + jh, j = 1, 2, · · · , M − 1}, Ωτ = {tn | tn = nτ, n = 0, 1, · · · , N }. Let (unj , vjn ) be the numerical approximations to the exact solutions (u, v) at the grid point (xj , tn ), the corresponding vector forms at any time level are then denoted by
U = (u1 , u2 , · · · , uM −1 )T ,
V = (v1 , v2 , · · · , vM −1 )T .
With these notations, we define the discrete inner product and the associated discrete maximum norm (L∞ -norm) as (U, V ) = h
M −1 X j=1
uj vj , kU k∞ =
sup 1≤j≤M −1
|uj |.
3
Journal Pre-proof n R o +∞ Lemma 3.1. [2] Suppose u ∈ L4+α (R) := u| −∞ (1 + |ξ|)4+α |b u(ξ)|dξ < ∞ , then for a fixed h, α −δhα u(x) = −(−∆) 2 u(x) + O h4 , 1 < α ≤ 2,
1 α α −α where δhα u(x) = 34 ∆α h u(x) − 3 ∆2h u(x), ∆h u(x) = h
From Lemma 3.1, we have δhα u(xj )
+∞ 1 X b(α) = α dk u(xj − kh), h k=−∞
with
(α) dbk
+∞ P
(α)
=
=
(α) 1 4 (α) , 3 dk − 3·2α d k 2 4 (α) 3 dk ,
k is even,
Then above approximation can be written as a matrix form, (α) (α) db0 db−1 b(α) (α) db0 α 1 d1 (−∆) 2 u(xj ) = (DU )j , with D = α . .. h .. . (α) (α) db db M −2
(−1)k Γ(α+1) . Γ( α −k+1 )Γ( α2 +k+1) 2
dk u(x − kh), dk
k=−∞
(
(α)
M −3
(3.1)
k is odd,
(3.2)
namely, ··· ··· .. . ···
(α) db−M +2 (α) db−M +3 .. . . (α) db
(3.3)
0
where the differential matrix D is an (M − 1) × (M − 1) symmetric matrix.
Explicit energy-preserving scheme
na lP repr oo f
3.2
By using the fourth-order fractional centered difference scheme to approximate the fractional Laplacian in (2.2), a semi-discrete system is given as Ut = V, Vt = −DU − ∇F (U ).
(3.4) (3.5)
By setting G(U ) = 12 U T DU + F (U ), then system (3.4)-(3.5) can be rewritten as Ut = V, Vt = −∇G(U ).
(3.6) (3.7)
Multiplying Vt and Ut on (3.6)-(3.7), respectively, and summing them together, we can derive semi-discrete system has the following energy conservation law 1 d ˜ ˜ H(U, V ) = 0, with H(U, V ) = V T V + G(U ). dt 2
Based on the discussions in Section 1, employing the centered difference scheme and AVF method for system (3.6)(3.7), respectively, we obtain an explicit fully-discrete scheme, namely
ur
U n+1 − U n−1 = V n, 2τ Z 1 V n+1 − V n =− ∇G(ηU n+1 + (1 − η)U n )dη. τ 0
(3.8) (3.9)
According to (3.8), equivalent form with one single variable U can be obtained
Jo
U n+2 − U n+1 − U n + U n−1 =− 2τ 2
Z
0
1
∇G(ηU n+1 + (1 − η)U n )dη.
(3.10)
Theorem 3.1. The fully-discrete scheme (3.10) has the following discrete energy conservation law H n+1 = H n ,
3 ≤ n < N − 1, with H n =
(U n+1 − U n )T (U n − U n−1 ) + G(U n ) 2τ 2
(3.11)
Proof. The proof process of this theorem is similar to that of (1.13) in Section 1, which implies that the scheme (3.10) is an explicit energy-preserving (EX-EP) scheme.
