and matrix-valued allen-cahn equations
TRANSCRIPT
ENERGY STABILITY OF STRANG SPLITTING FOR VECTOR-VALUED
AND MATRIX-VALUED ALLEN-CAHN EQUATIONS
DONG LI AND CHAOYU QUAN
Abstract. We consider the second-order in time Strang-splitting approximation for vector-valued and matrix-valued Allen-Cahn equations. Both the linear propagator and the nonlinearpropagator are computed explicitly. For the vector-valued case, we prove the maximum principleand unconditional energy dissipation for a judiciously modified energy functional. The modifiedenergy functional is close to the classical energy up to O(τ) where τ is the splitting step. Forthe matrix-valued case, we prove a sharp maximum principle in the matrix Frobenius norm. Weshow modified energy dissipation under very mild splitting step constraints. We exhibit severalnumerical examples to show the efficiency of the method as well as the sharpness of the results.
1. Introduction
In this work we investigate the stability of second-order in time Strang-splitting methodsapplied to two models: One is the vector-valued Allen-Cahn (AC) equation, and the other isthe matrix-valued Allen-Cahn system. The operator splitting methods have been extensivelyused in the numerical simulation of many physical problems, including phase-field equations[2, 3, 11, 12, 13, 15, 14, 10], Schrodinger equations [4, 5, 18], and the reaction-diffusion systems[6, 7]. A prototypical second order in time method is the Strang splitting approximation [8, 9].In a slightly more general set up, we consider the following abstract parabolic problem:
∂tu = Lu− f(u), t > 0;
u∣∣∣t=0
= u0.(1.1)
where u : [0,∞) → B and B is a real Banach space. The operator L : D(L) ⊂ B → B is adissipative closed operator which typically is the infinitesimal generator of a strongly-continuousdissipative semigroup. The domain D(L) is typically a dense subset of B. On the other handf : B→ B is a nonlinear operator. Take τ > 0 as the splitting time step. Define for t > 0
S(B)L (t) = etL (1.2)
which is the solution operator to the linear equation ∂tu = Lu. Define S(B)N (τ) as the nonlinear
solution operator to the system ∂tu = −f(u), 0 < t ≤ τ ;
u∣∣∣t=0
= b ∈ B.(1.3)
In yet other words, S(B)N (τ) is the map b → u(τ). The Strang-splitting approximation for (1.1)
takes the form
un+1 = S(B)L (τ/2)S(B)
N (τ)S(B)L (τ/2)un, n ≥ 0. (1.4)
Let uex be the exact solution to (1.1). In general it is expected that on a finite time interval [0, T ]
supnτ≤T
‖uex(nτ)− un‖ = O(τ2), (1.5)
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2 D. LI AND C.Y. QUAN
where ‖ · ‖ denotes a working norm which is allowed to be weaker than the norm endowed withthe Banach space B. On the other hand, the local truncation error is typically O(τ3), i.e.
‖Sex(τ)a− S(B)L (τ/2)S(B)
N (τ)S(B)L (τ/2)a‖ = O(τ3), (1.6)
where a ∈ B, and Sex(τ) is the exact solution operator to (1.1).Despite the remarkable effectiveness of the scheme (1.4) (cf. [3] for the case of Allen-Cahn
equations), there have been few rigorous works addressing the stability and regularity of theStrang splitting solutions. The general assertions (1.5)–(1.6) are hinged on nontrivial a prioriestimates of the numerical iterates in various Banach spaces. In a recent series of works [11,12, 13, 14, 15], we developed a new theoretical framework to establish convergence, stability andregularity of general operator splitting methods for a myriad of phase field models including theCahn-Hilliard equations, Allen-Cahn equations and the like. More pertinent to the discussionhere is the recent work [14] which settled the stability for a class of scalar-valued Allen-Cahnequations with polynomial or logarithmic potential nonlinearities. In this work we develop furtherthe program initiated in [14] and analyze the Strang-splitting for two types of models. The firstis the vector-valued Allen-Cahn equation
∂tu = ∆u + (1− |u|2)u, u ∈ Rm, (1.7)
where |u|2 =∑m
i=1 u2i for u = (u1, · · · , um)T (m ≥ 1 is an integer), and the Laplacian operator is
applied to u component-wise. The second is the matrix-valued Allen-Cahn equation ([16]):
∂tU = ∆U + U − UUTU, U ∈ Rm×m, (1.8)
where the Laplacian operator is applied to the matrix U entry-wise. For both models we shalldevelop the corresponding stability theory for the Strang-splitting approximation in the style of(1.4). Roughly speaking, our results can be summarized in the following table where τ is thesplitting time step.
L∞-stability Modified energy dissipation
Vector-valued AC 0 < τ <∞ 0 < τ <∞
Matrix-valued AC 0 < τ <∞ meτ (e2τ − 1) ≤ 0.43
We turn now to more precise formulation of the results. Our first result is on the vector-valued Allen-Cahn equation. An interesting feature is that the nonlinear propagator still enjoysa relatively simple explicit expression.
Theorem 1.1 (Unconditional stability of Strang-splitting for the vector-valued AC). SupposeΩ = [−π, π]d is the 2π-periodic d-dimensional torus in physical dimensions d ≤ 3. Consider thevector-valued AC for u : [0,∞)× Ω→ Rm, m ≥ 1:
∂tu = ∆u + (1− |u|2)u, (t, x) ∈ (0,∞)× Ω;
u∣∣∣t=0
= u0,(1.9)
where u0 : Ω→ Rm is the initial data. Define SL(t) = et∆ and for w ∈ Rm,
SN (t)w :=((e2t − 1)|w|2 + 1
)− 12 etw. (1.10)
Define for n ≥ 0 the Strang-splitting iterates
un+1 = SL (τ/2)SN (τ)SL (τ/2) un. (1.11)
The following hold.
