Accepted Manuscript

A new space-time discretization for the Swift-Hohenberg equation that strictly

respects the Lyapunov functional

Hector Gomez, Xesús Nogueira

PII: S1007-5704(12)00228-6

DOI: http://dx.doi.org/10.1016/j.cnsns.2012.05.018

Reference: CNSNS 2471

To appear in: Communications in Nonlinear Science and Numer‐

ical Simulation

Received Date: 13 October 2011

Revised Date: 14 May 2012

Accepted Date: 17 May 2012

Please cite this article as: Gomez, H., Nogueira, X., A new space-time discretization for the Swift-Hohenberg

equation that strictly respects the Lyapunov functional, Communications in Nonlinear Science and Numerical

Simulation (2012), doi: http://dx.doi.org/10.1016/j.cnsns.2012.05.018

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http://dx.doi.org/10.1016/j.cnsns.2012.05.018http://dx.doi.org/http://dx.doi.org/10.1016/j.cnsns.2012.05.018

A new space-time discretization for the

Swift-Hohenberg equation that strictly

respects the Lyapunov functional

Hector Gomez1 ∗ , Xesús Nogueira1

1: University of A CoruñaDepartment of Mathematical Methods

Campus de Elviña, s/n15192, A Coruña

Abstract

The Swift-Hohenberg equation is a central nonlinear model in modern physics. Orig-inally derived to describe the onset and evolution of roll patterns in Rayleigh-Bénardconvection, it has also been applied to study a variety of complex fluids and bio-logical materials, including neural tissues. The Swift-Hohenberg equation may bederived from a Lyapunov functional using a variational argument. Here, we intro-duce a new fully-discrete algorithm for the Swift-Hohenberg equation which inheritsthe nonlinear stability property of the continuum equation irrespectively of the timestep. We present several numerical examples that support our theoretical results andillustrate the efficiency, accuracy and stability of our new algorithm. We also com-pare our method to other existing schemes, showing that is feasible alternative tothe available methods.

Key words: Nonlinear stability, Time-integration, Unconditionally stable,Isogeometric Analysis, Swift-Hohenberg, Rayleigh-Bénard convection, Patternformation.

1 Introduction

The Swift-Hohenberg model is an evolutive nonlinear higher-order Partial Differential Equation(PDE) which develops complicated dynamics. Since it was proposed in the late seventies [1] asa model for the description of Rayleigh-Bénard convection [2,3], it has become one of the par-adigms of nonlinear dynamical system leading to complex pattern formation [4,5]. Apart fromfluid convection, the Swift-Hohenberg equation has also been employed to describe complex flu-ids and biological tissues [6]. The Swift-Hohenberg equation may be derived from a Lyapunovfunctional using a variational argument, which endows the theory with a nonlinear stability

∗ Correspondence to: University of A Coruña, Deparment of Mathematical MethodsEmail address: hgomez@udc.es (Hector Gomez1).

Preprint submitted to Elsevier 24 May 2012

property. If inadequate algorithms are employed, this important property of the model can belost after numerical discretization, leading to non-physical solutions. For example, an standardexplicit method would require ∆t ∼ ∆x4 for the discrete solution to be energy-decreasing. Thisimposes a severe restriction over the periods of time that can be simulated. Here we present afully discrete algorithm that inherits the nonlinear stability of the continuum model irrespec-tively of the mesh and time step sizes (in what follows, an algorithm verifying this propertywill be called unconditionally stable or thermodynamically consistent). Thus, our algorithmeliminates the restriction over the time step and opens the possibility to perform simulationsover very large periods of time with the additional guarantee that the numerical solution willsatisfy an important physical property of the model.