4
Journal Pre-proof Remark 3.1. It should be remark that the EX-EP scheme (3.10) is a four level method. Applying the AVF method for system (3.6)-(3.7), we can construct the following conservative scheme as the starting step to obtain U 1 and U 2 1 U n+1 − U n = V n+ 2 , τ Z 1 V n+1 − V n =− ∇G(ηU n+1 + (1 − η)U n )dη. τ 0
(3.12) (3.13)
Proof. By taking the discrete inner products of (1) and (2) with V n+1 − V n , U n+1 − U n , respectively, and summing them together, we can prove that the scheme can preserve the energy in discrete sense.
4
Numerical examples
We study system (1.1) with F(u)= 1 − cos(u) for any u ∈ R on domain (−20, 20) × (0, T ]. The initial conditions are chosen as ψ(x) = 0, φ(x) = ω4 sech ωx , x ∈ Ω. When α = 2, system (1.1) reduces to the standard wave equation with the h i √ √ analytical solution u(x, t) = 4 tan−1 sech x/ω sin(ω −1 ω 2 − 1t)/ ω 2 − 1 , ω > 1. The relative energy error is defined
-13
alpha=1.7
α=1.4
alpha=1.9
alpha=2
na lP repr oo f
as RH n = |(H n − H 0 )/H 0 |, where H n denotes the energy at t = nτ . To confirm our theoretical results, without loss of generality, we take ω = 1.1 and α = 1.4, 1.7, 1.9, 2.0. For 1 < α < 2, the numerical exact solution u is obtained by a fine mesh size h = 0.00125 and a small time step size τ = 0.00001. Fig.1.a and Fig.1.b show the time and space convergence orders, respectively, we find that the EX-EP scheme is of second-order accuracy in time, and is of fourth-order accuracy in space. The motivation of our work is to develop a more efficient structure-preserving scheme, thus, it is valuable to compare our EX-EP scheme with some newly developed linearlyimplicit energy-preserving schemes [1, 4] for the equation in computing efficiency. From the bar plot in Fig.1c, the costs of the EX-EP scheme are about 20 times cheaper than that of schemes IEQ/CN [1] and SAV/CN [4]. We also run a long time simulation till T = 1000 and plot the relative energy deviation in Fig.1.d for different α, which indicates that the EX-EP scheme can preserve the energy exactly in discrete sense. -6
slope=2
alpha=1.7
α=1.4
alpha=1.9
alpha=2
10-12
30
slope=4
EX-EP
-14
SAV/CN
IEQ/CN
α=1.4
α=1.7
α=1.9
α=2.0
-7
25
-17 -18 -19
20
-9
-10 -11
RH
-16
10-13
-8
Cost time (s)
L ∞ norm error in logscale
L ∞ norm error in logscale
-15
15
10-15
-12
-20
10-14
10
5
-13
-21 -22 -11
-10.5
-10
-9.5
-9
-8.5
-8
time step in logscale
a
-7.5
-7
-6.5
-14 -2
10-16
0
-1.95
-1.9
-1.85
-1.8
-1.75
-1.7
-1.65
-1.6
-1.55
τ =0.05
τ =0.025
τ =0.0125
τ =0.01
space step in logscale
b
c
0
100
200
300
400
500
600
700
800
900
1000
t
d
Fig. 1: a: Convergence orders in time; b: Convergence orders in space; c: CPU time for α = 2 with different time steps till T = 100 under h = 0.1; d: The relative energy errors for different α when h = 0.1, τ = 0.01.
5
Conclusions
ur
In this paper, we derive the Hamiltonian formulation of the nonlinear fractional wave equation, and construct an efficient difference scheme for the equation. Specifically, the scheme is explicit, and can preserve discrete energy exactly. Theoretical analysis and numerical experiments indicate that the scheme is efficient and has desirable energy conservation property.
Acknowledgments
Jo
This work is supported by the National Key Research and Development Project of China (Grant No. 2017YFC0601505, 2018YFC1504205), the National Natural Science Foundation of China (Grant No. 11771213, 61872422, 11971242), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant Nos. KYCX19 0776), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant No.18KJA110003).