ENERGY STABILITY OF STRANG SPLITTING METHOD 3
(1) The maximum principle. For any τ > 0 and any n ≥ 0, it holds that
‖|un+1|‖L∞x ≤ max1, ‖|un|‖L∞x . (1.12)
It follows that
supn≥1‖|un|‖L∞x ≤ max1, ‖|u0|‖L∞x . (1.13)
In particular if ‖|u0|‖L∞x ≤ 1, then
supn≥1‖|un|‖L∞x ≤ 1. (1.14)
(2) Modified energy dissipation. For any τ > 0, we have
E(un+1) ≤ E(un), ∀n ≥ 0. (1.15)
Here
un = SL (τ/2) un; (1.16)
E(u) =
∫Ω
(1
2τ
⟨(e−τ∆ − 1)u,u
⟩+G(u)
)dx; (1.17)
G(u) =1
2τ|u|2 − eτ
τ(e2τ − 1)
((1 + (e2τ − 1)|u|2
) 12 − 1
). (1.18)
In the above 〈a, b〉 =∑m
i=1 aibi for a = (a1, · · · , am)T, b = (b1, · · · , bm)T ∈ Rm.
Remark 1.1. The significance of the uniform stability result obtained in Theorem 1.1 is that itleads to all higher Sobolev estimates as well as convergence. For example, by using the techniquesdeveloped in [14], we can show uniform Sobolev bounds. Namely if u0 ∈ Hk0(Ω, Rm) for somek0 ≥ 1, then
supn≥1‖un‖Hk0 ≤ C1, (1.19)
where C1 > 0 depends only on (k0, d, ‖u0‖Hk0 , m). Moreover for any k ≥ k0, we have
supn≥ 1
τ
‖un‖Hk ≤ C2, (1.20)
where C2 > 0 depends only on (m, k, k0, d, ‖u0‖Hk0 ). Let uex be the exact solution to thevector-valued Allen-Cahn equation corresponding to initial data u0. If we assume u0 has highregularity (e.g. u0 ∈ Hk0 for some sufficiently large k0), then for any T > 0, it holds that
supn≥1,nτ≤T
‖un − uex(nτ, ·)‖L2(Ω) ≤ C · τ2, (1.21)
where C > 0 depends on (u0, T , d, m).
Remark 1.2. The modified energy for un has a close connection with the standard energy Est(un)defined by
Est(un) =
∫Ω
(1
2|∇un|2 +
1
4(|un|2 − 1)2 − 1
4
)dx. (1.22)
Note that here the integrand of Est(·) includes a harmless constant −14 . If u0 ∈ Hk0(Ω,Rm) for
sufficiently large k0, then for 0 < τ ≤ 1, we have
supn≥0|En − Est(un)| ≤ C3τ, (1.23)
where C3 > 0 depends only on (d, m, u0). This result can be proved by using the uniformSobolev estimates established in the preceding remark. We omit the elementary argument here forsimplicity.
4 D. LI AND C.Y. QUAN
In recent work [16], Osting and Wang considered the minimization problem
minA∈H1(Ω,Om)
1
2
∫Ω‖∇A‖2Fdx, (1.24)
where Om ⊂ Rm×m is the group of orthogonal matrices, and the gradient is taken as the usualsense when A is regarded as a matrix-valued function in Rm×m, i.e. not the covariant derivativesense in e.g. [1]. For a matrix A,B ∈ Rm×m, we use the usual Frobenius norm and Frobeniusinner product:
‖A‖2F =m∑
i,j=1
A2ij , 〈A,B〉F =
m∑i,j=1
AijBij . (1.25)
To enforce the hard constraintA ∈ H1(Ω, Om), one can employ two relaxed functionals parametrizedby 0 < ε 1:
Model 1 : minA∈H1(Ω,Rm×m)
∫Ω
(1
2‖∇A‖2F +
1
2ε2dist2(On, A)
)dx; (1.26)
Model 2 : minA∈H1(Ω,Rm×m)
∫Ω
(1
2‖∇A‖2F +
1
4ε2‖ATA− Im‖2F
)dx. (1.27)
As shown in [16], these in turn lead to the following gradient flows
Model 1 : ∂tA = ∆A− ε−2U(Σ− Im)V T, A = UΣV T is non-singular; (1.28)
Model 2 : ∂tA = ∆A− ε−2U(Σ2 − Im)σV T, A = UΣV T . (1.29)
The gradient flow in Model 2 can be further simplified as
∂tA = ∆A− ε−2A(ATA− Im). (1.30)
In [16], the authors introduced an energy-splitting method to find local minima of (1.26) and(1.27). These are nontrivial stationary solutions other than the trivial constant orthogonal matrix-valued function) of (1.28) and (1.30). The method can be rephrased as the following operator-splitting:
Un+1 = SProjN SL(τ)Un, (1.31)
where SL(τ) = eτ∆ is applied to the matrix entry-wise, and(SProjN A
)(x) = U(x)V T(x), if A(x) = U(x)Σ(x)V T(x). (1.32)
In this work, we take a direct approach to (1.30) and employ a Strang-splitting method tosolve (1.30) efficiently and accurately. For simplicity of presentation we shall take ε = 1 in (1.30).We have the following theorem.
