Effective Maximum Principles for Spectral Methods

2021-09-15 07:05:18DongLi

Annals of Applied Mathematics 2021年2期

Dong Li

Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology,Shenzhen 518055, Guangdong, China

Abstract. Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data, the initial conditions and the operator coefficients.Sharp/strict maximum principles insomuch of fundamental importance for the continuous problem often do not persist under numerical discretization. A lot of past research concentrates on designing fine numerical schemes which preserves the sharp maximum principles especially for nonlinear problems. However these sharp principles not only sometimes introduce unwanted stringent conditions on the numerical schemes but also completely leaves many powerful frequencybased methods unattended and rarely analyzed directly in the sharp maximum norm topology. A prominent example is the spectral methods in the family of weighted residual methods.

Key words: Spectral method, Allen-Cahn, maximum principle, Burgers, Navier-Stokes.

1 Introduction

In solving physical problems such as Allen-Cahn flows in interfacial dynamics, the maximum principle plays an important role since it gives strong point-wise control of the physical solutions in terms of the boundary data, the initial conditions and the operator coefficients. For practical numerical simulations, it is often the case that sharp/strict maximum principles for the continuous problem often do not persist under numerical discretization. A lot of past research is centered on designing fine numerical schemes which preserves the maximum in a sharp way especially for nonlinear problems. For linear parabolic equations, it is well known that central finite difference in space with backward Euler time stepping can preserve the sharp maximum principle(cf. Chapter 9 of[5]for a textbook analysis of 1D homogeneous heat equation). This is also the case if one employs lumped mass linear finite element in space using acute simplicial triangulation. Although preserving the sharp maximum principle is highly desirable for numerical simulations, these often introduce unwanted stringent conditions on the numerical schemes. Moreover it completely leaves out many powerfulL2-based methods unattended and rarely analyzed directly in the sharp maximum norm topology. In this respect a prominent example is the spectral methods in the family of weighted residual methods. In this work we introduce and develop a new framework of almost sharp maximum principles which allow the numerical solutions to deviate from the sharp bound by a controllable discretization error: we call them effective maximum principles. Our main models are Allen-Cahn equations in physical dimensionsd≤3, but we also discuss related models such as Burgers equations, Navier-Stokes equations. All these will be discussed in this introduction.

We begin by considering the Allen-Cahn equation in physical dimensionsd=1,2,3:

whereuis a scalar function which typically represents the concentration of one of the two metallic components of the alloy. For simplicity we consider the periodic boundary condition and assume the function to have period 1 in each spatial coordinate axis. The parameterν>0 controls the interfacial width which is small compared with the system size under study. The nonlinear term has the usual double well

The system (1.1) can be regarded as a gradient flow ofε(·) in theL2metric. On the other hand if one changes the topology toH?1then we obtain the usual Cahn-Hilliard system. Due to the gradient flow structure

Similar energy laws also exists for other phase field models such as the Cahn-Hilliard system. However for the Allen-Cahn system due to its particular structure one has an additional maximum principle which asserts that theL∞norm of the smooth solution is bounded by 1 if the initial data is bounded by 1. Verification of these two fundamental conservation laws are of pivotal role in designing robust and stable numerical schemes for the Allen-Cahn system.

In yet other words ΠNis the projection to the space

A prototypical implicit-explicit Fourier spectral scheme has the form:

whereτ>0 is the size of the time step, andundenotes the numerical solution at the time stept=nτ. Note that the linear part is treated implicitly whereas the nonlinear part is explicit which makes the practical computation very convenient.Thanks to the frequency projection the Fourier modes ofunare all trapped in the window [?N,N]. In later sections we also consider the full collocation case and aliasing errors. On the other hand, the system (1.2) in some sense captures the essential difficulties of the numerical analysis for the Fourier Galerkin method.

As was already mentioned we are concerned with theL∞maximum principle for the approximation system (1.2). For this purpose it is convenient to recast it as

Similar reformulation can also be written down for other discretization methods such as the finite difference scheme. One immediate problem which makes the analysis of(1.3)nontrivial in theL∞setting is the lack ofL∞preservation due to the frequency truncation ΠN. Indeed, even forn=0 and generic initial datau0with‖u0‖∞≤1,one can have

which is caused by the lack of positivity of the Dirichlet kernel. For this and similar other technical obstructions (cf. page 219 of [8]), there has been no discussion ofL∞maximum principle for the Fourier spectral method in the literature prior to this work. The very purpose of this work is to settle this issue and develop a new framework for spectral methods.

Our first result is concerned with a detailed analysis of the operatorTN,τν2.Albeit classical the main novelty here is the quantification of the parameters and the sharpness of the involved constants. To allow some generality we denote forN ≥1,β>0,TN,β=(Id?β?xx)?1ΠN. Also denote

The first proof (Proposition 2.3) is essentially based on the Poisson summation whereas the second proof (Proposition 2.6) uses the Fejer kernel which exploits the convexity of the Fourier coefficients in certain regimes. The proof therein naturally generalizes to certain convex trigonometric series.

Our next result generalizes Theorem 1.1 to dimensionsd≥1. In numerical computations, we usually work with the projection operators

Theorem 1.2 is a restatement of Theorem 2.4 in Section 2. The corresponding analysis and proof can be found therein. From a practical point of view, the statement (2) in Theorem 1.2 is most useful and effective in the regime 0<β ?1 since we only need to takeNmoderately large in order to obtain an almost sharp maximum principle. The logd(N+2) factor in the upper bound reflects the fact that the corresponding kernel is a tensor product of one-dimensional Dirichlet kernels.