Thermodynamically consistent algorithms have been extensively studied in solid [7–10] and fluidmechanics [11–14], but remain rather unexplored for complex pattern-forming PDE’s (someexceptions may be found in [15–24]). Previous work on the numerical simulation of the Swift-Hohenberg equation includes [25–31]. Remarkably, thermodynamically consistent algorithmsfor the Swift-Hohenberg equation have been proposed in [32,20]. Here we present a new space-time discretization for the Swift-Hohenberg equation that is unconditionally stable and second-order time-accurate. The space discretization of our algorithm is based on variational formswhose well-posedness requires the use of globally C1-continuous basis functions. We satisfy thisrequirement using Isogeometric Analysis [33,34], a recently proposed generalization of the FiniteElement Method, that permits generating higher-order and higher-continuity basis functions.Compared to standard finite element formulations based on mixed methods, our method leadsto half of the global number of degrees of freedom and exhibits better approximability propertiesthan widely used C0-continuous piecewise polynomials [35]. For these reasons, our method isvery attractive for the simulation of extended systems over large periods of time.

We present several numerical examples that support our theorems proving second-order timeaccuracy and unconditional stability. These computations are related to fluid convection onsquare and circular domains. The outline of this paper is as follows: In section 2, we describethe Swift-Hohenberg equation. Section 3 presents our algorithm for this equation. We presentnumerical examples in section 4. Finally, we draw conclusions in section 5.

2 The Swift-Hohenberg equation

The Swift-Hohenberg equation describes the onset and evolution of roll patterns in Rayleigh-Bénard convection [36–39]. It may be derived from the fundamental equations of fluid mechanicsin the limit of large Prandtl number, under the assumption of the Boussinesq approximation [1].However, for the purpose of this work, it is more useful to derive it as a dissipative evolutionequation of a non-conserved phase variable. By dissipative equation, we understand one forwhich a Lyapunov functional exists. In what follows, we introduce the Lyapunov functionalof the Swift-Hohenberg equation, and derive an evolution equation whose solutions lead to atime-decreasing Lyapunov functional.

2

2.1 Lyapunov functional

Let u be a scalar phase variable defined on Ω, an open subset of R3. Let us call Γ the boundaryof Ω. We assume Γ to have a continuous unit outward normal n. We define the followingfree-energy functional

F(u) =∫Ω

{Ψ(u) +

D

2

[(∆u)2 − 2k2|∇u|2 + k4u2

]}dx (1)

where D and k are real constants, and Ψ is a nonlinear function of u, defined as

Ψ(u) = − �2u2 − g

3u3 +

1

4u4 (2)

Here � and g are positive constants, which represent physically relevant quantities. In whatfollows, we introduce the Swift-Hohenberg model, and show that F is indeed a Lyapunovfunctional of the equation.

2.2 The Swift-Hohenberg equation

The Swift-Hohenberg equation may be written as

∂u

∂t= −δF

δu(3)

whereδFδu

= Ψ′(u) + Dk4u + 2Dk2∆u + D∆2u (4)

denotes the variational derivative of F with respect to variations δu that verify δu = ∇(δu)·n =0 on Γ. Let us introduce the real-valued function F defined as F (t) = F(u(·, t)). Multiplyingequation (3) with δF/δu, and integrating over the domain Ω, we obtain the expression

dF

dt= −

∫Ω

(δFδu

)2dx (5)

which leads to the inequalitydF

dt≤ 0. (6)

The expression (6) may be thought of as a purely mechanical version of the Clausius-Duheminequality [40] (continuum version of the second law of thermodynamics), and we consider it thefundamental stability property of the Swift-Hohenberg equation. The objective of this paper isto develop a fully discrete numerical method which inherits this property irrespectively of themesh and time step sizes.

2.3 Initial/boundary-value problem

We state the following initial/boundary-value problem for the Swift-Hohenberg equation overthe spatial domain Ω and the time interval (0, T ): given u0 : Ω 7→ R, find u : Ω × [0, T ] 7→ Rsuch that

3

∂u

∂t= −µ(u)−Dk4u− 2Dk2∆u−D∆2u in Ω× (0, T ) (7)

∇(2Dk2u + D∆u) · n = 0 on Γ× [0, T ] (8)∇u · n = 0 on Γ× [0, T ] (9)

u(x, 0) = u0(x) in Ω (10)

where µ(u) stands for Ψ′(u). Equations (8)–(9) may be considered natural boundary conditionsof the Swift-Hohenberg equation in a variational formulation. In practice, most calculationsinvolving the Swift-Hohenberg equation are performed using periodic boundary conditions inall directions.