References [1] Y. Fu, W. Cai, and Y. Wang. A linearly implicit structure-preserving scheme for the fractional sine-Gordon equation based on the IEQ approach. Preprint. 5
Journal Pre-proof [2] A. Xiao, J. Wang. Symplectic scheme for the Schr¨ odinger equation with fractional Laplacian. Appl. Numer. Math., 146:469-487, 2019. [3] C. C ¸ elik, M. Duman. Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J.Comput. Phys., 231:1743-1750, 2012. [4] W. Cai, C. Jiang, Y. Wang, and Y. Song. Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions. J.Comput. Phys., 395:166-185, 2019. [5] M. Li, X. Gu, C. Huang, and M. Fei. A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schr¨ odinger equations. J. Comput. Phys. 358:256-282, 2018. [6] P. Wang, C. Huang. Structure-preserving numerical methods for the fractional Schr¨ odinger equation. Appl. Numer. Math. 129:137-158, 2018. [7] J. Mac´ıas-D´ıaz, A. Hendy, and R. De Staelen. A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. Appl. Math. Comput., 325:1-14, 2018. [8] J. Xie, Z. Zhang. An effective dissipation-preserving fourth-order difference solver for fractional-in-space nonlinear wave equations J. Sci. Comput., 79:1753-1776, 2019.
na lP repr oo f
[9] J. Mac´ıas-D´ıaz. A numerically efficient dissipation-preserving implicit method for a nonlinear multidimensional fractional wave Equation. J. Sci. Comput., 77:1-26, 2018. [10] G. Alfimov, T. Pierantozzi, and L. V´ azquez. Numerical study of a fractional sine-Gordon equation. Fract. Differ. Appl., 4:153-162, 2004. [11] Q. Yang, F. Liu, and I. Turner. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model., 34:200-218, 2010. [12] L. Caffarelli, L. Silvestre. An extension problem related to the fractional Laplacian. Commun. Part. Diff. Eq, 26:159-180, 2009. [13] K. Feng, M. Qin. Symplectic Geometric Algorithms for Hamiltonian Systems. Springer Berlin Heidelberg, 2010. [14] W. Cai, H. Li, and Y. Wang. Partitioned averaged vector field methods. J. Comput. Phys., 370:25-42, 2018.
Jo
ur
[15] X. Zhao, Z. Sun, and H. Peng. A fourth-order compact ADI scheme for 2D nonlinear space fractional Schr¨ odinger equation. SIAM J. Sci. Comput., 36:A2865-A2886, 2014.
6
Yayun Fu, Wenjun Cai, Yushun Wang
PII: DOI: Reference:
S0893-9659(19)30447-1 https://doi.org/10.1016/j.aml.2019.106123 AML 106123
To appear in:
Applied Mathematics Letters
Received date : 1 October 2019 Revised date : 1 November 2019 Accepted date : 1 November 2019 Please cite this article as: Y. Fu, W. Cai and Y. Wang, An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106123. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
Journal Pre-proof
*Manuscript Click here to view linked References
An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation Yayun Fu,
Wenjun Cai,
Yushun Wang∗
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Abstract
na lP repr oo f
In this paper, we propose a new approach to construct an explicit structure-preserving scheme for the nonlinear fractional wave equation. First, we reformulate the equation as a canonical Hamiltonian system. Then, we utilize the fourth-order fractional centered difference formula to discretize the equation in space direction, and obtain a conservative semi-discrete system. Subsequently, we develop an explicit fully-discrete energy-preserving scheme for the equation by using the proposed approach. Numerical examples are provided to confirm our theoretical analysis in long time computations at last. AMS subject classification: 35R11, 65M06, 65M12
Keywords: Nonlinear fractional wave equation; Hamiltonian system; Explicit energy-preserving scheme; Fourth-order difference formula
1
Introduction