A natural generalization of the operator (Id?β?)?1ΠNis the truncated Bessel type operators (Id?β?)?sΠN, wheres>0. We denote the kernel function corresponding to (Id?β?)?sΠNasFN,sand note thatFN,s ∈C∞(Td) for each finiteN.Lets?∈[0,1] be the unique solution to the equation

In Lemma 2.12 of Section 2, we show thats?∈(0.308443,0.308444). The following group of results is proved in Section 2. Perhaps a bit surprisingly, for the sharp maximum principle to hold, the transition threshold occurs ats=s?<1.

Proposition 1.1.Let τ>0and consider the cubic polynomial fτ(x)=(1+τ)x?τx3for x∈R. Then the following hold:

Before we proceed further, we should point out one subtle technical difficulty with the above simplified system in the regime 0<τ ?1. Note thatθ0(α,τ)→1 asτ →0.Since the the damping factor containsN2τ, it follows that the thresholdNmust be takenτ-dependent in order to obtain contractive estimates which is hardly desirable in practice. In order to retain stability forτ →0 and build a stability analysis forNmoderately large andindependent of τ, a different line of argument is needed and indeed we develop a refined analysis in Section 3 (and later sections) to cover the regime 0<τ ?1.

whereηnaccounts for the spectral error and can be made sufficiently small. The next proposition quantifies the desired strong stability.

Proposition 1.2 (Strong stability of the prototype iterative system).Let τ>0and fτ(x)=(1+τ)x?τx3for x∈R. Consider the recurrent relation

where η>0.

Both Proposition 1.1 and Proposition 1.2 are proved in Section 3(see in particular Lemmas 3.2 and 3.3 therein). For the one-dimensional system (1.3), a complete theory ofL∞-stability and instability is worked out in Section 3 for all 0<τ<∞andN ≥2. We shall not reproduce all the details here and turn now to the general theory for dimensions 1≤d≤3 developed in Section 4. Consider

Concerning (1.5), the following group of stability results is proved in Section 4. For the first time we are able to establish effective maximum principles for the Fourier spectral methods applied on the nonlinear system.

Theorem 1.4 (Effective maximum principles for (1.5)).Consider(1.5)onTd=[0,1)d with1≤d≤3and ν>0. Then the following hold:

Remark 1.5. Forτ=2 andNnot large, there are counterexamples as shown in Proposition 3.5.

To understand the effect of pure spectral truncation onL∞-stability, we now consider the following model system whereU.3=((U0)3,···,(UN?1)3)T.

Concerning (1.7), the following results are proved in Section 5.

Theorem 1.6.Consider(1.7)with ν>0. Then the following hold:

where C1>0depends only on(uinit, ν).

Moreover we have the following result which shows the sharpness of our estimates above. There exists a function f:T→R, continuous at all ofT{x?}for some x?∈T(i.e., continuous at all x/=x?)and has the bound‖f‖∞≤1such that the following hold: for a sequence of even numbers Nm→∞, tm=ν?2N?2m and xm=jm/Nm with0≤jm ≤Nm?1, if?Um(t)∈RNm solves(1.7)with

where η?>0.001is an absolute constant.

Remark 1.6. For 0<ν ?1 the dependence ofNi(ν),i=1,2,3 is only power like which is a mild constraint in practice.

We now consider the fully discrete system.

where0

(a) E(Un+1)≤E(Un)for all n≥0.

(b) For all n≥2N?0.3/τ, we have

where c1>0is a constant depending only on α.

Moreover we have the following result which shows the sharpness of our estimates above. There exists a function f:T→R, continuous at all ofT{x?}for some x?∈T (i.e., continuous at all x/=x?)and has the bound ‖f‖∞≤1such that the following hold: for a sequence of even numbers Nm →∞,

where η?>0.001is an absolute constant.

Remark 1.7. In Statement(1)the cut-offτ=1.99 is for convenience only. One can replace it by any number less than 2 butN4will have to be adjusted correspondingly.

Concerning time discretization, all the numerical methods we discussed so far are only first order in time. With further work our effective maximum principles can be generalized to higher order in time methods. To showcase the theory, we consider Strang splitting with Fourier collocation for Allen-Cahn on T. The exact model equation is

It is not difficult to check that this particular Strang splitting method is second order in time. The following results are established in Section 6.

Theorem 1.8(Effective maximum principle for Strang splitting of Allen-Cahn with Fourier Collocation).Consider(1.10)with ν>0, τ>0. Then the following hold

where c1>0is a constant depending only on α.

where η?>0.001is an absolute constant.

Our analysis is certainly not restricted to the Allen-Cahn equation and be generalized and developed in many other directions. In Sections 7 and 8 we introduce effective maximum principles for the 1D Burgers equation and the 2D Navier-Stokes equation with periodic boundary conditions. Quite interestingly,these almost sharp maximum principles exhibit similar strong stability properties much as the Allen-Cahn case, although they are derived using slightly different mechanisms. In the following we give a short summary of the results obtained in Sections 7 and 8.

Consider the 1D Burgers equation:

Such pre-processing of initial data is quite easy to implement in practice.

wheref ∈L∞(T).

Theorem 1.10 (Effective maximum principle, Burgers with forward Euler and spectral truncation).Let ν>0and un be the solution to(1.12). We have for all0<τ<τ0=τ0(ν,‖f‖∞)and N ≥2,

where γ1>0depends only on ‖f‖2.

Remark 1.10. We stress that the condition on the initial functionfcan certainly be removed,see Theorem 7.4 for a much more general and technical result for rough initial data. Our results give a clear explanation and justification of almost sharpL∞bounds observed in practical numerical simulations. By using the machinery developed in this work, it is also possible to give a comprehensive analysis of the fully discrete scheme such as forward Euler in time with implicit treatment of the dissipation term, and Fourier collocation in space. All these will be addressed elsewhere.

Consider on the torus T2=[0,1)2the two dimensional Navier-Stokes system expressed in the vorticity form:

wheref ∈L∞(T2)and has mean zero. We shall work withωhaving zero mean which is clearly preserved in time. The velocityuis connected to the vorticityωthrough the Biot-Savart law:u=?⊥??1ω.