3 Numerical formulation for the Swift-Hohenberg equation

In this section we present our numerical formulation for the Swift-Hohenberg equation. Wefirst derive a semidiscrete form, and then introduce our unconditionally stable time-integrationscheme.

3.1 Semidiscrete formulation

Our starting point is the weak formulation of the continuous problem. At this point we assumeperiodic boundary conditions in all directions. Let us call V the space of trial and weightingfunctions which are assumed to be the same. We suppose V ⊂ H2, where H2 is the Sobolevspace of square integrable periodic functions with square integrable first and second derivatives.The problem may be stated as follows: find u ∈ V such that for all w ∈ V(

w,∂u

∂t+ µ(u) + Dk4u

)−(∇w, 2Dk2∇u

)+ (∆w,D∆u) = 0 (11)

where (·, ·) is the L2-inner product with respect to the domain Ω.

To perform the space discretization of (11) we employ Galerkin’s method. We approximate(11) by the following finite-dimensional problem over the finite element space V h ⊂ V : finduh ∈ V h such that for all wh ∈ V h(

wh,∂uh

∂t+ µ(uh) + Dk4uh

)−(∇wh, 2Dk2∇uh

)+(∆wh, D∆uh

)= 0 (12)

We define the discrete space V h as V h = span{NA}A=1,...,nb , where the NA’s are basis functionsyet to be defined, and nb is the dimension of the discrete space. As a consequence, u

h in equation(12) may be written as,

uh(x, t) =nb∑

A=1

uA(t)NA(x) (13)

where the uA’s are the coordinates of uh on V h. The function wh is defined analogously.

We emphasize that the condition V h ⊂ V requires the discrete space to be a subset of H2.Standard C0-continuous finite elements do not satisfy this requirement, and may not be utilizeddirectly in the variational formulation (12). To handle this situation, we employ Isogeometric

4

Analysis [33,34], which is a generalization of Finite Element Analysis [41] possessing severaladvantages [42–52]. Isogeometric Analysis is a recently introduced technology that is basedon the developments of Computer Aided Design (CAD). Ideally, it would permit generatingcomputational meshes directly from geometrical models encapsulated in CAD files, makinguse of the underlying parametrization of the CAD design. This holds promise to simplify, oreven eliminate altogether, the mesh generation and refinement process, currently the majorbottleneck of analysis. Following the isoparametric concept, the geometrical parametrizationis also employed to generate the discrete space used to approximate the solution. Geometricalmodels in CAD files are usually parametrized using Non-Uniform Rational B-Splines (NURBS).NURBS are projective transformations of B-Splines, which, in turn, are piecewise polynomials[53,54]. NURBS not only permit generating computational models from CAD designs, but havealso shown superior approximation capabilities compared to classical piecewise polynomials[50]. Even more importantly for this paper, the use of NURBS permits generating globallyC1-continuous basis functions easily, which leads to simple treatment of higher-order operatorsand has proved significantly accurate and robust [18,35,55,56]. In what follows we show how togenerate our basis functions and discrete spaces.

3.2 Basis functions and discrete space

Here we show how to generate the NURBS basis functions that we employ for spatial discretiza-tion. The first step is to define a one-dimensional B-Spline basis in parametric space. A B-Splinebasis is a set of n piecewise polynomial functions of order p denoted by {Bi,p}i=1,...,n. Thesefunctions are defined from a knot vector, which is an array containing n+ p+1 non-decreasingcoordinates in parametric space called knots. We consider the knot vector

Kξ = {ξ1, ξ2, . . . , ξn+p+1}. (14)

which defines the parametric space [ξ1, ξn+p+1]. Since the functions will be eventually mappedinto physical space, we may assume without loss of generality ξ1 = 0 and ξn+p+1 = 1. Givena knot vector, the B-Spline basis functions of order p are defined recursively from their lower-order counterparts. The process is started with the zero-th order functions {Bi,0}i=1,...,n givenby

Bi,0(ξ) =

1 if ξi ≤ ξ ≤ ξi+1,0 otherwise. (15)Then, the following algorithm is applied

Bi,a(ξ) =ξ − ξi

ξi+p − ξiBi,a−1(ξ) +

ξi+p+1 − ξξi+p+1 − ξi+1

Bi+1,a−1(ξ); i = 1, . . . , n; a = 1, . . . , p. (16)

The functions {Bi,p}i=1,...,n are C∞ everywhere except at knots. At a non-repeated knot, thefunctions have p − 1 continuous derivatives. If a knot is repeated k times the number of con-tinuous derivatives at that point is p− 1− k.

