Maximum bound principles for a class of semilinear parabolic ...For the ETD1 scheme, we prove it by...

Introduction and motivation Abstract framework for semilinear parabolic equations Conclusion

Maximum bound principles for a class of semilinearparabolic equations and ETD schemes

Zhonghua Qiao

Department of Applied Mathematics,The Hong Kong Polytechnic University

Joint work with

Qiang Du (COLUMBIA), Lili Ju (USC), Xiao Li (PolyU)

Shanghai Jiao Tong University, Shanghai, June 17, 2020


Outline

1 Introduction and motivationMaximum bound principle preserving exponential timedifferencing schemes for the nonlocal Allen-Cahn equation

2 Abstract framework for semilinear parabolic equationsModel equation and its MBPExamplesMBP-preserving ETD schemesExtension

3 Conclusion


Allen-Cahn equation

(Local) Allen-Cahn equation:

ut − ε2∆u+ u3 − u = 0. (LAC)

As an L2 gradient flow w.r.t. the free energy functional

Elocal(u) =

∫ (1

4(u(x)2 − 1)2 +

ε2

2|∇u(x)|2

)dx, (1)

energy stability:

Elocal(u(t2)) ≤ Elocal(u(t1)), ∀ t2 ≥ t1 ≥ 0. (2)

As a second order reaction-diffusion equation,

maximum bound principle:

‖u(·, 0)‖L∞ ≤ 1 ⇒ ‖u(·, t)‖L∞ ≤ 1, ∀ t > 0. (3)


Allen-Cahn equation (continued)

Energy stable schemes:

Stabilized semi-implicit (SSI) scheme [Shen-Yang, 2010]:find un+1 such that

un+1 − un

τ−ε2∆hu

n+1+(un)3−un+κ(un+1 − un) = 0. (4)

Exponential time differencing (ETD) scheme [Ju et al., 2015]:find un+1 = w(τ) with w(t) subject to

dw

dt+ (κ− ε2∆h)w + (un)3 − un − κun = 0, t ∈ (0, τ ],

w(0) = un.(5)

Both schemes are easy to implement and conditionally energystable.



F (u) =1

4(u2 − 1)2, f(u) := F ′(u) = u3 − u.

What is the condition for energy stability?

κ ≥ 1

2‖f ′(u)‖L∞ . (6)

However,f ′(u) = 3u2 − 1, unbounded in L∞!

If we have that u is bounded in L∞, then so does f ′(u).

Discrete maximum bound principle insures the L∞ boundedness ofun+1.



Maximum bound principle preserving schemes:

first order semi-implicit scheme [Tang-Yang, 2016]:

un+1 − un

τ−ε2∆hu

n+1 +(un)3−un+κ(un+1−un) = 0 (7)

condition for MBP:1

τ+ κ ≥ 2.

Crank-Nicolson scheme [Hou-Tang-Yang, 2017]:

un+1 − un

τ−ε2∆h

un+1 + un

2+

(un+1)3 + (un)3

2−u

n+1 + un

2= 0

(8)

condition for MBP: τ ≤ 1

2min

1,h2

ε2

.


Cahn-Hilliard equation

(Local) Cahn-Hilliard equation:

ut + ε2∆2u+ ∆(u3 − u) = 0. (LCH)

No maximum bound principle!

Li-Q-Tang, SINUM, 2016Li-Q, JSC, 2017 (IMEX Frouier Spectral)Song-Shu, JSC, 2018 (IMEX LDG)

A clean description on the size of the constant κ, in the sense thatκ is independent of the L∞ bound on the numerical solution.