Theorem 1.2 (Stability for matrix-valued AC). Suppose Ω = [−π, π]d is the 2π-periodic d-dimensional torus in physical dimensions d ≤ 3. Consider the matrix-valued AC for U : [0,∞)×Ω→ Rm×m, m ≥ 1:
∂tU = ∆U + U − UUTU, (t, x) ∈ (0,∞)× Ω;
U∣∣∣t=0
= U0,(1.33)
where U0 : Ω→ Rm×m is the initial data. Define SL(t) = et∆ and for A ∈ Rm×m,
SN (t)A :=((e2t − 1)AAT + I
)− 12 etA. (1.34)
Define for n ≥ 0 the Strang-splitting iterates
Un+1 = SL (τ/2)SN (τ)SL (τ/2)Un. (1.35)
ENERGY STABILITY OF STRANG SPLITTING METHOD 5
The following hold.
(1) The maximum principle. For any τ > 0 and any n ≥ 0, it holds that
‖‖Un+1‖F ‖L∞x ≤ max√m, ‖‖Un‖F ‖L∞x . (1.36)
It follows that
supn≥1‖‖Un‖F ‖L∞x ≤ max
√m, ‖‖U0‖F ‖L∞x . (1.37)
In particular if ‖‖U0‖F ‖L∞x ≤√m, then
supn≥1‖‖Un‖F ‖L∞x ≤
√m. (1.38)
(2) Modified energy dissipation for small time. Assume ‖‖U0‖F ‖L∞x ≤√m. If τ > 0 satisfies
meτ (e2τ − 1) ≤ 0.43, (1.39)
then
E(Un+1) ≤ E(Un), ∀n ≥ 0. (1.40)
Here
Un = SL (τ/2)Un; (1.41)
E(U) =
∫Ω
1
2τ
⟨(e−τ∆ − 1)U,U
⟩F
+ 〈G(U), I〉F dx; (1.42)
G(U) =1
2τUUT − eτ
τ(e2τ − 1)
((I + (e2τ − 1)UUT
) 12 − I
). (1.43)
In the above 〈A,B〉F = Tr(ATB) =∑
i,j AijBij denotes the usual Frobenius inner product.
Remark 1.3. We should point it out that the dynamics of the matrix-valued Allen-Cahn case isin general qualitatively different from the vector-valued Allen-Cahn case. In particular there doesnot appear to exist a simple procedure such that the vector-valued AC model can be embedded intothe matrix-valued AC model. A very tempting idea is to consider the following system
∂tU = ∆U + U − UUTU,
U∣∣∣t=0
= U0 = aaT,(1.44)
where a : Ω → Rm. In yet other words, we consider the matrix-valued AC model with rank oneinitial data. It is natural to speculate that U is connected with the solution to
∂tu = ∆u + u− |u|2u,u∣∣∣t=0
= a.(1.45)
However one can check that U 6= uuT for t > 0. The main reason is that
et∆(aaT
)6= et∆a(et∆a)T. (1.46)
If one drops the Laplacian and adopt only the nonlinear evolution, then one can show that U =uuT.
The rest of this article is organized as follows. In Section 2 we carry out the proof of Theorem1.1. In Section 3 we analyze the Strang-splitting for the matrix-valued Allen-Cahn equation.In Section 4 we showcase a few numerical simulations for the vector-valued Allen-Cahn and thematrix-valued Allen-Cahn equations.
6 D. LI AND C.Y. QUAN
2. Vector-valued Allen-Cahn
In this section we give the proof of Theorem 1.1. We consider the vector-valued Allen-Cahnequation for u = u(t, x) : [0,∞)× Ω→ Rm, m ≥ 1:
∂tu = ∆u + u− |u|2u, (t, x) ∈ (0,∞)× Ω;
u∣∣∣t=0
= u0, x ∈ Ω.(2.1)
Here |u|2 is the usual l2 norm, i.e. |u|2 =∑m
i=1 u2i for u = (u1, · · · , um)T. The spatial domain
Ω = [−π, π]d is the 2π-periodic torus in physical dimensions d ≤ 3.We proceed in several steps.
2.1. Properties of SL and SN . We first consider the pure nonlinear evolution. This is drivenby the following ODE system written for u = u(t) : [0,∞)→ Rm.
ddtu = (1− |u|2)u,
u∣∣∣t=0
= a ∈ Rm.(2.2)
Proposition 2.1 (The explicit nonlinear propagator SN (t)). Given a ∈ Rm, the unique smoothsolution U(t) to (2.2) is given by
SN (t)a := U(t) =((e2t − 1)|a|2 + 1
)− 12 eta, t > 0. (2.3)
Remark 2.1. If a is a matrix-valued function, i.e. a : Ω → Rm, then we naturally extend thedefinition of SN (t)a as (
SN (t)a)
(x) = SN (t)(a(x)), x ∈ Ω. (2.4)
This convention will be used without explicit mentioning.
Proof. Taking the l2-inner product on both sides of (2.2) gives us
1
2
d
dt|u|2 = (1− |u|2)|u|2. (2.5)
This is an ODE for |u|2 which has an explicit solution:
|u(t)|2 =e2t|a|2
(e2t − 1)|a|2 + 1. (2.6)
Plugging the above into (2.3), we obtain
d
dtu =
1− |a|2
(e2t − 1)|a|2 + 1u. (2.7)
It is not difficult to work out the explicit solution as
u(t) =eta
((e2t − 1)|a|2 + 1)12
. (2.8)
Given u = (u1, · · · , um)T : Ω→ Rm and t > 0, we define the linear propagator(SL(t)u
)i(x) = (et∆ui)(x), i = 1, · · · ,m. (2.9)
In yet other words, the operator SL(t) = et∆ is applied to the vector u entry-wise.
Theorem 2.1 (Maximum principle for SL and SN ). Let Ω = [−π, π]d be the 2π-periodic d-dimensional torus. For any τ > 0, the following hold.
