We consider the Allen-Cahn equation
where the unknown . The parameter is called the mobility coefficient and we fix it as a constant for simplicity. The nonlinear term takes the form , where is the standard double well. To minimize technicality, we take the spatial domain in (1.1) as the -periodic torus . With some additional work our analysis can be extended to physical dimensions . The system (1.1) arises as a -gradient flow of a Ginzburg-Landau type energy functional , where
The basic energy conservation law takes the form
It follows that
Besides the -type conservation law, there is also -type control. Due to special form of the nonlinearity, for smooth solutions we have the maximum principle
As a consequence the long-time wellposedness and regularity is not an issue for (1.1).
The objective of this work is to establish strong stability of a fourth order in time operating splitting algorithm applied to the Allen-Cahn equation (1.1). This is a part of our on-going program to develop a new theory for the rigorous analysis of stability and convergence of operator-splitting methods applied to dissipative-type problems. Due to various subtle technical obstructions, there were very few rigorous results on the analysis of the operator-splitting type algorithms for the Allen-Cahn equation, the Cahn-Hilliard equation and similar models. Prior to our recent series of works [17, 18], most existing results in the literature are conditional one way or another. To put things into perspective, we briefly review a few closely related representative works and more recent developments.
The work of Gidey-Reddy. Gidey and Reddy considered in  a convective Cahn-Hilliard model of the form
where . They adopted an operator-splitting of (1.6) into the hyperbolic part, nonlinear diffusion part and diffusion part respectively. Several conditional results concerning certain weak solutions were obtained.
The work of Weng-Zhai-Feng. In , Weng, Zhai and Feng studied a viscous Cahn-Hilliard model:
where the parameter . The authors employed a fast explicit Strang-type operator splitting and proved the stability and the convergence (see Theorem 1 on pp. 7 of ) under the assumption that , stay bounded, and satisfy a technical condition . Here denotes the numerical solution.
The work of Cheng-Kurganov-Qu-Tang. In , Cheng, Kurganov, Qu and Tang considered the Cahn-Hilliard equation
and the MBE equation
Concerning the Cahn-Hilliard equation, the authors considered a Strang-type splitting approximation of the form
and is the nonlinear propagator
Various conditional results were given in  but the rigorous analysis of energy stability was a long-standing open problem. This problem and several related open problems were settled in our recent proof .
In recent , we carried out the first energy-stability analysis of a first order operator-splitting approximation of the Cahn-Hilliard equation. More precisely denote as the exact PDE solution to the Cahn-Hilliard equation . We considered a splitting approximation of the form:
where and solves
We introduced a novel modified energy and rigorously proved monotonic decay of the new modified energy which is coercive in -sense. Moreover we obtained uniform control of higher Sobolev regularity and rigorously justified the first order convergence of the method on any finite time interval.
In , we settled the difficult open problem of energy stability of the Strang-type algorithm applied to the Cahn-Hilliard equation which was introduced in the work of Cheng, Kurganov, Qu and Tang . One should note that the new theoretical framework developed in  is completely different from the first order case . In the second-order case, for most generic data one no longer has strict energy-monotonicity at disposal and several new ideas such as dichotomy analysis, an absorbing-set approach were introduced in  in order to settle long time unconditional (i.e. independent of time-step or time interval) Sobolev bounds of the numerical solution.
In this work we develop further the program initiated in [17, 18] and turn to the analysis of higher order in time splitting methods. A folklore issue is that operator-splitting methods with order higher than two require negative time-stepping, i.e. backward time integration for each time step. In , Sheng considered dimensional splitting for the two-dimensional parabolic problem . Using semi-discretization one obtains the ODE system , where the matrices and correspond to the discretization of and respectively. One is then naturally led to the approximation of . In , Sheng showed that if the non-commuting matrices and have eigen-values in the left half of the complex-plane, and one employs the approximation
then the highest order of a stable approximation is two even if is chosen to be large. Put it differently, Sheng’s fundamental result states that for an -order () partitioned split-step schemes, at least one of the solution operators must be applied with negative time step, i.e. backwardly. In  Suzuki adopted an elegant time-ordering principle from quantum mechanics and proved a general nonexistence theorem of positive decomposition for high order splitting approximations. In , Goldman and Kaper strengthened these results further and showed that even with partitioned schemes, each solution operator within a convex partition must be performed with at least one negative/backward fractional time step.
For deterministic Hamiltonian-type systems, backward time stepping in general does not create instabilities and high-order operating splitting methods have shown promising effectiveness . On the other hand, due to the negative time stepping, there are some arguments that higher-than-three operator splitting methods cannot be applied to parabolic equations with diffusive terms ([33, 36, 37]). As we shall explain momentarily, these concerns are not unsubstantiated and even turn up in the well-defined-ness of these algorithms.
Although rigorous analysis of these issues were not available before, recently Cervi and Spiteri  considered three third-order operating-splitting methods and demonstrated via extensive numerical simulations the effectiveness of higher order splitting methods for representative cardiac electrophysiology simulations. These compelling numerical evidences propel us to re-examine in detail the stability property of general negative/backward time-stepping methods in parabolic problems. Indeed, the very purpose of this work is to break these aforementioned technical barriers and establish a new stability theory for these problems.