Nonlocal Allen-Cahn equation

Nonlocal Allen-Cahn (NAC) equation:

ut − ε2Lδu+ u3 − u = 0. (NAC)

As an L2 gradient flow w.r.t. the free energy functional

E(u) =

∫ (1

4(u(x)2 − 1)2 − ε2

2u(x)Lδu(x)

)dx, (9)

energy stability:

E(u(t2)) ≤ E(u(t1)), ∀ t2 ≥ t1 ≥ 0. (10)

Similar to the case of local Allen-Cahn equation, we can prove

maximum bound principle:

‖u(·, 0)‖L∞ ≤ 1 ⇒ ‖u(·, t)‖L∞ ≤ 1, ∀ t > 0. (11)


Nonlocal Allen-Cahn equation (continued)

Nonlocal diffusion operator (x ∈ Rd):

Lδu(x) =1

2

∫Bδ(0)

ρδ(|s|)(u(x+s) +u(x−s)− 2u(x)

)ds. (12)

Kernel ρδ : [0, δ]→ R is nonnegative and

1

2

∫Bδ(0)

|s|2ρδ(|s|) ds = d. (13)

Consistency of Lδ with L0 := ∆ via [Du et al., 2012]

maxx|Lδu(x)− L0u(x)| ≤ Cδ2‖u‖C4 . (14)

In particular, in 1-D case,

Lδu(x) =1

2

∫ δ

−δ|s|2ρδ(|s|)·

u(x+ s) + u(x− s)− 2u(x)

|s|2ds. (15)


Nonlocal Allen-Cahn equation (continued)

Du-Ju-Li-Q, SIAM J. Numer. Anal., 2019.

Consider the initial-boundary-value problem of the NAC equation

ut − ε2Lδu+ u3 − u = 0, x ∈ Ω, t ∈ (0, T ],

u(·, t) is Ω-periodic, t ∈ [0, T ],

u(x, 0) = u0(x), x ∈ Ω,

where Ω = (0, X)d is a hypercube domain in Rd.

Main theoretical results:

discrete maximum bound principle;

maximum-norm error estimates;

discrete energy stability.


Quadrature-based finite difference discretization

Uniform spatial mesh with the nodes xi.

The discretization of Lδ is defined by [Du-Tao-Tian-Yang, 2018]

Lδ,hu(xi) =1

2

∫Bδ(0)

Ih(u(xi + s) + u(xi − s)− 2u(xi)

|s|2|s|1)|s|2

|s|1ρδ(|s|) ds.

(16)

where Ih is the piecewise d-multi-linear interpolation.

The matrix Lδ,h is

symmetric and negative semi-definite;

weakly diagonally dominant with all negative diagonal entries.


Quadrature-based finite difference discretization(continued)

Introduce a stabilizing parameter κ > 0 and define

Lh := −ε2Lδ,h + κI, N(U) := κU + U − U .3. (17)

Then, we reachdU

dt+ LhU = N(U), (18)

whose solution satisfies

U(t+ τ) = e−LhτU(t) +

∫ τ

0e−Lh(τ−s)N(U(t+ s)) ds. (19)

The matrix Lh is

symmetric and positive definite;

strictly diagonally dominant with all positive diagonal entries,

which implies that ‖e−Lhτ‖∞ ≤ e−κτ for any κ, τ > 0.


ETD methods for the temporal integration

Uniform time step τ and the nodes tn = nτ.

At the time level t = tn, we have

U(tn+1) = e−LhτU(tn) +

∫ τ

0e−Lh(τ−s)N(U(tn + s)) ds. (20)

By

approximating N(U(tn + s)) by N(U(tn)) in s ∈ [0, τ ],

calculating the integral exactly,

we have the first order ETD scheme of (NAC):

Un+1 = e−LhτUn +

∫ τ

0e−Lh(τ−s)N(Un) ds

= e−LhτUn + L−1h (I − e−Lhτ )N(Un).

(ETD1)


ETD methods for the temporal integration (continued)

At the time level t = tn:

U(tn+1) = e−LhτU(tn) +

∫ τ

0e−Lh(τ−s)N(U(tn + s)) ds. (21)

By

approximating N(U(tn + s)) by a linear interpolation basedon N(U(tn)) and N(U(tn+1)),

we have the second order ETD Runge-Kutta scheme of (NAC):Un+1 = e−LhτUn +

∫ τ

0e−Lh(τ−s)

[(1− s

τ

)N(Un) +

s

τN(Un+1)

]ds,

Un+1 = e−LhτUn +

∫ τ

0e−Lh(τ−s)N(Un) ds.