Theorem 1.12 (Effective maximum principle for 2D Navier-Stokes with spectral Galerkin truncation).Let ω be the solution to(1.14). We have for all N ≥2,

wheref ∈L∞(T2) and has mean zero.

Theorem 1.13 (Effective maximum principle for 2D Navier-Stokes, Forward Euler with spectral truncation).Let ωn be the solution to(1.15). We have for all0<τ<τ0=τ0(‖f‖∞)and N ≥2,

where α1,α2>0depend only on ‖f‖∞.

Notation and preliminaries

For anyk=(k1,···,kd)∈Rd, we denote

We shall denoteX ?YifX ≤cYfor some sufficiently small constantc. The smallness of the constantcis usually clear from the context. The notationX ?Yis similarly defined. Note that our use of?and?here isdifferentfrom the usual Vinogradov notation in number theory or asymptotic analysis.

We adopt the following convention for Fourier transforms. Denote forf:Rd→C,g:Rd →C,

For any functionfdefined on the periodic torus Td=Rd/Zdwhich we identify as[0,1)d, denote Now we recall the usual Poisson summation formula which will be used sometimes without explicit mentioning.

Lemma 1.1 (Poisson summation).Let δ be the usual Dirac distribution. Then

Remark 1.11. This is just saying that under suitable decay assumptions the natural periodization of the original function inherits its Fourier coefficients. An immediate useful estimate is: if

where equality holds whenfhas a definite sign.

2 Linear setting: 1D torus continuous case

Consider the periodic 1D torus T=R/Z which can be identified as[0,1). For periodic functionf:T→C and integerN ≥1, recall

Note that ΠNf=DN ?f, whereDNis the usual Dirichlet kernel given by

Proposition 2.1.Let N ≥1. Then

The bound is sharp in the following sense. For each N ≥1, there exists fN ∈C∞(T),such that

where c1>0is an absolute constant.

Proof.This is rather standard. For the lower bound define

A better bound is available. See below.

Denote (belowφandφ1are the same functions used in the definition ofA(x))

Clearly forNlarge,

Now observe that on T:

Thus the lower bound for‖A‖L1(T)is shown whenNis large.

Now ifNis of constant order, we can use the interpolation inequality

Thus, we complete the proof.

In Proposition 2.3, the lower bound onK>Ncan also be obtained from a more general result, see Proposition 2.4 below.

We first need a simple lemma.

Lemma 2.1.For any θ0∈[0,1), there exits α1(α1may depend on θ0)with0<10?6<α1<1?10?6, such that

This heuristic computation then leads to the following proposition.

Sinceφis a Schwartz function, the second piece above can be easily bounded byN?10which is negligible asNtends to infinity. In the computation below we shall completely ignore this piece. Then

In the next few propositions, we outline an alternative approach to obtain theL1-norm bound onK>N. The advantage is that it can be used on more general trigonometric series whose coefficients satisfy certain convexity properties.

whereν>0 is sufficiently small. Noting that

we obtain a similar lower bound forK>N0.

So, we complete the proof.

where β1is an absolute constant.

whereγ1is an absolute constant. Note that we lose a logarithm here compared to the optimal bound.

2.1 Generalization to convex sequences

The simple method used in the proof of Proposition 2.5 is quite robust and can be generalized. For example, consider a sequence of real numbers (ck)k≥0such that supk≥0|ck|<∞and the following hold:

1. for somek0≥0,

2. limk→∞(ck+1?ck)k=0.

3. limk→∞cklogk=0.

Remark 2.2. Condition (1) and (2) implies that

Remark 2.3. Condition (3) cannot be deduced from (1) and (2). For example, let

then condition (1) and (2) are satisfied, but not condition (3).

Remark 2.4. Condition (2) and (3) is convenient for extracting convergence rates.These conditions can be weakened further provided one works with other norms such as total variational norms and so on. However, we do not dwell on this issue here.

Now let

Theorem 2.1.Suppose the sequence(ck)k≥0satisfies the conditions(1),(2)and(3).Then GN converges in L1to a function G∞as N tends to infinity. Furthermore the following upper and lower estimates hold:

where α1>0is an absolute constant.

Remark 2.5. This theorem is quite handy in practical applications. For example,one can takeck=k?α,k ≥1, whereα>0 (note that we do not requireα>1!). It is easy to check that conditions (1), (2) and (3) are satisfied. Then corresponding to this sequence (ck)k≥0we have

for all largeN.

Proof.Note that for 4≤N

Since (by Remark 2.2)

it follows easily thatGNis Cauchy inL1and converges to anL1function asN →∞. The upper and lower bounds are similarly estimated as before. We omit the details.

2.2 PDE results

Letν>0 and consider the following linear elliptic PDE posed on the 1D torus T=[0,1):

where we recall ΠNdefined as

This equation is perhaps one of the simplest cases for the spectral Galerkin method.By using the results derived earlier, we have the following theorem.

Theorem 2.2.Let f ∈L∞(T). Then the unique solution u to(2.4)satisfies the following:

where α2>0is an absolute constant.

Proof.The first and fourth result follow from Proposition 2.8. The second and third results follow from Proposition 2.3.

2.3 The case d≥2

We now discuss some higher dimensional analogues of the previous results. In practical numerical computations, we usually use the projection

wheref=F?1(〈2πξ〉??) is the usual Bessel potential. Note that even thoughfhas exponential decay, the decay of the Fourier coefficient is not enough to apply the classic version of the Poisson summation formula. To resolve this the natural idea is to truncate the LHS and in some sense Lemma 2.4 assures the equivalence of different two truncations. The convergence of the Fourier series on the LHS can then be understood inL1x(Td) sense. Stronger point-wise asymptotics are also available but we will not dwell on this issue here.

Consider the operator

Note that it is spectrally localized to the sector{k:|k|∞>N}. Denote the corresponding kernel asK>N,β.