ENERGY STABILITY OF STRANG SPLITTING METHOD 7
(1) For any measurable vector-valued a : Ω→ Rm, we have
‖|SL(τ)a|‖L∞x ≤ ‖|a(x)|‖L∞x . (2.10)
(2) For any w ∈ Rm, we have
|SN (τ)w| ≤ max1, |w|. (2.11)
Proof. We first show (2.10). Clearly for any vector v ∈ Rm, we have
|v| = supv∈Rm: |v|≤1
〈v, v〉, (2.12)
where 〈, 〉 denotes the usual l2-inner product. With no loss we may assume ‖|a(x)|‖L∞x ≤ 1. Fixx0 ∈ Ω. It suffices for us to show ∣∣∣∣∫
Ωk(x0 − y)a(y)dy
∣∣∣∣ ≤ 1, (2.13)
where k(·) is the scalar-valued kernel corresponding to SL(τ). By (2.12), we only need to checkfor any v with |v| ≤ 1, ∫
Ωk(x0 − y)〈a(y), v〉dy ≤ 1. (2.14)
But this is obvious since∫
Ω k(x0 − y)dy = 1 and 〈a(y), v〉 ≤ ‖|a(x)|‖‖L∞x |v| ≤ 1.We turn now to (2.11). By (2.6), we have
|SN (t)w|2 =e2t|w|2
(e2t − 1)|w|2 + 1. (2.15)
Consider the scalar function
ϕ(λ) =e2tλ
(e2t − 1)λ+ 1. (2.16)
It is not difficult to check that ϕ is monotonically increasing on [0,∞). Furthermore if λ ≥ 1,then
ϕ(λ) ≤ e2tλ
(e2t − 1) + 1= λ. (2.17)
The desired result clearly follows.
Lemma 2.1. Let τ > 0 and consider G : Rm → R defined as
G(u) =1
2τ|u|2 − eτ
τ(e2τ − 1)
((1 + (e2τ − 1)|u|2
) 12 − 1
). (2.18)
For any u,v ∈ Rm, it holds that
−〈(∇G)(u), v − u〉 ≤ G(u)−G(v) +1
2τ|v − u|2. (2.19)
Here 〈a, b〉 =∑m
i=1 aibi for a = (a1, · · · , am)T, b = (b1, · · · , bm)T ∈ Rm.
Proof. We first examine the auxiliary function
h(u) = −(1 + |u|2)12 . (2.20)
Clearly
∂ih = −(1 + |u|2)−12ui; (2.21)
∂ijh = (1 + |u|2)−32uiuj − (1 + |u|2)−
12 δij (2.22)
= (1 + |u|2)−12
( ui√1 + |u|2
uj√1 + |u|2
− δij). (2.23)
8 D. LI AND C.Y. QUAN
Thusm∑
i,j=1
ξiξj∂ijh ≤ 0, ∀ ξ = (ξ1, · · · , ξm)T ∈ Rm, ∀u ∈ Rm. (2.24)
In yet other words, the function h is concave. Our desired result then easily follows from Taylorexpanding G(u + θ(v − u)) for θ ∈ [0, 1].
Theorem 2.2 (Unconditional modified energy dissipation for vector-valued Allen-Cahn). Sup-pose Ω = [−π, π]d is the 2π-periodic d-dimensional torus in physical dimensions d ≤ 3. Letu0 : Ω→ Rm satisfy ‖|u0(x)|‖‖L∞x ≤ 1. Recall SL(t) = et∆ and for w ∈ Rm,
SN (t)w :=((e2t − 1)|w|2 + 1
)− 12 etw. (2.25)
Define for n ≥ 0 the Strang-splitting iterates
un+1 = SL (τ/2)SN (τ)SL (τ/2) un. (2.26)
For any τ > 0, we have
E(un+1) ≤ E(un), ∀n ≥ 0, (2.27)
where
un = SL (τ/2) un; (2.28)
E(u) =
∫Ω
(1
2τ
⟨(e−τ∆ − 1)u,u
⟩+G(u)
)dx; (2.29)
G(u) =1
2τ|u|2 − eτ
τ(e2τ − 1)
((1 + (e2τ − 1)|u|2
) 12 − 1
). (2.30)
In the above 〈a, b〉 =∑m
i=1 aibi for a = (a1, · · · , am)T, b = (b1, · · · , bm)T ∈ Rm.
Proof. By definition we have
e−τ∆un+1 =eτ un
((e2τ − 1)|un|2 + 1)12
. (2.31)
We rewrite it as
1
τ(e−τ∆ − 1)un+1 +
1
τ(un+1 − un) =
1
τ
(eτ un
((e2τ − 1)|un|2 + 1)12
− un
). (2.32)
Note that
(∇G)(u) =u
τ− eτu
τ ((e2τ − 1)|u|2 + 1)12
. (2.33)
By Lemma 2.1, we have ⟨1
τ
( eτ un
((e2τ − 1)|un|2 + 1)12
− un), un+1 − un
⟩(2.34)
≤ G(un)−G(un+1) +1
2τ|un+1 − un|2. (2.35)
Taking the l2-inner product with (un+1 − un) and integrating in x on both sides of (2.32), wehave
E(un+1)− E(un) ≤ − 1
2τ
∫Ω|un+1 − un|2dx ≤ 0. (2.36)
ENERGY STABILITY OF STRANG SPLITTING METHOD 9
3. Matrix-valued Allen-Cahn equation
In this section we carry out the proof of Theorem 1.2 in several steps. We study the matrix-valued Allen-Cahn equation for U = U(t, x) : [0,∞)× Ω→ Rm×m:
∂tU = ∆U + U − UUTU. (3.1)
The spatial domain Ω = [−π, π]d is the 2π-periodic torus in physical dimensions d ≤ 3.
3.1. Definition and properties of SN and SL. We consider first the pure nonlinear part, i.e.the following ODE for U = U(t) : [0,∞)→ Rm×m:
ddtU = U − UUTU,
U∣∣∣t=0
= U0 ∈ Rm×m.(3.2)
Remarkably, we find that the above ODE admits an explicit solution.