(ETDRK2)


Discrete maximum principle

For the ETD1 scheme, we prove it by induction:

‖U0‖∞ ≤ ‖u0‖L∞ ≤ 1;

assume ‖Uk‖∞ ≤ 1, prove ‖Uk+1‖∞ ≤ 1.

We have

‖Uk+1‖∞ ≤ ‖e−Lhτ‖∞‖Uk‖∞+

∫ τ

0‖e−Lh(τ−s)‖∞ ds ·‖N(Uk)‖∞.

We can prove

‖e−Lhτ‖∞ ≤ e−κτ for any κ, τ > 0;

‖N(Uk)‖∞ ≤ κ when κ ≥ 2.

Then,

‖Uk+1‖∞ ≤ e−κτ · 1 +1− e−κτ

κ· κ = 1.


Discrete maximum principle (continued)

For the ETDRK2 scheme, we have

‖Uk+1‖∞ ≤ ‖e−Lhτ‖∞‖Uk‖∞

+

∫ τ

0‖e−Lh(τ−s)‖∞

∥∥∥(1− s

τ

)f(Uk) +

s

τf(Uk+1)

∥∥∥∞

ds.

Note that Uk+1 is exactly the solution to ETD1 scheme, so

‖Uk+1‖∞ ≤ 1 ⇒ ‖f(Uk+1)‖∞ ≤ κ.

For s ∈ [0, τ ],∥∥∥(1− s

τ

)f(Uk)+

s

τf(Uk+1)

∥∥∥∞≤(

1− sτ

)‖f(Uk)‖∞+

s

τ‖f(Uk+1)‖∞ ≤ κ.

Then,

‖Uk+1‖∞ ≤ e−κτ · 1 +1− e−κτ

κ· κ = 1.


Discrete energy stability

We define the discretized energy Eh:

Eh(U) =

dN∑i=1

F (Ui)−ε2

2UTLδ,hU, F (s) =

1

4(s2 − 1)2. (22)

Discrete energy stability of the ETD1 scheme

Under the condition κ ≥ 2, for any τ > 0, we have

Eh(Un+1) ≤ Eh(Un).


Energy stability for ETD1

Step 1. We have

F (Un+1)− F (Un) = f(Un)(Un+1 − Un) +1

2f ′(ξ)(Un+1 − Un)2,

where ‖f ′(ξ)‖∞ = ‖3ξ2 − 1‖∞ ≤ 2 since ‖ξ‖∞ ≤ 1 due to DMP.Then, we obtain

Eh(Un+1)− Eh(Un) ≤ (Un+1 − Un)T (LhUn+1 − f(Un)).

Step 2. Solve N(Un) from (ETD1) to get

N(Un) = (I − e−Lhτ )−1Lh(Un+1 − Un) + LhUn,

and then,LhU

n+1 −N(Un) = B1(Un+1 − Un)

with B1 = Lh − (I − e−Lhτ )−1Lh symmetric and negative definite.So,

Eh(Un+1)− Eh(Un) ≤ (Un+1 − Un)TB1(Un+1 − Un) ≤ 0.


Numerical experiments

We consider the 2-D case.

Setting

Ω = (0, 2π)× (0, 2π), ε = 0.1;

kernel: ρδ(r) =6

πδ3r, r > 0;

N = 512, τ = 0.01;

random initial data ranging from −0.9 to 0.9 uniformly;

δ = 0, δ = 3ε, δ = 4ε.


Numerical experiments (continued)

From left to right: δ = 0 (local), δ = 3ε, δ = 4ε.Top: maximum norms; bottom: energies.