Thus the desired lower bound holds also whenNis of order 1.

Remark 2.7. An interesting problem is to investigate the maximum principle for the spherical operator

Proof.The first two follow from Theorem 2.3. We only need to show(3). Note that the cased=1 is already settled before with explicit dependence of constants. The cased=2 is proved in the appendix.

2.4 Truncation of general Bessel case

We first recall the usual Bessel potential on Rd: fors>0,

Denote forβ>0,N ≥2,

2.4.1 Complete analysis for 1D case

We first investigate the 1D case in complete detail. For convenience throughout this subsection we shall identify

then FN,s(x)>0.

whereηNis a linear interpolation such thatηN(k)=0 for|k|≤NandηN(k)=1 for|k|≥N+1. In particular

We only need to show

It then suffices to consider

The desired estimate then follows. One should observe that this line of computation is in some sense analogous to (2.6) whereas the latter can be viewed as discrete integration by parts.Remark 2.10. In the preceding remark, one may also takeηNto be a smooth function as

f1(0.308443)≈?1.92202×10?6<0, f1(0.308444)≈5.43492×10?7>0.Thus, we complete the proof.

The following theorem follows immediately from the preceding series of lemmas and computations.

Remark 2.13. It is also possible to characterize the lack of positivity ofFN,svia real-space representation. For example, for 0

where {x}=x?[x]denotes the usual fractional part (i.e.,[x]denotes the largest integer less than or equal to x so that {x}∈[0,1)). For example {?3.1}=0.9and{3.1}=0.1.

where a1and α2are absolute constants.

where Γ(·) is the usual Gamma function andc2>0 is another absolute constant.The desired result then follows easily.

Remark 2.16. Letβ>0,N ≥2. Letd≥2, 0

These will be investigated elsewhere.

We summarize the results obtained in this section as the following theorem.

Theorem 2.7 (Sharp maximum principle).Let d≥1and β>0. Then the following hold

1. If s>d and

(cs,d>0depends only on(s, d)), then FN,s is a positive function and

(α>0is an absolute constant), then FN,1is positive and has unit L1mass.

where cs,d>0depends only on(d, s).

Proof of Theorem2.8.The proof is a simple adaptation of the proof of Theorem 2.3. We omit the details.

3 1D torus case for Allen-Cahn:semi-discretization

Consider the scheme for Allen-Cahn on the torus T=[0,1):

Hereτ>0 is the size of the time step.

The following lemma will be used later which we will often refer to as a standard discrete energy estimate.

Proof.Denote by (,) the usualL2pairing of functions. Then

where we have denotedf(z)=F′(z)=z3?z.

Now note that

whereξlies betweenunandun+1. The desired inequality then easily follows sincef′(z)=3z2?1.

Remark 3.3. The constraintα0>0 already comes from the fact thatu0=ΠNu0in general does not preserve the strict upper bound as the Dirichlet kernel changes its sign. On the other hand, even if we assume‖u0‖∞≤1 (noteu0=ΠNu0!), this maximum is still not preserved(for smallN)as one can see from the counter-example in Proposition 3.2.

The inductive assumption is

In the following we shall showun+1≤1+α0. By repeating the argument for?un+1we also get the lower bound.

Denoteun=1+ηn. Then clearly by the inductive assumption

ifN ≥N0(here note that in the last inequality we need the lower bound onτso thatN0remains independent ofτ). Note that in the second inequality above we have used theL1bound for the operator

which was established in the previous section.

Proposition 3.2(Lack of strict maximum principle).There exist ν>0,τ>0,N≥1,u0with ‖u0‖∞≤1such that

Remark 3.4. More elaborate (covering more regimes of the parameters) construction of counter-examples is possible but we shall not dwell on this issue here.

In yet other words the true maximum 1 is asymptotically preserved up to the spectral truncation error! One should also note that the decay is exponential inn.

Proof.We adopt the same notation as in Proposition 3.1 and the proof is a minor variation. First by using the proof of Proposition 3.1 withα0=1, one has the weak bound

We should emphasize here thatN0can be taken to be independent ofu0since we assumed‖u0‖∞≤2.

Define

Loosely speaking, with the help of Lemma 3.2, one can reduce the problem of estimating‖un+1‖∞in terms of‖un‖∞(whenτ ≥τ?is not close to zero) to the recurrent relation

whereη>0 denotes the spectral error. The following Lemma plays a key role in the proof of maximum principle later. From a practical point of view, we do not state the results in its most general form. For example, belowα0corresponds to the upper bound of‖u0‖∞in the original numerical scheme, and it is convenient to assumeα0to be around 1 in practice.

Lemma 3.3 (Prototype iterative system for the maximum principle).Let τ>0and p(x)=(1+τ)x?τx3. Consider the recurrent relation

where θ=1?2τ.

Then there exists a constant η0>0depending only on ?0, such that if0≤η≤η0,then for all n≥1, we have

Remark 3.7. Forτ=2, even ifα0=1, we haveαn →∞asn→∞. Also note that in Case (2) we do not discuss the decay estimate at all since in practice we are only interested in the regime where the maximum is around 1.

Proof.(1) Observe that forη>0 sufficiently small the values ofαnare trapped in[1,2]by a simple induction argument. By another induction we obtainαn≥1+ηfor alln≥1.

Remark 3.12. The bound

can be replaced by any number

The result then follows from Lemma 3.3 and induction.

3.1 The case 0<τ<τ?

We now consider the analogue of Proposition 3.1 for the case 0<τ<τ?.

Proof.It is necessary to modify the proof of Proposition 3.1 and we shall outline the main changes.

The main inductive assumption is:

We tacitly assume thatu?1=u0and start the induction forn≥0. Note that‖u0‖H1≤‖u0‖H1andE(u0)E(u0) sinceu0=ΠNu0.