Proposition 3.1 (The explicit nonlinear propagator SN (t)). Given U0 ∈ Rm×m, the uniquesmooth solution U(t) to (3.2) is given by
SN (t)U0 := U(t) =((e2t − 1)U0U
T0 + I
)− 12 etU0, t > 0. (3.3)
Remark 3.1. If U0 is a matrix-valued function, i.e. U0 : Ω→ Rm×m, then we naturally extendthe definition of SN (t)U0 as(
SN (t)U0
)(x) = SN (t)(U0(x)), x ∈ Ω. (3.4)
This convention will be used without explicit mentioning.
Proof. We begin by noting that
d
dt
((e2t − 1)U0U
T0 + I
)= 2e2tU0U
T0 . (3.5)
This clearly commutes with (e2t − 1)U0UT0 + I. In particular we have
d
dt
(((e2t − 1)U0U
T0 + I)−
12
)= −e2tU0U
T0 ((e2t − 1)U0U
T0 + I)−
32 (3.6)
= −((e2t − 1)U0UT0 + I)−
12 e2tU0U
T0 . (3.7)
With the above we obtain
U ′(t) =((e2t − 1)U0U
T0 + I
)− 12 etU0 −
((e2t − 1)U0U
T0 + I
)− 32 e3tU0U
T0 U0. (3.8)
Note that((e2t − 1)U0U
T0 + I
)− 12 and U0U
T0 commute. It follows that
UUTU =((e2t − 1)U0U
T0 + I
)− 12 e2tU0U
T0
((e2t − 1)U0U
T0 + I
)−1etU0
=((e2t − 1)U0U
T0 + I
)− 32 e3tU0U
T0 U0.
(3.9)
Therefore, U(t) satisfiesd
dtU = U − UUTU. (3.10)
Given U : Ω→ Rm×m and t > 0, we define the linear propagator(SL(t)U
)ij
(x) =(et∆Uij
)(x), i, j = 1, · · ·m. (3.11)
In yet other words, the operator SL(t) = et∆ is applied to the matrix U entry-wise.
10 D. LI AND C.Y. QUAN
Theorem 3.1 (Maximum principle for SL and SN ). Let Ω = [−π, π]d be the 2π-periodic d-dimensional torus. For any τ > 0, the following hold.
(1) For any measurable matrix-valued A : Ω→ Rm×m, we have
‖‖SL(τ)A‖F ‖L∞x ≤ ‖‖A(x)‖F ‖L∞x . (3.12)
(2) For any B ∈ Rm×m with ‖B‖F ≤√m, we have
‖SN (τ)B‖F ≤√m. (3.13)
Remark 3.2. In [16, Prop. 3.2.], Osting and Wang proved a maximum principle for SL(τ) underthe assumption that A is a continuous function with ‖A‖F = 1 for every x ∈ Ω. We do not needsuch a stringent assumption here. Our result here is optimal and the proof appears to be simpler.
Proof. We first show (3.12). Recall the usual Frobenius inner product:
〈M1, M2〉F =m∑
i,j=1
(M1)ij(M2)ij = Tr(M1MT2 ). (3.14)
For any matrix M ∈ Rm×m, we clearly have
‖M‖F = supM∈Rm×m: ‖M‖F≤1
〈M, M〉F . (3.15)
With no loss we may assume ‖‖A(x)‖F ‖L∞x ≤ 1. Fix x0 ∈ Ω. It suffices for us to show
‖∫
Ωk(x0 − y)A(y)dy‖F ≤ 1, (3.16)
where k(·) is the scalar-valued kernel corresponding to SL(τ). By (3.15), we only need to check
for any M with ‖M‖F ≤ 1, ∫Ωk(x0 − y)〈A(y), M〉Fdy ≤ 1. (3.17)
But this is obvious since∫
Ω k(x0 − y)dy = 1 and 〈A(y), M〉F ≤ ‖‖A(x)‖F ‖L∞x ‖M‖F ≤ 1.Next we show (3.13). Denote U = U(t) = SN (t)B. Clearly
∂tU = U − UUTU. (3.18)
Taking the L2 Frobenius inner with U on both sides of the above equation, we obtain
1
2∂tα(t) = α(t)− ‖U(t)U(t)T‖2F , (3.19)
where we have denoted
α(t) = 〈U(t), U(t)〉F = Tr(U(t)U(t)T). (3.20)
Note that
α = Tr(UUT) = 〈UUT, I〉F ≤ ‖UUT‖F√m. (3.21)
Thus
‖U(t)U(t)T‖2F ≥1
mα(t)2. (3.22)
It follows that
1
2∂t
(1
mα(t)
)≤ 1
mα(t)−
(1
mα(t)
)2
. (3.23)
ENERGY STABILITY OF STRANG SPLITTING METHOD 11
It is not difficult to check that 1mα(t) is a continuously-differentiable function of t defined for all
t ≥ 0, nonnegative and 1mα(0) ≤ 1. By a simple argument-by-contradiction, we can show that
for any δ1 > 0
supt≥0
1
mα(t) ≤ 1 + δ1. (3.24)
Sending δ1 to zero then yields the desired estimate.