To set the stage and minimize technicality, we consider the Allen-Cahn equation (1.1) posed on the -periodic torus . To build some intuition, let and consider
and let solve the equation
An explicit formula for is readily available, indeed it is not difficult to check that
Define a second-order Strang-type propagator
Following Yoshida , we consider a 4th order integrator obtained by a symmetric repetition (product) of the 2nd order integrator:
Somewhat surprisingly, we first show that the propagator is ill-defined if one does not introduce a judiciously chosen spectral cut-off.
Proposition 1.1 (Ill-definedness of with a spectral cut-off).
The propagator is ill-defined in general.
The proof of Proposition 1.1 is given in Section 2. Armed with this important observation, we are led to introduce a spectral cut-off condition for the propagators. Let be an integer and we shall consider the projection operator defined for via the relation
where denotes the Fourier coefficient of . We introduce the following spectral cut-off condition.
Definition (Spectral condition). Let be the spectral truncation parameter. We shall say satisfy the spectral condition if for some .
This constraint in is reminiscent of the CFL condition in hyperbolic problems. Here in the parabolic setting we require that the operator to remain bounded when restricted to the spectral cut-off .
Let . We now consider the following modified propagators:
Theorem 1.1 (Stability of negative-time splitting methods).
Let , and consider the Allen-Cahn equation (1.1) on the one-dimensional -periodic torus . Assume the spectral cut-off condition for some . Assume the initial data ( is an integer). Let and define
There exists a constant depending only on , and , such that if , then
where depends on (, , , ).
One can also show the fourth-order convergence of the operator splitting approximation. Namely if and let be the exact PDE solution corresponding to initial data . Let bet the same as in Theorem 1.1. Then for any , we have
where depends on (, , , ). The regularity assumption on initial data can be lowered. We shall refrain from stating such a pedestrian result here and leave its justification to interested readers as exercises.
The rest of this paper is organized as follows. In Section we set up the notation and collect some preliminary lemmas. In Section we give the proof of Theorem 1.1.
2 Notation and preliminaries
For any two positive quantities and , we shall write or if for some constant whose precise value is unimportant. We shall write if both and hold. We write if the constant depends on some parameter . We shall write if and if .
We shall denote if for some sufficiently small constant . The smallness of the constant is usually clear from the context. The notation is similarly defined. Note that our use of and here is different
from the usual Vinogradov notation in number theory or asymptotic analysis.
For any , we denote , and . Also occasionally we use the Japanese bracket notation:
We denote by the usual -periodic torus. For and any function , we denote the Lebesgue -norm of as
If is a sequence of complex numbers and is the index set, we denote the discrete -norm as
For example, . If
is a vector-valued function, we denote, and . We use similar convention for the corresponding discrete norms for the vector-valued case.
We use the following convention for the Fourier transform pair:
and denote for ,
Proof of Proposition 1.1.
To prove Proposition 1.1, it suffices for us to examine the following statement. Consider where is the periodic Dirac comb on . Let and consider
Claim: for any .
Proof of Claim. Observe that the Fourier coefficients of are all positive and
Clearly the desired conclusion follows. ∎
3 Proof of Theorem 1.1
Lemma 3.1 (One-step stability).
Let , and assume the spectral condition for some constant . Suppose , and . Let . There exists sufficiently small such that if , then
where depends on (, , , , , ).
Later we shall apply Lemma 3.1 -times. In particular we need to bound the expression
Since the constant depends on the -norm of the iterates, it is of importance to give uniform control of the -norm. To resolve this issue, we first choose , and such that
We choose such that
Clearly in the course of iteration, the -norm of the iterates never exceeds . In particular
Proof of Lemma 3.1.
We begin by noting that for the ODE
we have the estimate (note thatis an algebra for )
It follows that for sufficiently small,
The desired estimates then easily follows from the above using the spectral condition. ∎
Lemma 3.2 (Multi-step stability and higher regularity).
Let , and assume the spectral condition for some constant . Suppose and for some constant . Define
where , . For , define
where . There exist and such that if (we may assume ), then
where depends on (, , ).
Proof of Lemma 3.2.
To establish (3.16), we note that for ,
provided . We then rewrite as
where . One can then bootstrap the higher regularity from this. We omit further details. ∎
It is not difficult to check that under the assumption of uniform high Sobolev regularity, we have (below we assume )
It follows that
Lemma 3.3 (Control of the energy flux).
Let and . Suppose satisfies and
where depends only on . Furthermore if and where is a sufficiently small constant depending on (, ), then
where depends only on (, ).
Lemma 3.4 (One-step strict energy dissipation with nontrivial energy flux).
Let , and . Suppose with and satisfies
where is a given constant. There exists sufficiently small such that if , then
Proof of Lemma 3.4.
By using Remark 3.2, it suffices for us to show
for some . Denote and observe that
It follows that for sufficiently small,
where is a constant. Now we clearly have
The desired result clearly follows. ∎
Let , and . Assume and . Define and
There exists sufficiently small such that if , then
where depends only on (, , ).
Proof of Theorem 3.1.
where the constants , are the same as in Lemma 3.3. In the argument below we shall assume is sufficiently small. The needed smallness (i.e. the existence of ) can be easily worked out by fulfilling the conditions needed in Lemma 3.2, Lemma 3.3 and Lemma 3.4.
By Lemma 3.2, we can find such that
Claim: We have
To prove the claim we argue by contradiction. Suppose is the first integer such that
By Lemma 3.3 we must have
Since , we have for some integer . By using smoothing estimates we obtain
which is clearly a contradiction to (3.36). Thus we have proved the claim. ∎
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