We now justify the inductive assumption forun+1. This is divided into the following steps.

Step 1: Energy stability. By a standard discrete energy estimate, we have

Recallp(x)=(1+τ)x?τx3and By inductive assumption we have‖un‖∞≤2. Then by takingτ?>0 sufficiently small(depending only onνand‖u0‖H1), we clearly haveE(un+1)≤E(un).

Step 2:L∞-estimate. This step is similar to that in Proposition 3.1. Denoteun=1+ηn. Then from the inductive assumption

Thus, we complete the proof.

Now we discuss the decay estimate for the case 0<τ<τ?. Note that in the followingN0is independent ofα0.

Corollary 3.2.Assume u0∈H1(T)and ‖u0‖L∞(T)≤1+α0for some0<α0≤1.For some τ?=τ?(ν,‖u0‖H1)>0the following holds for any0<τ<τ?: If N ≥N0=N0(ν,‖u0‖H1), then for any n≥1,

Proof.First by takingα0=1 in Proposition 3.9 we can achieve forN ≥N0=N0(ν,‖u0‖H1) that

Now denoteαn=maxηn=max(un?1). Then by repeating the proof in Proposition 3.9, we obtain

where the constantCν,‖u0‖H1>0 depends only onνand‖u0‖H1. Similar estimates also hold for ?αn=max(?1?un). Thus forθ=1?2τ

Thus, we complete the proof.

3.2 New energy stability results for <τ<τ1≈0.86

Consider the equation

Denote

where η(?0)is the same as in(3.3). Then for N ≥N0(ν,?0), we have

which is exactly (3.3).

4 General theory for dimension d≥1

Consider the scheme for Allen-Cahn on the torus T=[0,1)din the physical dimensionsd≤3:

whereν>0.

Remark 4.2. Forτ=2 andNnot large, there are counterexamples as shown in Proposition 3.5.

Proof of Theorem4.3.For the first statement, we repeat the proof of Proposition 3.8. Note that for general dimensiond≤3,we use Theorem 2.4 to control the operator(I?ν2τ?)?1(Id?ΠN). The second statement follows from a similar argument as in Proposition 3.7. Here we also use the sharp maximum principle established in Theorem 2.4 for dimensiond=1,2.

whereν>0.

Remark 4.3. The assumptionu0∈Hsis needed so that we can have a uniform choice of the spectral cut-offN0. One can also use Theorem 4.2 and takeτ →0 to derive the result ford≤3.

Proof of Theorem4.4.First observe that for someT0=T0(s,d,ν,‖u0‖Hs)>0, we can construct a unique local solution which satisfies ifN ≥B2.

Thus (4.3) holds. Furthermore by a bootstrapping estimate, we have

whereB3>0,B4>0 depend only on (s,d,ν,u0). Also observe that

Our desired result then follows by running again the maximum principle argument(similar to the proof of (4.3)) and choosingNsufficiently large.

We now drop the assumptionu0∈Hsand derive another form of the maximum principle.

Theorem 4.5 (Maximum principle for the continuous in time system with spectral truncation).Consider(4.2)with ν>0, d≥1. Assume ‖u0‖∞≤1. Then for N ≥N1=N1(d,ν,u0), we have

where C2>0depends only on(d, ν, u0).

Proof of Theorem4.5.The proof is similar to that in Theorem 4.4 and we shall only sketch the needed modifications. Since ΠNis the product of one-dimensional Fourier multipliers, it is not difficult to verify that

whereB1>0,B2>0 are constants depending on (d,ν,u0,q). Clearly fort1

whereB3>0 depends on (d,ν,u0). Note that after this step is finished we can fix the value ofq. By using the smoothing estimates and similar maximum principle estimates as used in Theorem 4.4 we also have

whereB4>0 depends on (d,ν,u0).

Now note that for 0≤t≤t1, we have

5 1D Allen-Cahn: fully discrete case using DFT

is real-valued. Note that

On the Fourier side,

In yet other words, for smoothg:T→R we have

Remark 5.1. Note that the operatorQNis also defined for boundedg: T→R by using only the expression

Thus the mere condition‖U‖∞≤1 cannot guarantee a good bound onQNU. In a similar vein, one can also consider the function

Forτ>0 sufficiently small (τcan depend onN), we have

One can view(5.4)as the time discretization of the continuous-in-time ODE system:

For smallτ>0 one can study the nearness of the solutions to(5.5)and(5.6). However we shall not explore it here. On the other hand,we can consider the following system

Suppose f ∈RN, then

whereα2>0 is a small absolute constant.

Our next set of results will be concerned with generalized maximum principles.We first consider the ODE system (5.6).

where C1>0depends only on(uinit, ν). In the above u is the solution to(5.7)with QNuinitas initial data.

Remark 5.3. Since we are concerned with the generalized maximum principe, the dependence ofN1onνwill only be power-type which is much better than the dependence in Theorem 5.1. These can be easily inferred from the computations in the proof below. However we shall not state the explicit dependence here for the ease of notation.

It follows easily that the system (5.6) admits a global solution.

Step 2.H1Estimate ofu. We first consider the initial data. Observe that

where the implied constant depends on (uinit,ν).

Now we consider the nonlinear term. By using Step 2, we have‖u(t)3‖H11 uniformly int≥0. Clearly we have

where C2>0depends only on(uinit, ν). In the above u is the solution to(5.7)with QNuinit=uinitas initial data.

Remark 5.4. The dependence ofN2onνis only power-like.

where c1>0is a constant depending only on α.

where η?>0.001is an absolute constant.

Remark 5.6. One should compare our last result with the Fourier spectral Galerkin truncation case (see for example Theorem 4.5) where we dealt with initial data bounded by one and proved a sharp maximum principle. There thanks to spectral localization the initial data is smooth. Here in the collocation case, our counterexample shows that there is non-approximation if we work with mereL∞initial data which is bounded by one and everywhere continuous except at one point. This is certainly connected with the non-smooth cut-offin the Fourier space.