From the physical perspective, the advances in science and engineering have shown that the use of fractional models may result in better descriptions of various complex phenomena with long-range temporal cumulative memory [11, 12]. The nonlinear fractional wave equation is one of the most important models in describing a physical system of interacting particles [9]. It can be regarded as fractional extension of the well known sine-Gordon and Klein-Gordon equations from relativistic quantum mechanics. In recent years, discussions regarding this equation have received much attention. For example, Alfimov [10] obtained the fractional sine-Gordon equation via replacing the Laplacian operator by the Riesz fractional differential operator, and proposed the energy functional of the equation. It is generally difficult to give the explicit forms of the analytical solutions of the nonlinear fractional wave equation, thus many numerical schemes have been discovered to solve this equation, the readers can refer to Refs. [1, 7, 8] and references therein for more details. In this paper, we consider the following initial and boundary value problems of the space nonlinear fractional wave equation α
0
(x, t) ∈ R × (0, T ],
utt + (−∆) 2 u + F (u) = 0,
x ∈ R,
u(x, 0) = ψ(x), ut (x, 0) = φ(x),
x ∈ R\Ω,
ur
u(x, t) = 0,
Ω = (xa , xb ),
(1.1) (1.2) t ∈ (0, T ],
(1.3)
0
Jo
where 1 < α ≤ 2, u(x, t) represents the wave displacement at the position x and time t, and F = ∇F (u) is the derivative of a smooth potential energy F (u) with respect to u. When α = 2, the equation reduces to the classical nonlinear wave equation. The fractional Laplacian is defined as a pseudo-differential operator with the symbol |ξ|α in the Fourier space [12] α
ˆ(ξ, t)), −(−∆) 2 u(x, t) = −F −1 (|ξ|α u
(1.4)
where F is the Fourier transform and u ˆ(ξ, t) = F[u(x, t)]. Yang [11] proposed that the fractional Laplacian is equivalent to the Riesz fractional derivative in one dimension, namely Z Z +∞ i α −1 ∂2 h x 1−α −(−∆) 2 u(x, t) = (x − s) u(s, t)ds + (s − x)1−α u(s, t)ds . (1.5) απ 2 2Γ(2 − α)cos( 2 ) ∂x −∞ x ∗ Correspondence
author. Email:[email protected].
1
Journal Pre-proof In the last few years, some numerical schemes have been devoted to approximating the Riesz fractional derivative, see the details in [15] and references therein. As well known, many continuous systems possess some physical quantities that naturally arise from the physical context, such as energy, momentum and mass. Following a similar argument for the fractional sine-Gordon equation in Ref. [10], we can derive that system (1.1) with the boundary condition (1.3) has the following fractional energy conservation law [7] Z α 2 dE(t) 1 = 0, with E(t) = u2t + (−∆) 4 u + 2F (u) dx. (1.6) dt 2 Ω
na lP repr oo f
Structure-preserving algorithms are achieved by constructing numerical methods which can preserve some properties of continuous systems. These methods have long time computing stability and many advantages over other numerical methods [13], and have been successfully applied to fractional differential equations [5, 6]. The conservation of energy is a crucial property for the nonlinear fractional wave equation, thus constructing various energy-preserving schemes for this equation exerted a tremendous fascination on scholars. For instance, Mac´ıas-D´ıaz [7] studied the energy conservation law and proposed energy-preserving difference schemes for the problem. Xie et al. [8] developed dissipation-preserving schemes for the fractional nonlinear wave equation with damping. All mentioned energy-preserving schemes for the equation are fully-implicit, one has to use iterations to solve algebraic system at each time step, which brings a large number of calculations. To overcome the nonlinear difficulty of these methods [7, 8], scholars developed linearly-implicit energy-preserving scheme [1, 4] for the equation. Compared with the fully-implicit scheme, the linearly-implicit scheme can reduce the computational complexity in practical computation. However, this is still a computationally expensive procedure in long time numerical simulation. The explicit scheme is very efficient, but rarely preserves the structure of the equation. Therefore, it is important to develop explicit structure-preserving schemes for the equation. Up to now, there exist few works focusing on the above aspect. Our aim is to present and analyse an explicit energy-preserving scheme for the nonlinear fractional wave equation. To this end, we consider the following second-order differential equation q¨ = −∇G(q).
