AND MATRIX-VALUED ALLEN-CAHN EQUATIONS

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.


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)


Un+1 = SL (τ/2)SN (τ)SL (τ/2)Un. (1.35)


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.


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.


(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


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)


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)


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


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)


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.


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)


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)


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)


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)


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)


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)


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.


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.


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).


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.


∆E = |E − E| for the matrix-valued AC equation with initial condition (4.7) inSection 4.2.

References

[1] T. Batard and M.Bertalmio. On covariant derivatives and their applications to image regularization. SIAMJournal on Imaging Sciences 7, no. 4: 2393-2422, 2014.

[2] Y. Cheng, A. Kurganov, Z. Qu, and T. Tang. Fast and stable explicit operator splitting methods for phase-fieldmodels. Journal of Computational Physics, 303:45–65, 2015.


Figure 5. Dynamics of line defect for the matrix-valued AC equation at t =0, 0.1, 0.2, 0.4, 0.8, and 1 respectively with initial condition (4.10) in Section 4.2.


∆E = |E − E| for the matrix-valued AC equation with initial condition (4.10) inSection 4.2.

[3] Z. Weng and L. Tang. Analysis of the operator splitting scheme for the allen–cahn equation. Numerical HeatTransfer, Part B: Fundamentals, 70(5):472–483, 2016.

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[5] M. Thalhammer. Convergence analysis of high-order time-splitting pseudospectral methods for nonlinearSchrodinger equations. SIAM Journal on Numerical Analysis, 50(6):3231–3258, 2012.

[6] S. Descombes. Convergence of a splitting method of high order for reaction-diffusion systems. Mathematics ofComputation, 70(236):1481–1501, 2001.

[7] S. Zhao, J. Ovadia, X. Liu, Y. Zhang, and Q. Nie. Operator splitting implicit integration factor methods forstiff reaction–diffusion–advection systems. Journal of Computational Physics, 230(15):5996–6009, 2011.

[8] G. Strang. On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis,5(3):506–517, 1968.

[9] G. I Marchuk. Splitting and alternating direction methods. Handbook of Numerical Analysis, 1:197–462, 1990.[10] Y. Li, H. G. Lee, D. Jeong, and J. Kim. An unconditionally stable hybrid numerical method for solving the

Allen–Cahn equation. Computers & Mathematics with Applications, 60(6):1591–1606, 2010.[11] D. Li and C. Quan. The operator-splitting method for Cahn-Hilliard is stable. arXiv:2107.01418, 2021.[12] D. Li and C. Quan. On the energy stability of Strang-splitting for Cahn-Hilliard. arXiv:2107.05349, 2021.[13] D. Li and C. Quan. Negative time splitting is stable. arXiv:2107.07332, 2021.[14] D. Li, C. Quan, and T. Tang. Energy dissipation of Strang splitting method for Allen–Cahn equations.

arXiv:2108.05214, 2021.[15] D. Li. Effective Maximum Principles for Spectral Methods. Ann. Appl. Math., 37 (2021), pp. 131–290.[16] B. Osting and D. Wang. A diffusion generated method for orthogonal matrix-valued fields. Mathematics of

Computation, 89(322):515–550, 2020.[17] A. S Lewis and H. S Sendov. Nonsmooth analysis of singular values. Part I: Theory. Set-Valued Analysis,

13(3):213–241, 2005.[18] B. Li and Y. Wu. A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrodinger

equation. Numer. Math. (to appear), arXiv:2101.03728[19] M. Fei, F. Lin, W. Wang, and Z. Zhang. Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-

Sternberg problem. arXiv:2106.08293, 2021.

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]

http://arxiv.org/abs/2107.01418






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