Remark 5.7. The dependence ofN3onνis only power-like.

In the argument below we shall takeN ≥N0whereN0is a sufficiently large absolute constant. The needed largeness ofN0can be easily worked out from the argument.

Step 1. Local in time estimate. Introduce the norm

By carefully tracking the constants and similar estimates as done in the above, it is not difficult to check that forT=T0>0(T0is a sufficiently small absolute constant)we have contraction in the ball{‖u‖XT ≤C1}whereC1is an absolute constant.

Step 2. Refined estimate. For any 0

The remaining results then follow from Theorem B.1, Theorem B.2 and Theorem B.3 proved in the appendix.

where ε1>0depends only on ν.

Remark 5.8. The cut-offτ=1.99 is for convenience only. One can replace it by any number less than 2 butN4will have to be adjusted correspondingly.

Proof.Denote

Then (5.10) follows from Lemma 3.3 and induction.

The proof of (5.11a)–(5.11b) is similar to that in the proof of Proposition 3.10.The main point is to use the inequality

It is not difficult to check that forn≥1,

wheref(z)=z?z3andf(U)=U ?U.3.

Observe that for anyj ≥1 andτ>0, we have

whereC1>0 is an absolute constant.

Now first forn=1, we have

Our next result is concerned with the regime 0<τ ≤N?0.3. Letun=un(x):T→R solve the system

Theorem 5.7 (Almost sharp maximum principle for (5.4), case 0<τ ≤N?0.3).Consider(5.4). Suppose U0∈RN satisfies ‖U0‖∞≤1. Assume0<τ ≤N?0.3. If N ≥N6(ν)>0(the dependence of N6on ν is only power-like), then it holds that

where0

1. E(Un+1)≤E(Un)for all n≥0.

2. For some T0=T0(ν)>0sufficiently small and for all n≥T0/τ, we have

where un solves(5.17).

3. For all n≥2N?0.3/τ, we have

4. For1≤n≤2N?0.3/τ, we have

where c>0is an absolute constant, and ωf is defined in(B.5).

In the argument below we shall assumeN ≥N6=N6(ν) whereN6is a sufficiently large constant depending only onν. The needed largeness ofN6as well as the precise (power) dependence onνcan be worked out from the context with more detailed computations. However for simplicity of notation we shall not present it here.

Step 1. Local estimate. We show that forT0=T0(ν)>0 sufficiently small and alln≥1 withnτ ≤T0, it holds that

whereC0>0 is a constant depending only onν.

Forn=1, we shall start with

Clearly

where in the above we usebito denote various constants depending only onν,and we takeT0sufficiently small andNsufficiently large. Similarly, by using the inequality

we have

Hence (5.19) is proved.

Step 2. Refined estimate. First we note that

Step 3. The regimenτ ≥2N?0.3. Setn0=[2N?0.3/τ]≥2 where [x] denotes the usual integer part of the real numberx>0. Clearlyn0τ ～N?0.3and

Thus forn0≤n≤T0/τ,

This implies for alln0≤n≤T0/τ,

Now consider the system

If we assume‖un‖∞≤1+δ, then (note again that forp(x)=x+τ(x?x3),p′′(x)<0 forxnear±1 so thatO(δ2) does not appear below)

provided we takeδ≥O(N?0.25).

Thus it is clear that for alln0≤n≤T0/τ,

Forn≥T0/τ, note that‖?xun‖21. By another maximum principle argument, we obtain

Note that we did not optimize the exponent(inN)here for simplicity of presentation.These can certainly be improved by more refined estimates but we do not dwell on this issue here.

Step 4. The regimenτ ≤2N?0.3. In this regime we observe that

The remaining results then follow from Theorem C.4 proved in the appendix and a simple perturbation argument.

6 Strang splitting with Fourier collocation for Allen-Cahn

In this section we consider the Strang splitting with Fourier collocation for Allen-Cahn on T. The PDE is again

We shall adopt the same notation as in Section 5. In particular we slightly abuse the notation and denote forU ∈RN

We consider the time splitting as follows. Letτ>0 be the time step. Consider the ODE

We define the solution operatorNτ:RN →RNas the mapa→U(τ). Thanks to the explicit form off(U), we have (a=(a0,···,aN?1)Tand with no loss we shall assume thatN ≥2)

Lemma 6.1.Consider(6.1). We have

Furthermore assuming ‖U0‖2≤1, there are τ0=τ0(ν)>0, t0=t0(ν)>0sufficiently small such that the following hold: If τ ≥τ0, then

where C1>0depends only on ν. If0<τ<τ0, then

where C2>0depends only on (ν, t0).

Suppose ‖U0‖∞≤1, then for all τ>0, we have

where C3>0depends only on ν. Also for any t1≥τ and nτ ≤t1, we have

where C4>0depends only on ν.

Proof.For the basicL2bound we can just use the ODE. Clearly

Thus‖U(t)‖2≤max{1,‖U(0)‖2}. This immediately yields (6.2).

For (6.3), we note that sinceτ1, it follows easily that

For (6.4), we first rewrite

By (6.7), we have

Note that in the above, the empty summation whenn=1 is defined to be zero. For 1≤j ≤n?1, note that

A simple induction argument then yields the desired estimate. Finally the estimate(6.6) follows easily from (6.5).

Theorem 6.1 (Almost sharp maximum principle for (6.1), caseN?0.4≤τ<∞).Consider(6.1). Suppose U0∈RN satisfies ‖U0‖∞≤1. Assume N?0.4≤τ<∞. If N ≥N1(ν)>0(the dependence of N1on ν is only power-like), then for all n≥0, we have

whereC1>0 is an absolute constant.