(1.7)
Introducing a conjugate momentum p = q, ˙ the system can be rewritten as q˙ = p, p˙ = −∇G(q).
(1.8) (1.9)
Multiplying p and q˙ on the above system respectively, and summing them together, we can derive the energy conservation law d H(p, q) = 0, dt
with H(p, q) =
1 T p p + G(q). 2
Employing the centered difference scheme and averaged vector field (AVF) method [14] for system (1.8)-(1.9), respectively, we obtain q n+1 − q n−1 = pn , 2τ Z 1 pn+1 − pn =− ∇G ηq n+1 + (1 − η)q n dη. τ 0
ur
Equivalent form with one single variable q is given by Z 1 q n+2 − q n+1 − q n + q n−1 = − ∇G ηq n+1 + (1 − η)q n dη. 2τ 2 0
(1.10) (1.11)
(1.12)
Jo
Multiplying both sides of the above scheme with q n+1 − q n , we have q n+2 − q n+1 T q n+1 − q n q n+1 − q n T q n − q n−1 − 2τ 2 2τ 2 Z 1 = − q n+1 − q n ∇G ηq n+1 + (1 − η)q n dη Z
0
1
d =− G ηq n+1 + (1 − η)q n dη dη 0 = − G q n+1 − G (q n ) .
(1.13)
2
Journal Pre-proof Consequently, H n+1 = H n ,
with
Hn =
q n+1 − q n
T
2τ 2
q n − q n−1
+ G (q n ) .
(1.14)
The above scheme is a four level explicit energy-preserving scheme, we can obtain q 1 , q 2 by using the conservative scheme for the first two steps The outline of this paper is as follows. In Section 2, we reformulate the nonlinear fractional wave equation as a canonical Hamiltonian system. In Section 3, we construct an explicit difference scheme for the equation and prove that the scheme can preserve the discrete energy. Numerical examples are presented in Section 4 to demonstrate the theoretical results. We draw some conclusions in Section 5.
2
Hamiltonian formulation and conservation law
The Hamiltonian structure is important to analyse the conservative systems and further to construct numerical schemes for them. In this section, we will derive the Hamiltonian formulation of system (1.1) with homogeneous boundary conditions. Let v = ut , system (1.1) can be rewritten as a first-order system ut = v,
(2.1) α 2
0
vt = −(−∆) u − F (u).
(2.2)
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Taking the inner products of (2.1)-(2.2) with vt , v, respectively, we derive the energy conservation law as follows Z α 2 d 1 2 H = 0, with H = v + (−∆) 4 u + 2F (u) dx. dt Ω 2 Lemma 2.1. [1, 6] For a functional G[ρ] with the following form Z α G[ρ] = g ρ(η), (−∆) 4 ρ(η) dη, Ω
(2.3)
(2.4)
where g is a smooth function on the Ω, the variational derivative of G[ρ] is given as follows α ∂g δG ∂g = + (−∆) 4 α . δρ ∂ρ ∂ (−∆) 4 ρ
(2.5)
Based on the fractional variational derivative formula in Lemma 2.1, we obtain the following result straightforwardly. Theorem 2.1. The system (2.1) is an infinite-dimensional canonical Hamiltonian system ut 0 1 δH/δu = , vt −1 0 δH/δv where the energy functional H is given in (2.3).
3
Construction of the explicit energy-preserving scheme
3.1
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In this section, we construct an explicit energy-preserving difference scheme for system (2.1).