Remark 3.3. An alternative proof of (3.13) goes as follows. Since ‖B‖F ≤√m, we have
Tr(BBT) =
m∑i=1
λi ≤ m, (3.25)
where λi ≥ 0 are the eigenvalues of BBT. By (3.3), we have
SN (t)B =((e2t − 1)BBT + I
)− 12 etB; (3.26)
‖SN (t)B‖2F = e2tTr(
((e2t − 1)BBT + I)−12BBT((e2t − 1)BBT + I)−
12
)= e2tTr
(((e2t − 1)BBT + I)−1BBT
)(3.27)
=
m∑i=1
e2t((e2t − 1)λi + 1
)−1λi︸ ︷︷ ︸
=:ϕ(λi)
. (3.28)
Clearly for any λ ≥ 0,
ϕ′(λ) = e2t((e2t − 1)λ+ 1
)−2 ≥ 0; (3.29)
ϕ′′(λ) = −2e2t(e2t − 1)((e2t − 1)λ+ 1
)−3 ≤ 0. (3.30)
In particular ϕ is a concave function on [0,∞). By Jensen’s inequality and the fact that 1m
∑mi=1 λi ≤
1 , we have
1
m
m∑i=1
ϕ(λi) ≤ ϕ(1
m
m∑i=1
λi) ≤ ϕ(1) = 1. (3.31)
Thus ‖SN (t)B‖2F ≤ m.
3.2. Modified energy dissipation. In this subsection we shall often use (sometimes withoutexplicit mentioning) the obvious identity
Tr(ABT) = 〈A, B〉F =
m∑i,j=1
AijBij , ∀A,B ∈ Rm×m, (3.32)
where 〈, 〉F denotes the usual Frobenius inner product. In particular
Tr(A) = 〈A, I〉F . (3.33)
It follows that if A = A(s), s ∈ [0, 1] is a continuously differentiable matrix-valued function, then
d
dsTr(A(s)) = 〈A′(s), I〉F = Tr(A′(s)). (3.34)
Other formulae follow similarly from the above identities.
Lemma 3.1. Suppose B = B(s) : s ∈ [0, 1]→ Rm×m is continuously differentiable with
max0≤s≤1
‖B(s)‖F < 1. (3.35)
12 D. LI AND C.Y. QUAN
For any s ∈ [0, 1] and any B1 ∈ Rm×m, it holds that∣∣∣∣Tr( dds
((I +B(s))−
12
)B1
)∣∣∣∣ ≤ 1
2(1− ‖B(s)‖F )−
32 ‖B′(s)‖F ‖B1‖F . (3.36)
Proof. It suffices for us to bound ‖ dds(
(I + B(s))−12
)‖F . Recall the power series expansion for a
real number |x| < 1
(1− x)−12 =
∑k≥0
Ckxk, (3.37)
1
2(1− x)−
32 =
∑k≥1
Ckkxk−1. (3.38)
where the coefficients Ck are all positive. For integer k ≥ 1, we note that
d
ds
(Bk)
= B′Bk−1 +BB′B · · ·B +BBB′B · · ·B + · · ·+Bk−1B′. (3.39)
In particular we do not assume the matrix B′ commutes with B. On the other hand, since thematrix Frobenius norm is sub-multiplicative, we have
‖ dds
(Bk)‖F ≤ k‖B‖k−1
F ‖B′‖F . (3.40)
It follows that ∥∥∥∥ dds((I +B(s))−12
)∥∥∥∥F
(3.41)
≤∑k≥1
Ck
∥∥∥ dds
(B(s)k)∥∥∥F
(3.42)
≤∑k≥1
Ckk‖B(s)‖k−1F ‖B′(s)‖F (3.43)
=1
2(1− ‖B(s)‖F )−
32 ‖B′(s)‖F . (3.44)
The desired result then easily follows.
Lemma 3.2. Denote by Rm×msp the set of symmetric positive-definite matrices in Rm×m. Suppose
B = B(s) : s ∈ [0, 1]→ Rm×msp is continuously differentiable with
ξTB(s)ξ ≥ η1 > 0, ∀ ξ ∈ Rm, ∀ s ∈ [0, 1]. (3.45)
Then
d
dsTr(B(s)
12 ) =
1
2Tr(B(s)−
12B′(s)
). (3.46)
Remark 3.4. Later we shall take B(s) = I + εφ(s)φ(s)T with ε > 0 sufficiently small andφ(s) ∈ Rm×m. In that case we can directly make use of the power series expansion and derive(3.46) for small ε. The strength of Lemma 3.2 is that the smallness of ε is not needed.
Proof. We begin by noting that for any integer k ≥ 2,
d
dsTr(B(s)k) = Tr
(B′Bk−1 +BB′B · · ·B + · · ·+Bk−1B′
)(3.47)
= kTr(B(s)k−1B′(s)
). (3.48)
ENERGY STABILITY OF STRANG SPLITTING METHOD 13
It follows that for any α0 ∈ R,
d
dsTr(eα0B(s)
)= α0Tr
(eα0B(s)B′(s)
). (3.49)
Note that
B(s)12 =
1
Γ(12)
∫ ∞0
e−tB(s)t−12dt, (3.50)
where Γ(·) is the usual Gamma function. In view of the strict positivity assumption (3.45), theconvergence in (3.50) is out of question. Clearly
Tr(B(s)12 ) =
1
Γ(12)
∫ ∞0
Tr(e−tB(s)
)t−
12dt. (3.51)
The desired result then easily follows.