Our main induction hypothesis is‖Un‖∞≤1+δ, whereδ=O(N?0.5). Then it suffices to verify the inequality:

This amounts to showing

Sinceδ=O(N?0.5) andτ ≥N?0.4, this last inequality is obvious forNsufficiently large.

Lemma 6.2.Assume N ≥2. Suppose u is a smooth solution to the equation:

where α>0is an absolute constant.

Clearly for|l|≤2 andk,k1,k2∈J, we have

Define

It is then not difficult to show that

whereC>0 is an absolute constant. The desired result then follows easily by takingtsufficiently small.

where c>0is an absolute constant, and ωf is defined in(B.5).

where c1>0is a constant depending only on α.

whereVτis the time-τsolution operator to the problem:

Since‖U0‖∞≤1 andn0τ ～N?0.3, it easily follows that

Thus

Then forn≥n0, we consider the problem

By Lemma 6.2 (note thatτ

By using Lemma 6.1, we have

If we assume that‖u(0)‖∞≤1+δ0for someδ0=O(N?0.29), then a simple maximum principle argument yields that

Consequently

Finally we consider the regime 1≤n≤n0. Clearly we have

We are then in a situation similar to that in Theorem 5.4. The argument is then similar and thus omitted.

Remark 6.1. We should point out that, by using the argument in the preceding proof, it is also possible to obtain the result

for the caseN?0.4≤τ ≤τ1, whereτ1?1 is a constant determined in Lemma 6.2(τ1=t0in the notation therein). We briefly sketch the argument as follows. Again considerun+1,un:T→R defined via the relation:

whereVτis the time-τsolution operator to the problem:

Then we consider the problem

By using Lemma 6.1 and Lemma 6.2, we have

In the above last inequality, the first term is due to the caseτ ～1, and the second term accounts for the caseτ ?1. Since we assumedτ ≥N?0.4, we then obtain

A maximum principle argument then yields that‖un‖∞≤1+O(N?c).

Remark 6.2. Denoteθ=e?2τand foru:T→R,

Clearly we haveQNNτU=QNTθQNUfor anyU ∈RN. It is not difficult to check that for anyf ∈H1(T), we have

By takingNsufficiently large, we obtain

Thus in the regimet0?1 one does not have uniform-in-timeH1-norm bounds.

7 Maximum principle for spectral Burgers

Consider

wheref ∈L∞(T).

Theorem 7.1.Let ν>0and u be the solution to(7.1). We have for all N ≥2,

where c>0is an absolute constant, and α>0depends only on(‖f‖∞, ν).

Such pre-processing of initial data is quite easy to implement in practice.

Proof.Denoteas the average ofuon T which is clearly preserved in time. Letv=u?Then

Thus

By using this and standard smoothing estimates, we then obtain

where the constantC1>0 depend only on (‖f‖∞,ν),c1>0 is an absolute constant,and 0

wherec>0 is an absolute constant,C2>0 depends only on(‖f‖∞,ν), and 0

whereFobeys

Since by spectral truncationu0is clearly smooth, the functionuis also smooth.Consider

where

Clearly for anyt0>0 andt≥t0, we havegis smooth and

A simple maximum principle argument then yields

wheref ∈L∞(T).

Theorem 7.2.Let ν>0and un be the solution to(7.2). We have for all0<τ<τ0=τ0(ν,‖f‖∞)and N ≥2,

where α1,α2>0depend only on(‖f‖∞, ν).

Remark 7.3. For convenience we do not spell out the explicit dependence ofτ0onν.

Proof.To ease the notation we shall setν=1. Rewrite (7.2) as

We shall proceed in several steps.

Step 1. Local smoothing estimates. Lett0>0 be chosen sufficiently small (depending only on‖f‖∞) so that the nonlinear part is dominated by the linear part whennτ ≤t0. Choose 1≤n0∈Z such thatn0τ ～t0. By using (7.3) and discrete smoothing estimates, we have

whereC1>0 depends only on‖f‖∞.

Step 2.L2estimate. Multiplying both sides of (7.4)byvn+1and integrating,we obtain

then

By a similar estimate and under the same condition (7.6), we also obtain

Step 3. Induction. Forn≥n0, we inductively assume

whereA1is a suitably large constant. Throughout the argument we shall assumeτ>0 is sufficiently small, in particular it has to satisfy

and some additional mild constraints in the argument below. Now clearly by Step 2 we have‖un+1‖2≤C1. By using(7.3),we have(belowβidenotes absolute constants)

where in the last step we chooseA1≥2β1C1and takeτsufficiently small. By a similar estimate, it is not difficult to check that

This then completes the induction proof and we obtain

It follows that

whereβ>0 is an absolute constant, andC2>0 depends only on‖f‖∞.

Step 4. Time-global estimates. By using the estimates derived in Step 1 to Step 3 and further bootstrapping estimates, we then obtain

where 00 are absolute constants, andC3>0 depends only on‖f‖∞. By using (7.2), we have

where 00 depends only on‖f‖∞. Also

where 00 is an absolute constant, andC6>0 depends only on‖f‖∞. Now forn≥1 we rewrite the equation forun+1as

Clearly by a maximum principle argument, we have

Iterating inn, we obtain

whereC7,C8>0 are constants depending only on‖f‖∞. Finally we observe that

where 00 is an absolute constant. Our desired result then follows easily.

Remark 7.4. At this point we should point out a subtle technical difficulty associated with the analysis of (7.7) and the equivalent system (7.8). Namely in general we have

or in terms ofu:

An example can be constructed as follows. LetN=6k0≥6 wherek0is an integer.Setm=N/3 andu(x)=sin2πmx. It is not difficult to check that

Clearly then

where γ1>0depends only on ‖f‖2, and u solves(7.8).

where0

1. For any t>0, we have

where c>0is an absolute constant, and ωf is defined in(B.5).

where c1>0is a constant depending only on α.

where η?>0.001is an absolute constant.