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Recall the proof of the discrete maximum principle

The crucial results are

‖e−Lhτ‖∞ ≤ e−κτ for any κ, τ > 0,

(This is the result of the strictly diagonal dominance of Lh.)

and

‖N(U)‖∞ ≤ κ when κ ≥ 2, for any U such that ‖U‖∞ ≤ 1.

(This comes from the property of the function f(u) = u− u3.)


Outline



3 Conclusion


Domains

Consider the domain Ω ⊂ Rd in the following two situations.

(D1) space-continuous case:

Ω is an open, connected and bounded set;

∂Ω is the Lipschitz boundary of Ω;

Ωc is a closed connected set disjoint with Ω but ∂Ω ⊂ Ωc;

Ω = Ω ∪ ∂Ω and Ω = Ω ∪ Ωc.

We have Ωc = ∂Ω for classic differential operators and Ωc isusually a nonempty volume for nonlocal integral operators.

(D2) corresponds to a discrete version of (D1).


Banach spaces and operators

Ω∗ = Ω, Ω∗c = ∂Ω, if Ωc = ∂Ω;

Ω∗ = Ω, Ω∗c = Ωc \ ∂Ω, otherwise.

Let X = C(Ω) ∩ Cb(Ω∗c) be the Banach space equipped with

‖w‖ = supx∈Ω

|w(x)|, w ∈ X .

Let

f : Cb(Ω∗)→ Cb(Ω

∗) be a nonlinear operator;

L : D(L)→ Cb(Ω∗) be a linear operator with D(L) ⊂ X ;

L0 = L|D(L0) : D(L0)→ X, where D(L0) ⊂ D(L), X ⊂ X ,both related to boundary conditions.

(C1) Dirichlet boundary condition;(C2) periodic boundary condition.


Model equation

The model problem is a class of semilinear parabolic equations

ut = Lu+ f [u], t > 0, x ∈ Ω∗, (SPE)

where u : [0,∞)× Ω→ R is the unknown function subject to

initial value condition

u(0,x) = u0(x), x ∈ Ω;

for Case (C1): Dirichlet boundary condition

u(t,x) = g(t,x), t ≥ 0, x ∈ Ω∗c

with g ∈ C([0,∞);Cb(Ω∗c)) and g(0, ·) = u0 on Ω∗c ;

for Case (C2): periodic boundary condition with u0 ∈ X.


Linear operators L and L0

Main idea: L should be a generalization of ∆.

Assumption 1

(a) for any w ∈ D(L) and x0 ∈ Ω∗,

w(x0) = supx∈Ω

w(x) ⇒ Lw(x0) ≤ 0;

(b) the domain D(L0) is dense in X;

(c) there exists λ0 > 0 such that λ0I − L0 is surjective.

Lemma 1

Under Assumption 1, L0 generates a contraction semigroupSL0(t)t≥0, i.e.,

‖SL0(t)‖B(X) ≤ 1.


Nonlinear operator f

Main idea: f should be a generalization of f(u) = u− u3.

Assumption 2

There exists f0 ∈ C1(R) such that

f [w](x) = f0(w(x)), ∀w ∈ Cb(Ω∗), ∀x ∈ Ω∗,

and there exists β > 0 such that

f0(β) ≤ 0 ≤ f0(−β).

If f0(m) ≥ 0 ≥ f0(M) for some m < M , carry out an affinetransform.


Nonlinear operator f (ctd.)

Introduce a stabilizing constant κ ≥ 0, and then we obtain

ut + κu = Lu+N [u],

where N := κI + f .

Requirement on the stabilizing constant:

κ ≥ max|ξ|≤β

|f ′0(ξ)|. (K)

Write N0(ξ) = κξ + f0(ξ).

Lemma 2

Under Assumption 2 and the requirement (K), it holds that(i) |N0(ξ)| ≤ κβ for any ξ ∈ [−β, β];(ii) |N0(ξ1)−N0(ξ2)| ≤ 2κ|ξ1 − ξ2| for any ξ1, ξ2 ∈ [−β, β].