Lemma 3.3. Let U0, H ∈ Rm×m satisfy ‖U0‖F ≤√m and ‖U0 + H‖F ≤
√m. Let τ > 0. For
s ∈ [0, 1], define
φ = φ(s) = U0 + sH; (3.52)
h(s) = Tr( 1
2τφφT − eτ
τ(e2τ − 1)
((I + (e2τ − 1)φφT)
12 − I
)). (3.53)
We have
h′(0) =1
τTr(U0H
T)− eτ
τTr(
(I + (e2τ − 1)U0UT0 )−
12U0H
T). (3.54)
If eτ (e2τ − 1)m ≤ ε0 < 1, then
max0≤s≤1
h′′(s) ≤ 1
τ
(1 + (1− ε0)−
32 ε0
)‖H‖2F . (3.55)
If eτ (e2τ − 1)m ≤ ε0 and (1− ε0)−32 ε0 ≤ 1, then
−h′(0) ≤ h(0)− h(1) +1
τ‖H‖2F . (3.56)
Remark 3.5. If we take ε0 = 0.43, then
(1− ε0)−32 ε0 ≈ 0.99209 < 1. (3.57)
Proof. Observe that for all s ∈ [0, 1]
‖φ(s)‖F = ‖s(U0 +H) + (1− s)U0‖F ≤√m. (3.58)
By Lemma 3.2, we have
h′(s) = Tr( 1
2τ
(φ′φT + φ(φ′)T
)− eτ
τ(e2τ − 1)· 1
2(I + (e2τ − 1)φφT)−
12 (e2τ − 1)(φ′φT + φ(φ′)T)
)(3.59)
= Tr(1
τφHT − eτ
τ· 1
2(I + (e2τ − 1)φφT)−
12 (HφT + φHT)
). (3.60)
The equality (3.54) follows from the fact that if A ∈ Rm×m is symmetric, then
Tr(ABT ) = Tr(BA) = Tr(AB), ∀B ∈ Rm×m. (3.61)
By direction computation, we also have
h′′(s) = Tr(1
τHHT)− eτ
τTr(
(I + (e2τ − 1)φφT)−12HHT
)(3.62)
− eτ
2τTr
(d
ds
((I + (e2τ − 1)φφT)−
12
)(HφT + φHT)
). (3.63)
14 D. LI AND C.Y. QUAN
Clearly
Tr(
(I + (e2τ − 1)φφT)−12HHT
)(3.64)
=Tr(HT(I + (e2τ − 1)φφT)−
12H)≥ 0. (3.65)
Note that
(e2τ − 1)‖φφT‖F ≤ (e2τ − 1)m < ε0 < 1. (3.66)
By Lemma 3.1 we have
eτ
2τTr
(d
ds
((I + (e2τ − 1)φφT)−
12
)(HφT + φHT)
)(3.67)
≤ eτ
2τ· 1
2
(1− (e2τ − 1)‖φφT‖F
)− 32(e2τ − 1)‖HφT + φHT‖2F (3.68)
≤ 1
τ(1− ε0)−
32 eτ (e2τ − 1)m‖H‖2F . (3.69)
Since Tr(HHT) = ‖H‖2F , it follows that
h′′(s) ≤ 1
τ
(1 + (1− ε0)−
32 eτ (e2τ − 1)m
)‖H‖2F . (3.70)
The inequality (3.56) follows from a simple Taylor expansion of h(s), namely
h(1) ≤ h(0) + h′(0) +1
2max
0≤s≤1h′′(s). (3.71)
Theorem 3.2 (Modified energy dissipation for matrix-valued AC with mild splitting step con-straint). Suppose Ω = [−π, π]d is the 2π-periodic d-dimensional torus in physical dimensionsd ≤ 3. Let U0 : Ω → Rm×m satisfy ‖‖U0(x)‖F ‖L∞x ≤
√m. Recall SL(t) = et∆ and for
A ∈ Rm×m,
SN (t)A :=((e2t − 1)AAT + I
)− 12 etA. (3.72)
Define for n ≥ 0 the Strang-splitting iterates
Un+1 = SL (τ/2)SN (τ)SL (τ/2)Un. (3.73)
If τ > 0 satisfies meτ (e2τ − 1) ≤ 0.43, then
E(Un+1) ≤ E(Un), ∀n ≥ 0, (3.74)
where
Un = SL (τ/2)Un; (3.75)
E(U) =
∫Ω
1
2τ
⟨(e−τ∆ − 1)U,U
⟩F
+ 〈G(U), I〉F dx; (3.76)
G(U) =1
2τUUT − eτ
τ(e2τ − 1)
((I + (e2τ − 1)UUT
) 12 − I
). (3.77)
In the above 〈A,B〉F = Tr(ATB) =∑
i,j AijBij denotes the usual Frobenius inner product.
Proof. Observe that
e−τ∆Un+1 =(
(e2τ − 1)Un(Un)T
+ I)− 1
2eτ Un. (3.78)
ENERGY STABILITY OF STRANG SPLITTING METHOD 15
We rewrite the above as
1
τ(e−τ∆ − 1)Un+1 +
1
τ(Un+1 − Un) =
1
τ
(((e2τ − 1)Un(Un)
T+ 1)− 1
2eτ Un − Un
). (3.79)
Taking the Frobenius inner product with Un+1 − Un on both sides of (3.79), we obtain
1
τ〈(e−τ∆ − 1)Un+1, Un+1 − Un〉F +
1
τ‖Un+1 − Un‖2F
=1
τ
⟨((e2τ − 1)Un(Un)
T+ 1)− 1
2eτ Un − Un, Un+1 − Un
⟩F
. (3.80)
It is not difficult to check that∫Ω〈(e−τ∆ − 1)Un+1, Un+1 − Un〉Fdx (3.81)
=1
2
∫Ω〈(e−τ∆ − 1)Un+1, Un+1〉Fdx−
1
2
∫Ω〈(e−τ∆ − 1)Un, Un〉Fdx (3.82)
+1
2
∫Ω〈(e−τ∆ − 1)(Un+1 − Un), Un+1 − Un〉Fdx (3.83)
≥ 1
2
∫Ω〈(e−τ∆ − 1)Un+1, Un+1〉Fdx−
1
2
∫Ω〈(e−τ∆ − 1)Un, Un〉Fdx. (3.84)
By Lemma 3.3 and taking ε0 = 0.43 therein, we have∫Ω
1
τ
⟨((e2τ − 1)Un(Un)
T+ I)− 1
2eτ Un − Un, Un+1 − Un
⟩F
dx (3.85)
≤∫
ΩG(Un)dx−
∫ΩG(Un+1)dx+
1
2τ
(1 + (1− ε0)−
32 ε0
)∫Ω‖Un+1 − Un‖2Fdx (3.86)
≤∫
ΩG(Un)dx−
∫ΩG(Un+1)dx+
1
τ
∫Ω‖Un+1 − Un‖2Fdx. (3.87)
It follows that
E(Un+1) ≤ E(Un). (3.88)
4. Numerical experiments
4.1. Vector-valued AC equation. Consider the vector-valued AC equation∂tu = ∆u + u− |u|2u, (t, x) ∈ (0,∞)× Ω;
u∣∣∣t=0
= u0, x ∈ Ω.(4.1)
on the 1-periodic torus Ω =[−1
2 ,12
]2. We use the Strang splitting method given to this equation
with a fixed splitting time step τ = 10−4. For the spatial discretization, we use the pseudo-spectral method with 256× 256 Fourier modes. We draw v0 from a uniformly distribution in Ωand take the initial condition
u0 =
0.8 v0
|v0| , if |v0| 6= 0;
0, otherwise.(4.2)
In this way u0 has a fixed magnitude 0.8 with randomly distributed directions.