Structure-preserving spatial discretization
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−xa T and τ := N be the space size and time step, respectively. We choose two positive integers M and N , let h := xbM Denoting Ωh = {xj | xj = xa + jh, j = 1, 2, · · · , M − 1}, Ωτ = {tn | tn = nτ, n = 0, 1, · · · , N }. Let (unj , vjn ) be the numerical approximations to the exact solutions (u, v) at the grid point (xj , tn ), the corresponding vector forms at any time level are then denoted by
U = (u1 , u2 , · · · , uM −1 )T ,
V = (v1 , v2 , · · · , vM −1 )T .
With these notations, we define the discrete inner product and the associated discrete maximum norm (L∞ -norm) as (U, V ) = h
M −1 X j=1
uj vj , kU k∞ =
sup 1≤j≤M −1
|uj |.
3
Journal Pre-proof n R o +∞ Lemma 3.1. [2] Suppose u ∈ L4+α (R) := u| −∞ (1 + |ξ|)4+α |b u(ξ)|dξ < ∞ , then for a fixed h, α −δhα u(x) = −(−∆) 2 u(x) + O h4 , 1 < α ≤ 2,
1 α α −α where δhα u(x) = 34 ∆α h u(x) − 3 ∆2h u(x), ∆h u(x) = h
From Lemma 3.1, we have δhα u(xj )
+∞ 1 X b(α) = α dk u(xj − kh), h k=−∞
with
(α) dbk
+∞ P
(α)
=
=
(α) 1 4 (α) , 3 dk − 3·2α d k 2 4 (α) 3 dk ,
k is even,
Then above approximation can be written as a matrix form, (α) (α) db0 db−1 b(α) (α) db0 α 1 d1 (−∆) 2 u(xj ) = (DU )j , with D = α . .. h .. . (α) (α) db db M −2
(−1)k Γ(α+1) . Γ( α −k+1 )Γ( α2 +k+1) 2
dk u(x − kh), dk
k=−∞
(
(α)
M −3
(3.1)
k is odd,
(3.2)
namely, ··· ··· .. . ···
(α) db−M +2 (α) db−M +3 .. . . (α) db
(3.3)
0
where the differential matrix D is an (M − 1) × (M − 1) symmetric matrix.
Explicit energy-preserving scheme
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3.2
By using the fourth-order fractional centered difference scheme to approximate the fractional Laplacian in (2.2), a semi-discrete system is given as Ut = V, Vt = −DU − ∇F (U ).
(3.4) (3.5)
By setting G(U ) = 12 U T DU + F (U ), then system (3.4)-(3.5) can be rewritten as Ut = V, Vt = −∇G(U ).
(3.6) (3.7)
Multiplying Vt and Ut on (3.6)-(3.7), respectively, and summing them together, we can derive semi-discrete system has the following energy conservation law 1 d ˜ ˜ H(U, V ) = 0, with H(U, V ) = V T V + G(U ). dt 2
Based on the discussions in Section 1, employing the centered difference scheme and AVF method for system (3.6)(3.7), respectively, we obtain an explicit fully-discrete scheme, namely
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U n+1 − U n−1 = V n, 2τ Z 1 V n+1 − V n =− ∇G(ηU n+1 + (1 − η)U n )dη. τ 0
(3.8) (3.9)
According to (3.8), equivalent form with one single variable U can be obtained
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U n+2 − U n+1 − U n + U n−1 =− 2τ 2
Z
0
1
∇G(ηU n+1 + (1 − η)U n )dη.
(3.10)
Theorem 3.1. The fully-discrete scheme (3.10) has the following discrete energy conservation law H n+1 = H n ,
3 ≤ n < N − 1, with H n =
(U n+1 − U n )T (U n − U n−1 ) + G(U n ) 2τ 2
(3.11)
Proof. The proof process of this theorem is similar to that of (1.13) in Section 1, which implies that the scheme (3.10) is an explicit energy-preserving (EX-EP) scheme.
