Finite difference methods for the time fractional order differential equations Zhi-zhong Sun Department of Mathematics, Southeast University, Nanjing 210096, P R China (e-mail: [email protected],) Joint work with Wanrong Cao, Rui Du, Guanghua Gao, Jincheng Ren Xiaonan Wu, Yanan Zhang, Xuan Zhao Fractional PDEs Confernce June 3-5, 2013
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Finite difference methods for the time fractionalorder differential equations
Zhi-zhong SunDepartment of Mathematics, Southeast University,
Finite difference methods for the multi-term time fractionaldiffusion-wave equation
ADI methods for the multi-dimensional time fractional equationsFractional sub-diffusion equationFractional diffusion-wave equation
1.1 Definition of the Caputo fractional derivative
For a given positive real number γ, n − 1 < γ 6 n, the Caputofractional derivative with the order of γ, is defined by
C0 D
γt f (t) =
1
Γ(n − γ)
∫ t
0
f (n)(ξ)
(t − ξ)γ−n+1dξ.
I Case γ ∈ (0, 1) :
C0 D
γt f (t) =
1
Γ(1− γ)
∫ t
0
f ′(ξ)
(t − ξ)γdξ.
I Case γ ∈ (1, 2) :
C0 D
γt f (t) =
1
Γ(2− γ)
∫ t
0
f ′′(ξ)
(t − ξ)γ−1dξ.
1.1 Definition of the Caputo fractional derivative
For a given positive real number γ, n − 1 < γ 6 n, the Caputofractional derivative with the order of γ, is defined by
C0 D
γt f (t) =
1
Γ(n − γ)
∫ t
0
f (n)(ξ)
(t − ξ)γ−n+1dξ.
I Case γ ∈ (0, 1) :
C0 D
γt f (t) =
1
Γ(1− γ)
∫ t
0
f ′(ξ)
(t − ξ)γdξ.
I Case γ ∈ (1, 2) :
C0 D
γt f (t) =
1
Γ(2− γ)
∫ t
0
f ′′(ξ)
(t − ξ)γ−1dξ.
1.2 Approximation of the fractional derivative: γ = 12
Theorem [Sun and Wu 2004 ANM]
Suppose f (t) ∈ C 2[0, tn] Let
R(f (tn)) ≡ C0 D
12t f (tn)−
τ−12
Γ(2− 12)
[a0f (tn)−
n−1∑k=1
(an−k−1 − an−k
)f (tk)− an−1f (t0)
],
then
|R(f (tn))| 61
6√π
(10√
2− 11)
max06t6tn
|f ′′(t)|τ32 ,
where al =√
l + 1−√
l , l > 0.
1.3 Approximation of the fractional derivative: γ ∈ (0, 1)
Theorem [Sun and Wu 2006 ANM]
Suppose f (t) ∈ C 2[0, tn] and γ ∈ (0, 1). Let
R(f (tn)) ≡ C0 D
γt f (tn)−
τ−γ
Γ(2− γ)
[a0f (tn)−
n−1∑k=1
(an−k−1 − an−k
)f (tk)− an−1f (t0)
],
then
|R(f (tn))| 61
Γ(2− γ)
[1− γ
12+
22−γ
2− γ−(1+2−γ)
]max
06t6tn|f ′′(t)|τ2−γ ,
where al = (l + 1)1−γ − l1−γ , l > 0.
1.4 Approximation of the fractional derivative: γ ∈ (1, 2)
Theorem [Sun and Wu 2006 ANM]
Suppose f (t) ∈ C 3[0, tn] and γ ∈ (1, 2). Let
R(f (tn)) ≡1
2
[C0 D
γt f (tn) + C
0 Dγt f (tn−1)
]−
τ1−γ
Γ(3− γ)
[b0δt f
n− 12 −
n−1∑k=1
(bn−k−1 − bn−k
)δt f
k− 12 − 1
2bn−1f
′(t0)],
then
|R(f (tn))| 61
Γ(3− γ)
(2− γ
12+
23−γ
3− γ− 21−γ − 5
6
)max
06t6tn|f ′′′(t)|τ3−γ ,
where
bl = (l+1)2−γ−l2−γ , l > 0, δt fk− 1
2 =f (tk)− f (tk−1)
τ, 1 6 k 6 n.