Maximum bound principle (MBP)

Theorem 1

Given any constant T > 0. Under Assumptions 1 and 2, if

|u0(x)| ≤ β, ∀x ∈ Ω, (IC)

then the equation (SPE) subject to either the periodic boundarycondition or the Dirichlet boundary condition with

|g(t,x)| ≤ β, ∀ t ∈ [0, T ], ∀x ∈ Ω∗c (BC)

has a unique solution u ∈ C([0, T ];X ) and it satisfies ‖u(t)‖ ≤ βfor any t ∈ [0, T ].


Maximum bound principle (MBP) (ctd.)

Sketch of the proof.Denote Xβ = w ∈ X : ‖w‖ ≤ β and

Cg([0, t];Xβ) = w ∈ C([0, t];Xβ) : w|[0,t]×Ω∗c= g.

For a fixed t1 > 0 and a given v ∈ Cg([0, t1];Xβ), let us definew : [0, t1]→ X being the solution of the linear problem

wt + κw = Lw +N [v], t ∈ (0, t1], x ∈ Ω∗,

w(t,x) = g(t,x), t ∈ [0, t1], x ∈ Ω∗c ,

w(0,x) = u0(x), x ∈ Ω.

Step 1. Prove w ∈ Cg([0, t1];Xβ) for given v ∈ Cg([0, t1];Xβ).

Step 2. Prove A : v 7→ w is a contraction if t1 is small sufficiently.

Step 3. Repeat the same argument on [t1, 2t1], [2t1, 3t1], . . . .


Examples of the nonlinear function f0

Example 1. Consider the function

f0(s) = λs(1− sp),

where λ > 0 and p ∈ N+.

f0 satisfies f0(m) ≥ 0 ≥ f0(M) with m ∈ [0, 1] and M ≥ 1;

for even p, one can choose β ≥ 1 to meet Assumption 2.

Special cases:

Case p = 2 with λ = 1 gives

f0(s) = s− s3,

the derivative of −F with F (s) = 14(s2 − 1)2.

Choosing β = 1, the requirement (K) becomes κ ≥ 2.


Examples of the nonlinear function f0 (ctd.)

Example 2. Consider the Flory-Huggins free energy

F (s) =θ

2[(1 + s) ln(1 + s) + (1− s) ln(1− s)]− θc

2s2,

where θ and θc are two constants satisfying 0 < θ < θc, and

f0(s) = −F ′(s) =θ

2ln

1− s1 + s

+ θcs.

Denote by ρ the positive root of f0(ρ) = 0, i.e.,

1

2ρln

1 + ρ

1− ρ=θcθ.

Then f0 satisfies Assumption 2 with β ∈ [ρ, 1).


Examples of the nonlinear function f0 (ctd.)

Example 3. The Helmholtz free-energy density:

F (s) = RTs(ln s−1)−RTs ln(1− bs) +as

2√

2bln

1 + (1−√

2)bs

1 + (1 +√

2)bs,

and f0(s) = −F ′(s), i.e.,

f0(s) = −RT lns

1− bs− RTbs

1− bs− a

2√

2bln

1 + (1−√

2)bs

1 + (1 +√

2)bs+

as

1 + 2bs− b2s2,

which has two zero points m and M satisfying 0 < m < M < 1/b.

The corresponding model, Peng-Robinson equation of state, iswidely used in the oil industries and petroleum engineering.


Examples of the linear operator L

1. Infinite dimensional examples

Example 4. Second-order elliptic differential operator

Lw(x) = A(x) : ∇2w(x) + q(x) · ∇w(x),

where q ∈ C(Ω;Rd) and A ∈ C(Ω;Rd×d) is symmetric andpositive definite uniformly. Here, Ωc = ∂Ω.