4
Journal Pre-proof Remark 3.1. It should be remark that the EX-EP scheme (3.10) is a four level method. Applying the AVF method for system (3.6)-(3.7), we can construct the following conservative scheme as the starting step to obtain U 1 and U 2 1 U n+1 − U n = V n+ 2 , τ Z 1 V n+1 − V n =− ∇G(ηU n+1 + (1 − η)U n )dη. τ 0
(3.12) (3.13)
Proof. By taking the discrete inner products of (1) and (2) with V n+1 − V n , U n+1 − U n , respectively, and summing them together, we can prove that the scheme can preserve the energy in discrete sense.
4
Numerical examples
We study system (1.1) with F(u)= 1 − cos(u) for any u ∈ R on domain (−20, 20) × (0, T ]. The initial conditions are chosen as ψ(x) = 0, φ(x) = ω4 sech ωx , x ∈ Ω. When α = 2, system (1.1) reduces to the standard wave equation with the h i √ √ analytical solution u(x, t) = 4 tan−1 sech x/ω sin(ω −1 ω 2 − 1t)/ ω 2 − 1 , ω > 1. The relative energy error is defined
-13
alpha=1.7
α=1.4
alpha=1.9
alpha=2
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as RH n = |(H n − H 0 )/H 0 |, where H n denotes the energy at t = nτ . To confirm our theoretical results, without loss of generality, we take ω = 1.1 and α = 1.4, 1.7, 1.9, 2.0. For 1 < α < 2, the numerical exact solution u is obtained by a fine mesh size h = 0.00125 and a small time step size τ = 0.00001. Fig.1.a and Fig.1.b show the time and space convergence orders, respectively, we find that the EX-EP scheme is of second-order accuracy in time, and is of fourth-order accuracy in space. The motivation of our work is to develop a more efficient structure-preserving scheme, thus, it is valuable to compare our EX-EP scheme with some newly developed linearlyimplicit energy-preserving schemes [1, 4] for the equation in computing efficiency. From the bar plot in Fig.1c, the costs of the EX-EP scheme are about 20 times cheaper than that of schemes IEQ/CN [1] and SAV/CN [4]. We also run a long time simulation till T = 1000 and plot the relative energy deviation in Fig.1.d for different α, which indicates that the EX-EP scheme can preserve the energy exactly in discrete sense. -6
slope=2
alpha=1.7
α=1.4
alpha=1.9
alpha=2
10-12
30
slope=4
EX-EP
-14
SAV/CN
IEQ/CN
α=1.4
α=1.7
α=1.9
α=2.0
-7
25
-17 -18 -19
20
-9
-10 -11
RH
-16
10-13
-8
Cost time (s)
L ∞ norm error in logscale
L ∞ norm error in logscale
-15
15
10-15
-12
-20
10-14
10
5
-13
-21 -22 -11
-10.5
-10
-9.5
-9
-8.5
-8
time step in logscale
a
-7.5
-7
-6.5
-14 -2
10-16
0
-1.95
-1.9
-1.85
-1.8
-1.75
-1.7
-1.65
-1.6
-1.55
τ =0.05
τ =0.025
τ =0.0125
τ =0.01
space step in logscale
b
c
0
100
200
300
400
500
600
700
800
900
1000
t
d
Fig. 1: a: Convergence orders in time; b: Convergence orders in space; c: CPU time for α = 2 with different time steps till T = 100 under h = 0.1; d: The relative energy errors for different α when h = 0.1, τ = 0.01.
5
Conclusions
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In this paper, we derive the Hamiltonian formulation of the nonlinear fractional wave equation, and construct an efficient difference scheme for the equation. Specifically, the scheme is explicit, and can preserve discrete energy exactly. Theoretical analysis and numerical experiments indicate that the scheme is efficient and has desirable energy conservation property.
Acknowledgments
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This work is supported by the National Key Research and Development Project of China (Grant No. 2017YFC0601505, 2018YFC1504205), the National Natural Science Foundation of China (Grant No. 11771213, 61872422, 11971242), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant Nos. KYCX19 0776), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant No.18KJA110003).
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