2.1 Dirichlet boundary problem of the sub-diffusionequation
Consider the following one-dimensional problem
C0 Dα
t u(x , t) = κα∂2u(x , t)
∂x2+ f (x , t), a < x < b, 0 < t 6 T ,
(1)
u(x , 0) = ψ(x), a 6 x 6 b, (2)
u(a, t) = ϕ1(t), u(b, t) = ϕ2(t), 0 < t 6 T , (3)
where α ∈ (0, 1).
The fractional equation (1) is called the time fractionalsub-diffusion equation.
2.1 Dirichlet boundary problem of the sub-diffusionequation
For finite difference approximation, discretize equally the interval[a, b] with xi = a + ih (0 6 i 6 M), [0,T ] withtk = kτ (0 6 k 6 N), where h = 1/M and τ = T/N are thespatial and temporal step sizes, respectively. First the followingnotations are introduced.
δxui− 12
=1
h(ui − ui−1), δ2xui =
1
h
(δxui+ 1
2− δxui− 1
2
),
‖u‖∞ = max0≤i≤M
|ui |, Aui =1
12(ui−1 + 10ui + ui+1), 1 6 i 6 M − 1,
In addition, denote a discrete fractional derivative operator Dατ
Dατ uk
i =1
µ
[uki −
k−1∑j=1
(ak−j−1−ak−j)uji−ak−1u
0i
], 0 6 i 6 M, 1 6 k 6 N.
Define the grid function
Uki = u(xi , tk), 0 6 i 6 M, 0 6 k 6 N.
2.1 Dirichlet boundary problem of the sub-diffusionequation
In 2006, we constructed the following difference scheme
Dατ uk
i = καδ2xu
ki + f k
i , 1 6 i 6 M − 1, 1 6 k 6 N, (4)
u0i = ψ(xi ), 0 6 i 6 M, (5)
uk0 = ϕ1(tk), uk
M = ϕ2(tk), 1 6 k 6 N. (6)
We proved that
2.1 Dirichlet boundary problem of the sub-diffusionequation
Theorem (Stability) [Sun and Wu 2006 ANM]
The finite difference scheme (4)-(6) is unconditionally stable to theinitial value ψ and the right hand term f .
Theorem (Convergence) [Sun and Wu 2006 ANM]
Assume that u(x , t) ∈ C4,2x ,t ([a, b]× [0,T ]) is the solution of
(1)-(3) and uki | 0 6 i 6 M, 0 6 k 6 N is solution of the finite
difference scheme (4)-(6), respectively. Then there exists a positiveconstant C such that
‖Uk − uk‖∞ 6 C (τ2−α + h2), 0 6 k 6 N.
2.1 Dirichlet boundary problem of the sub-diffusionequation
Theorem (Stability) [Sun and Wu 2006 ANM]
The finite difference scheme (4)-(6) is unconditionally stable to theinitial value ψ and the right hand term f .
Theorem (Convergence) [Sun and Wu 2006 ANM]
Assume that u(x , t) ∈ C4,2x ,t ([a, b]× [0,T ]) is the solution of
(1)-(3) and uki | 0 6 i 6 M, 0 6 k 6 N is solution of the finite
difference scheme (4)-(6), respectively. Then there exists a positiveconstant C such that
‖Uk − uk‖∞ 6 C (τ2−α + h2), 0 6 k 6 N.
2.1 Dirichlet boundary problem of the sub-diffusionequation
In 2011, we established the following difference scheme
ADατ uk
i = καδ2xu
ki +Af k
i , 1 6 i 6 M − 1, 1 6 k 6 N, (7)
u0i = ψ(xi ), 0 6 i 6 M, (8)
uk0 = ϕ1(tk), uk
M = ϕ2(tk), 1 6 k 6 N. (9)
We proved that
2.1 Dirichlet boundary problem of the sub-diffusionequation
Theorem (Stability) [Gao and Sun 2011 JCP]
The finite difference scheme (7)-(9) is unconditionally stable to theinitial value ψ and the right hand term f .