Proof.We proceed in several steps.

Step 1. We note that statement (1) follows from the same proof as in Theorem 7.3.

Step 2. Local in time estimate. To establish the remaining results, we first perform a local in time estimate. First we show that for someT0=T0(‖U0‖2)sufficiently small, we have contraction using the norm

Indeed by using the estimate

we have

Thus forT0=T0(‖U0‖2) sufficiently small, we have

Now for any 0

For 0

Step 4. The regimet

whereUn?hUntakes point-wise product ofUnand?hUn. We shall address these and similar other schemes elsewhere.

8 Two dimensional Navier-Stokes

Consider on the torus T2=[0,1)2the two dimensional Navier-Stokes system expressed in the vorticity form:

wheref ∈L∞(T2) and has mean zero. Here and below we assumeωhas mean zero which is clearly preserved in time. The velocityuis connected to the vorticityωthrough the Biot-Savart law:u=?⊥??1ω.

Lemma 8.1.Let the dimension d ≥1. Suppose g:Td →Ris in Cγ(Td)for some0<γ<1. Then for all N ≥2, it holds that

where C1>0depends only on(d, γ).

Proof.By using smooth Fourier cut-offs, we can rewrite

wherePN,PNare smooth Fourier projectors localized to the regimes{|k|～N}and{|k|N}respectively. The result then follows from Theorem 2.4 together with the fact that

Thus, we complete the proof.

Theorem 8.1.Let ω be the solution to(8.1). We have for all N ≥2,

where α>0depends only on ‖f‖∞.

Proof.First we compute theL2-norm. Clearly

Using this and further smoothing estimates, we obtain

wheref ∈L∞(T2) and has mean zero.

Theorem 8.2.Let ωn be the solution to(8.2). We have for all0<τ<τ0=τ0(‖f‖∞)and N ≥2,

where α1,α2>0depend only on ‖f‖∞.

Proof.Rewrite (8.2) as

whereA=(Id?τ?)?1.

We shall proceed in several steps.

Step 1.L2estimate. Multiplying both sides of (8.2) byωn+1and integrating,we obtain

The second term above vanishes thanks to incompressibility. Then

Thus if

then

Step 2. Induction.

Forn=0, we have

whereC1>0,C2>0 are absolute constants. Clearly if

then

Now forn≥0, we assume

Then forn+1 by using (8.3), we have

This then completes the induction proof and we obtain for sufficiently smallτ,

It follows that

whereβ>0 is an absolute constant, andγ1>0 depends only on‖f‖∞.

Step 3. Smoothing estimates. Lett0>0 be chosen sufficiently small (depending only on‖f‖∞) so that the nonlinear part is dominated by the linear part whennτ ≤t0. Choose 1≤n0∈Z such thatn0τ ～t0. By using (8.3)and discrete smoothing estimates, we have

whereC>0 is an absolute constant. Our desired result then follows easily.

Remark 8.1. We should mention that similar to the Burgers case, we can also develop the corresponding analysis for the Fourier collocation method applied to the Navier-Stokes system and many other similar fluid models. All these and related issues will be investigated elsewhere.

Appendix A: some auxiliary estimates

Lemma A.1.Consider for z>0,

Proof of Statement(3)of Theorem2.4, 2D case.For simplicity we takeβ=1. We first consider the cased=2. It suffices for us to work with the expression

Thus the contribution of this piece is acceptable for us.

By a computation similar to that in (A.1), it is not difficult to check that the contribution due to the termc1χ(z)+F(z)(1?χ(z)) is acceptable for us. We now consider the contribution due to the termf1(|z|)χ(z). For this we need to work with the expression

By repeating a similar integration by parts argument in (A.1) and using Lemma A.1, we then only need to control the main error term (the boundary terms are easily controlled)

Thus the contribution ofI1is acceptable for us. Finally we consider the contribution due to the term?χ(z)log|z|. For this we need to work with the expression

Consider 0<|x|?1. We have

For (A.2), sinceχ(y?Mx) is compactly supported, it follows easily that

For (A.5), one can use integration by parts to obtain that

Collecting the estimates, it is then not difficult to check that for 0<|x|?1 andMsufficiently large,

wherec1>0 is an absolute constant.

Next consider the case|x|～1. In this case we note thatM|x|=O(M). In estimating (A.2),we note that when|y?Mx|1,we have|y|=O(M)which implies that

On the other hand, it is not difficult to check that

Collecting the estimates, we obtain that for the case|x|～1,

Collecting all the estimates from Case 1 and Case 2, we then obtain the desired conclusion for 2D.

Appendix B: estimate of eν2t?xxQNU0

Recall that

We now focus on analyzing the quantity

First we considerk0=1. Fork0=1, a rigorous numerical computation gives

Next we shall give more precise bounds onA(k0) depending on the size ofk0.

B.1 Estimate of A(k0) when k0 is large

We first give a bound onA(k0) whenk0≥1. Rewrite

This bound is particularly effective whenais large.

B.2 Estimate of A(k0) when k0 is small

In this subsection we give the precise form ofA(k0) whenk0is small. First we rewrite

wheref(s)=e?as2. Recall the Dirichlet kernel

Alternatively if one does not need the quantitative convergence rate,one can directly obtain

Clearly

Thus we obtain

Remark B.5. More generally, supposea

Note that the above derivation is a natural extension of the usual identity for Dirac comb:

where c1>0is a constant depending only on α.

Proof.With no loss we assumeν=1. Supposefsatisfies‖f‖∞≤1. By Theorem B.1 we only need to consider the regimeN2t<100logN. For 0≤l ≤N ?1 andN2t<100logN, by using the proof of Theorem B.1, we have

the desired result then follows easily.

Appendix C: estimate of (Id?ν2τ?xx)?nQNU0and related estimates

Lemma C.1.Let τ>0and s≥1. Then for any1≤p