Second-order elliptic differential operator in the divergence form:

Lw(x) = ∇ · (A(x)∇w(x)) + q(x) · ∇w(x)

with A ∈ C(Ω;Rd×d) ∩C1(Ω;Rd×d) and q ∈ C(Ω;Rd). This formcould be written in a non-divergence form by setting q = ∇·A+ q.


Examples of the linear operator L (ctd.)

2. Finite dimensional examples

Example 7. Central difference operator for Laplacian

Lhw(xi) =1

h2

(w(xi−1)− 2w(xi) + w(xi+1)

).

Example 8. Quadrature-based difference operator

Lhw(xi) =∑

0<|sj |<δ

w(xi + sj) + w(xi − sj)− 2w(xi)

|sj |2|sj |1βδ(sj),

where βδ(sj) ≥ 0.

Example 9. Fractional difference operator (for Example 6).

Example 10. Mass-lumping finite element approximation for ∆.


Equivalent form of the model equation

Using the uniform time step τ and the nodes tn = nτ.We focus the equivalent equation on [tn, tn+1].

The function w(s,x) := u(tn + s,x) satisfiesws + κw = Lw +N [w], s ∈ (0, τ ], x ∈ Ω∗,

w(s,x) = g(tn + s,x), s ∈ [0, τ ], x ∈ Ω∗c ,

w(0,x) = u(tn,x), x ∈ Ω.

ETD schemes:

Approximating N [w(s)] by polynomial interpolations.


ETD1 scheme and discrete MBP

First-order ETD: N [u(tn + s)] ≈ N [u(tn)].

For n ≥ 0 and given vn, find wn : [0, τ ]→ X solvingwns + κwn = Lwn +N [vn], s ∈ (0, τ ], x ∈ Ω∗,

wn(s,x) = g(tn + s,x), s ∈ [0, τ ], x ∈ Ω∗c ,

wn(0,x) = vn(x), x ∈ Ω,

and vn+1 = wn(τ) gives the ETD1 solution.

Theorem 2 (Discrete MBP of the ETD1 scheme)

Suppose that Assumptions 1–2, (K), (IC) and (BC) hold. TheETD1 scheme preserves the discrete MBP unconditionally, i.e., forany time step size τ > 0, the ETD1 solution satisfies ‖vn‖ ≤ β.


Higher-order ETDRK schemes

Let Pr(s) be an interpolation of N [u(tn + s)] on sk := kr τ

rk=0:

Pr(s) =

r∑k=0

`r,k(s)N [vn+ kr ], s ∈ [0, τ ],

where vn+ kr is an approximated value of u(tn + sk).

The MBP would be preserved if

‖vn+ kr ‖ ≤ β, ∀ k ⇒ ‖Pr(s)‖ ≤ κβ.

The unique satisfactory interpolation corresponds to r = 1, i.e.,

P1(s) =(

1− s

τ

)N [vn] +

s

τN [vn+1], s ∈ [0, τ ].


ETDRK2 scheme and discrete MBP

Second-order ETD: N [u(tn + s)] ≈(

1− s

τ

)N [u(tn)] +

s

τN [u(tn+1)].

For n ≥ 0 and given vn, find wn : [0, τ ]→ X solvingwnt + κwn = Lwn +

(1− s

τ

)N [vn] +

s

τN [vn+1],

wn(s,x) = g(tn + s,x),

wn(0,x) = vn(x),

and vn+1 = wn(τ) gives the ETDRK2 solution, where vn+1 isgenerated by the ETD1 scheme.

Theorem 3 (Discrete MBP of the ETDRK2 scheme)

Suppose Assumptions 1–2, (K), (IC) and (BC) hold. The ETDRK2scheme preserves the discrete MBP unconditionally, i.e., for anytime step size τ > 0, the ETDRK2 solution satisfies ‖vn‖ ≤ β.


Energy stability of ETD schemes for phase field models

Phase field models are derived as the gradient flows w.r.t.