Theorem (Convergence) [Gao and Sun 2011 JCP]
Assume that u(x , t) ∈ C6,2x ,t ([a, b]× [0,T ]) is the solution of
(1)-(3) and uki | 0 6 i 6 M, 0 6 k 6 N is solution of the finite
difference scheme (7)-(9), respectively. Then there exists a positiveconstant C such that
‖Uk − uk‖∞ 6 C (τ2−α + h4), 0 6 k 6 N.
2.1 Dirichlet boundary problem of the sub-diffusionequation
Theorem (Stability) [Gao and Sun 2011 JCP]
The finite difference scheme (7)-(9) is unconditionally stable to theinitial value ψ and the right hand term f .
Theorem (Convergence) [Gao and Sun 2011 JCP]
Assume that u(x , t) ∈ C6,2x ,t ([a, b]× [0,T ]) is the solution of
(1)-(3) and uki | 0 6 i 6 M, 0 6 k 6 N is solution of the finite
difference scheme (7)-(9), respectively. Then there exists a positiveconstant C such that
‖Uk − uk‖∞ 6 C (τ2−α + h4), 0 6 k 6 N.
2.1 Dirichlet boundary problem of the sub-diffusionequation
In (1)-(3), let a = 0, b = 1, T = 1, κγ = 1,
f (x , t) = ex[(1 + γ)tγ − Γ(2 + γ)
Γ(1 + 2γ)t2γ
],
ϕ1(t) = t1+γ , ϕ2(t) = et1+γ , u(x , 0) = 0.
Then the exact solution is
u(x , t) = ex t1+γ .
2.1 Dirichlet boundary problem of the sub-diffusionequation
Table: Convergence orders of the difference scheme (7)-(9) in temporaldirection with h = 1
2.3 Space unbounded domain problem for the timefractional sub-diffusion equation
Using the Laplace transform, the original problem on the spaceunbounded domain is reduced to the initial-boundary valueproblem on a space bounded domain, i.e.,
u(x , y , t) = ϕ(x , y , t), (x , y) ∈ ∂Ω, 0 < t 6 T , (62)
where 0 < α < 1, ∆ is the two-dimensional Laplacian operator, thedomain Ω = (0, L1)× (0, L2), and ∂Ω is the boundary,ϕ(x , y , t), ψ(x , y) and f (x , y , t) are known smooth functions.
5.1 ADI methods for 2D time fractional sub-diffusionequation
Taking two positive integers M1,M2, let xi = ih1 and yj = jh2 withh1 = L1/M1 and h2 = L2/M2. Define Ωh1 = xi | 0 6 i 6 M1 andΩh2 = yj | 0 6 j 6 M2, then the domain Ω is covered byΩh = Ωh1 × Ωh2 . For any mesh functionu = uij | 0 6 i 6 M1, 0 6 j 6 M2 defined on Ωh1 × Ωh2 , denote
u(x , y , t) = ϕ(x , y , t), (x , y) ∈ ∂Ω, 0 < t 6 T , (71)
where 1 < γ < 2, ∆ is the two-dimensional Laplacian operator, thedomain Ω = (0, L1)× (0, L2), and ∂Ω is the boundary,ϕ(x , y , t), ψ(x , y), φ(x , y) and f (x , y , t) are known smoothfunctions.
5.2 ADI methods for 2D time fractional diffusion-waveequation
We constructed the following Crank-Nicolson scheme
Dγτ u
n− 12
ij = ∆hun− 1
2ij − Γ(3− γ)
4τ1+γδ2xδ
2yδtu
n− 12
ij + fn− 1
2ij ,
(xi , yj) ∈ Ωh, 1 ≤ n ≤ N, (72)
unij = φ(xi , yj , tn), (xi , yj) ∈ ∂Ωh, 1 ≤ n ≤ N, (73)
u0ij = ψ(xi , yj), (xi , yj) ∈ Ωh. (74)
The difference scheme (72) can be decomposed into the ADI form.