16 D. LI AND C.Y. QUAN
Figure 1 shows the computed vector field u at t = 0, 0.004, 0.008, 0.016, 0.032, and 0.05respectively. Define the standard energy and the modified energy:
E(u) =
∫Ω
(1
2|∇u|2 +
1
4(|u|2 − 1)2
)dx; (4.3)
E(u) =
∫Ω
(1
2τ
⟨(e−τ∆ − 1)u,u
⟩+G(u) +
1
4
)dx; (4.4)
=
∫Ω
(1
2τ
⟨(e−τ∆ − 1)u,u
⟩+
1
2τ|u|2 − eτ
τ(e2τ − 1)
((1 + (e2τ − 1)|u|2
) 12 − 1
)+
1
4
)dx.
(4.5)
It should be noted that a harmless constant 1/4 is added in the definition of E to ensure theconsistency with the standard energy.
It can be observed that the initial disordered state becomes ordered quickly. Figure 2 plots
the evolution of the standard and modified energies as well as their difference ∆E = |E − E|.Reassuringly both energy functionals decrease monotonically in time.
4.2. Matrix-valued AC equation. Consider the matrix-valued AC equation∂tU = ∆U + U − UUTU, (t,x) ∈ (0,∞)× Ω;
U∣∣∣t=0
= U0.(4.6)
The spatial domain Ω = [−π, π]2 is the 2π-periodic torus in dimension two. By a slight abuse ofnotation, we set the initial condition in polar coordinates as
U0(r, θ) =
[
cosα − sinαsinα cosα
]if r < 0.6π + 0.12π sin(6θ);[
cosα sinαsinα − cosα
]otherwise,
(4.7)
Here α(x, y) = π2 sin(x+y), and (r, θ) is the polar coordinate of x = (x, y). For spatial discretiza-
tion we use the pseudo-spectral method with 256× 256 Fourier modes. The splitting time step isfixed as τ = 0.01. In Figure 3, the domain is colored by the sign of the determinant of U , that is,
yellow if det(U(t, x, y)) > 0;
blue if det(U(t, x, y)) < 0.(4.8)
The vector field is generated by the first column vector of the matrix U(t, x, y). Note that fort = 0 this is just (
cosαsinα
). (4.9)
It can be observed that the initial star-shaped line defect shrinks in time. The evolution of the
standard and the modified energy as well as their difference ∆E = |E −E| are plotted in Figure4. Clearly these two energy functionals are in good agreement for small τ > 0.
Next, we consider the initial condition given by the following.
U0(r, θ) =
[
cosα − sinαsinα cosα
]if |x| > 0.5π| sin(1.25y)|+ 0.4π,[
cosα sinαsinα − cosα
]otherwise,
(4.10)
where α = y. The splitting time step is τ = 0.01 and we take 256 × 256 Fourier modes. Thedynamics of the line defect and the evolution of the energy are illustrated in Figure 5 and 6respectively. It can be observed that the modified energy dissipation indeed holds in this case.
ENERGY STABILITY OF STRANG SPLITTING METHOD 17
Figure 1. Vector field u at t = 0, 0.004, 0.008, 0.016, 0.032, and 0.05 respec-tively for the vector-valued AC equation.
Figure 2. Evolution of the original and modified energy as well as their difference
∆E = |E − E| for the vector-valued AC equation.
Acknowledgements
The research of C. Quan is supported by NSFC Grant 11901281, the Guangdong Basic andApplied Basic Research Foundation (2020A1515010336), and the Stable Support Plan Programof Shenzhen Natural Science Fund (Program Contract No. 20200925160747003).
18 D. LI AND C.Y. QUAN
Figure 3. Dynamics of line defect for the matrix-valued AC equation at t =0, 0.4, 0.8, 1.6, 2.4, and 3 respectively with initial condition (4.7) in Section 4.2.
Figure 4. Evolution of the original and modified energy as well as their difference
∆E = |E − E| for the matrix-valued AC equation with initial condition (4.7) inSection 4.2.
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ENERGY STABILITY OF STRANG SPLITTING METHOD 19
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D. Li, SUSTech International Center for Mathematics, and Department of Mathematics, South-ern University of Science and Technology, Shenzhen, P.R. China
Email address: [email protected]
C.Y. Quan, SUSTech International Center for Mathematics, and Department of Mathematics,Southern University of Science and Technology, Shenzhen, P.R. China
Email address: [email protected]