E[u] = −1

2(u,Lu)L2(Ω) +

∫ΩF (u(x)) dx,

with F : R→ R subject to f0 = −F ′. We have the energy law:

E[u(t2)] ≤ E[u(t1)], ∀ t2 ≥ t1 ≥ 0.

Proposition (Energy stability of ETD1 and ETDRK2 schemes)

(i) The ETD1 solution vnn≥0 satisfies

E[vn+1] ≤ E[vn], ∀ τ > 0;

(ii) The ETDRK2 solution vnn≥0 satisfies

E[vn] ≤ E[v0] + C(|Ω|, T, κ), τ ∈ (0, 1].


Real vector-valued equation

Ginzburg–Landau model (without electric effect)

φt = (∇+ iA)2φ+ (1− |φ|2)φ.

If we let ψ = eiA·xφ, then

ψt = ∆ψ + (1− |ψ|2)ψ.

In more general, we consider the vector-valued equation

ut = ∆u+ (1− |u|2)u, in (0, T ]× Ω,

where u : [0, T ]× Ω→ Rm is subject to the periodic, Dirichlet, orhomogeneous Neumann boundary conditions.


Real vector-valued equation (ctd.)

Introducing the constant κ ≥ 0 as before, we obtain

ut + κu = ∆u+N0(u),

where N0(ξ) := κξ + (1− |ξ|2)ξ.

Requirement: κ ≥ 2.

Corollary

The ETD1 and ETDRK2 schemes of the vector-valued equationboth preserve the discrete MBPs unconditionally.


Numerical experiments

Example 1. Scalar equation

ut = 0.01∆u+ f0(u)

with logarithmic potential

f0(u) = 0.4 ln1− u1 + u

+ 1.6u,

where

Ω = (0, 2π)× (0, 2π);

h = 2π/512; u0 is random in [−0.9, 0.9];

periodic and homogeneous Neumann boundary conditions;

FFT-based algorithms for computing matrix exponentials.


Numerical experiments (ctd.)

Example 1. Scalar equation with logarithmic potential

the root of f0(ρ) = 0 is ρ ≈ 0.9575.

0 2 4 6 8 10 12 14 16 18 20

Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sup

rem

um n

orm

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20

Time

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sup

rem

um n

orm

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 170

Time

-15

-10

-5

0

5

10

15

20

25

Ene

rgy

0 5 10 15 20-10

0

10

20

30

150 155 160 165 170-10.4

-10.2

-10

-9.8

0 50 100 150 200 250 300 350 400 450 500 550

Time

-15

-10

-5

0

5

10

15

20

25

Ene

rgy

0 5 10 15 20-10

0

10

20

30

530 535 540 545 550-10.35

-10.3

-10.25

-10.2



Example 2. Vector-valued Allen–Cahn equation

ut = 0.005∆u+ (1− |u|2)u.

nonhomogeneous Dirichlet BC;

piecewise linear finite elementwith mass-lumping;

triangular mesh with 2210nodes and 4158 elements;

Krylov subspace method forcomputing matrix exponentials.




0 10 20 30 40 50 60 70 80 90 100

Time

0.75

0.8

0.85

0.9

0.95

1

Max

imum

val

ue o

f the

mod

ule

0 0.5 1 1.5 2 2.5 30.75

0.8

0.85

0.9

0.95

1

0 10 20 30 40 50 60 70 80 90 100

Time

0

5

10

15

Ene

rgy

0 0.1 0.2 0.3 0.40

5

10

15

70 80 90 1000.094375

0.094377


Outline



3 Conclusion


Conclusion

Du-Ju-Li-Q, SIAM Review, Accepted, 2020.

ut = Lu+ f [u]

MBP

assumption on L

assumption on f

requirement on κ

MBP-preservingETD schemes

Higher-order linear (or explicit) MBP-preserving schemes?

Thanks for your attention!

Maximum bound principles for a class of semilinear parabolic ...For the ETD1 scheme, we prove it by...

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