5.2 ADI methods for 2D time fractional diffusion-waveequation
Theorem 10(Stability) [Zhang, Sun and Zhao 2012 SINUM]
The finite difference scheme (72)-(74) is unconditionally stable tothe initial values ψ, φ and the right hand term f .
Theorem 11(Convergence) [Zhang, Sun and Zhao 2012SINUM]
Assume that the problem (69)-(71) has smooth solutionu(x , y , t) in the domain Ω× [0,T ] andun
ij | (xi , yj) ∈ Ωh, 1 ≤ n ≤ N be the solution of the differenceschemes (72)-(74). Then there exists a positive constant C suchthat
|Un − un|H1 6 C (τ3−γ + h21 + h2
2), 1 ≤ n ≤ N.
5.2 ADI methods for 2D time fractional diffusion-waveequation
Theorem 10(Stability) [Zhang, Sun and Zhao 2012 SINUM]
The finite difference scheme (72)-(74) is unconditionally stable tothe initial values ψ, φ and the right hand term f .
Theorem 11(Convergence) [Zhang, Sun and Zhao 2012SINUM]
Assume that the problem (69)-(71) has smooth solutionu(x , y , t) in the domain Ω× [0,T ] andun
ij | (xi , yj) ∈ Ωh, 1 ≤ n ≤ N be the solution of the differenceschemes (72)-(74). Then there exists a positive constant C suchthat
|Un − un|H1 6 C (τ3−γ + h21 + h2
2), 1 ≤ n ≤ N.
5.2 ADI methods for 2D time fractional diffusion-waveequation
We presented the following compact scheme
AxAyDγτ u
n− 12
ij = (Ayδ2x +Axδ
2y )u
n− 12
ij − Γ(3− γ)
4τ1+γδ2xδ
2yδtu
n− 12
ij
+AxAy fn− 1
2ij , (xi , yj) ∈ Ωh, 1 ≤ n ≤ N, (75)
unij = φ(xi , yj , tn), (xi , yj) ∈ ∂Ωh, 1 ≤ n ≤ N, (76)
u0ij = ψ(xi , yj), (xi , yj) ∈ Ωh. (77)
The difference scheme (75) can be decomposed into the ADI form.
5.2 ADI methods for 2D time fractional diffusion-waveequation
Theorem 12(Stability) [Zhang, Sun and Zhao 2012 SINUM]
The finite difference scheme (75)-(77) is unconditionally stable tothe initial values ψ, φ and the right hand term f .
Theorem 13(Convergence) [Zhang, Sun and Zhao 2012SINUM]
Assume that the problem (69)-(71) has smooth solutionu(x , y , t) in the domain Ω× [0,T ] andun
ij | (xi , yj) ∈ Ωh, 1 ≤ n ≤ N be the solution of the differencescheme (75)-(77). Then there exists a positive constant C suchthat
|Un − un|H1 6 C (τ3−γ + h41 + h4
2), 1 ≤ n ≤ N.
5.2 ADI methods for 2D time fractional diffusion-waveequation
Theorem 12(Stability) [Zhang, Sun and Zhao 2012 SINUM]
The finite difference scheme (75)-(77) is unconditionally stable tothe initial values ψ, φ and the right hand term f .
Theorem 13(Convergence) [Zhang, Sun and Zhao 2012SINUM]
Assume that the problem (69)-(71) has smooth solutionu(x , y , t) in the domain Ω× [0,T ] andun
ij | (xi , yj) ∈ Ωh, 1 ≤ n ≤ N be the solution of the differencescheme (75)-(77). Then there exists a positive constant C suchthat
|Un − un|H1 6 C (τ3−γ + h41 + h4
2), 1 ≤ n ≤ N.
5.2 ADI methods for 2D time fractional diffusion-waveequation
In (36)-(38), let Ω = (0, π)× (0, π),
f (x , y , t) = sin x sin y[Γ(3 + γ)
2t2 − 2t2−γ
],
u(x , y , 0) = 0, ut(x , y , 0) = 0, ϕ(x , y , t) = 0.Then the exact solution is
u(x , y , t) = sin x sin y t2−γ .
5.2 ADI methods for 2D time fractional diffusion-waveequation
Table: Convergence order of difference scheme (72)-(74) in temporaldirection with h = π
200 .
scheme (72)-(74) scheme (75)-(77)γ τ e∞(h, τ) Order e∞(h, τ) Order
In this review, I report some works on the difference method forthe time fractional differential equations. At first, two discretefractional numerical differential formulae with their truncationerrors are presented. Then some difference schemes areconstructed for the Dirichlet problem, Neumann problem of thesubdiffusion equation and diffusion-wave equation, respectively.For the 2d problem, we concentrate on the ADI schemes. At lastthe multi term problems are considered. Both spatial second orderand fourth order difference schemes are established for eachproblem. The stability and convergence of the difference schemesare proved. The main tool for analyzing the difference schemes isthe energy method. Some numerical examples are provided andthe numerical results are accordance with the theoretical results.
References
Jincheng Ren, Zhi-zhong Sun, Numerical algorithm with high spatialaccuracy for the fractional diffusion-wave equation with Neumannboundary conditions, J Sci Comput., Published online: 20 January 2013
Guang-hua Gao, Zhi-zhong Sun, The finite difference approximation for aclass of fractional sub-diffusion equations on a space unbounded domain,Journal of Computational Physics, 236 (2013) 443-460
Jincheng Ren, Zhi-zhong Sun, Xuan Zhao, Compact difference schemefor the fractional sub-diffusion equation with Neumann boundaryconditions, Journal of Computational Physics, 232 (2013) 456-467
Ya-nan Zhang, Zhi-zhong Sun, and Xuan Zhao. Compact AlternatingDirection Implicit Scheme for the Two-Dimensional FractionalDiffusion-Wave Equation. SIAM J. Numer. Anal., 2012, 50, 1535-1555
Guang-hua Gao, Zhi-zhong Sun, Ya-nan Zhang, A finite differencescheme for fractional sub-diffusion equations on an unbounded domainusing artificial boundary conditions, Journal of Computational Physics231£7¤(2012), 2865-2879
Guang-hua Gao, Zhi-zhong Sun, A Finite Difference Approach for theInitial-Boundary Value Problem of the Fractional Klein-Kramers Equationin Phase Space, Cent. Eur. J. Math., 2012, 10(1), 101-115
Ya-nan Zhang, Zhi-zhong Sun, Hong-wei Wu, Error estimates ofCrank-Nicolson type difference schemes for the sub-diffusion equation,SIAM J. Numer. Anal. 49 (2011), no. 6, 2302–2322.
Xuan Zhao, Zhi-zhong Sun, box-type scheme for fractional sub-diffusionequation with Neumann boundary conditions, Journal of ComputationalPhysics 230£15¤(2011), pp. 6061-6074
Guang-hua Gao, Zhi-zhong Sun, A compact finite difference scheme forthe fractional sub-diffusion equations, Journal of Computational Physics,230 £3¤(2011), 586-595
R. Du, W.R. Cao, Z.Z. Sun, A compact difference scheme for thefractional diffusion wave equation, Appl. Math. Modelling 34£10¤(Oct.2010), 2998-3007
Zhi-zhong Sun, Xiaonan Wu, A fully discrete difference scheme for adiffusion-wave system. Appl. Numer. Math. 56 (2006), no. 2, 193–209.
Xiaonan Wu, Zhi-Zhong Sun, Convergence of difference scheme for heatequation in unbounded domains using artificial boundary conditions.Appl. Numer. Math. 50 (2004), no. 2, 261–277.