A three-dimensional B-Spline basis is defined taking tensor products of one-dimensional basisin three orthogonal parametric directions. Therefore, given three polynomial orders p, q, r, andthree knot vectors, Kξ, Kη, Kζ of lengths n + p + 1, m + q + 1, l + r + 1, we can compute

Bijk(ξ, η, ζ) = Bi,p(ξ)Bj,q(η)Bk,r(ζ). (17)

5

Analogously to the one-dimensional case, the knot vectors define the parametric space, whichwe denote Ξ. Again, for simplicity, we take Ξ = [0, 1]3. Using the three-dimensional B-Splinebasis functions we can generate a geometric mapping F : Ξ 7→ Ω

F (ξ, η, ζ) =n+p+1∑

i=1

m+q+1∑j=1

l+r+1∑k=1

CijkBijk(ξ, η, ζ) (18)

which defines the geometric object Ω. The values Cijk ∈ R3 are called control variables.

At this point, we may define NURBS geometrical objects in Rd, which are projective trans-formations of B-Spline geometrical entities in R(d+1). Let Ĉijk ∈ R3 be a set of control pointsin three-dimensional space and ωijk a set of positive real numbers called weights such that

(Ĉijk, wijk) ∈ R4. We define the following B-Spline geometrical object in R4 as,

Ω̂ = F̂ (Ξ) (19)

where

F̂ (ξ, η, ζ) =n+p+1∑

i=1

m+q+1∑j=1

l+r+1∑k=1

(Ĉijk, wijk)Bijk(ξ, η, ζ), (ξ, η, ζ) ∈ Ξ (20)

The NURBS object ΩR is defined as

ΩR = F R(Ξ) (21)

where the geometrical mapping F R takes the form,

F R(ξ, η, ζ) =n+p+1∑

i=1

m+q+1∑j=1

l+r+1∑k=1

Ĉijkwijk

wijkBijk(ξ, η, ζ)n+p+1∑

a=1

m+q+1∑b=1

l+r+1∑c=1

wabcBabc(ξ, η, ζ)

, (ξ, η, ζ) ∈ Ξ (22)

Denoting,

Cijk =Ĉijkwijk

, W (ξ, η, ζ) =n+p+1∑

i=1

m+q+1∑j=1

l+r+1∑k=1

wijkBijk(ξ, η, ζ), Rijk(ξ, η, ζ) =wijkBijk(ξ, η, ζ)

W (ξ, η, ζ)

(23)we have that

F R(ξ, η, ζ) =n+p+1∑

i=1

m+q+1∑j=1

l+r+1∑k=1

CijkRijk(ξ, η, ζ), (ξ, η, ζ) ∈ Ξ (24)

We will call Rijk NURBS functions in parametric space.

NURBS functions in physical space are defined as the push forward of the functions Rijk.Thus, the discrete space that we use for our numerical method is the space spanned by thosefunctions, namely

Vh = span{Rijk ◦ F−1R }. (25)

Note that we invoke the isoparametric concept, because the geometrical mapping F R is definedin terms of NURBS functions.

6

3.3 Time integration

Here we present our time integration algorithm for the Swift-Hohenberg equation. Let us dividethe time interval [0, T ] into N subintervals In = (tn, tn+1); n = 0, . . . , N − 1, where t0 = 0 andtN = T . We call u

hn the discrete approximation of u

h(tn), where we have omitted the dependenceon the spatial coordinate for simplicity. Our time stepping algorithm is defined as follows: givenuhn, find u

hn+1 ∈ V h such that for all wh ∈ V h

(wh,

JuhnK∆tn

)+

(wh,

1

2

(µ(uhn+1) + µ(u

hn))− Ju

hnK2

12µ′′(uhn)

)+

(wh, Dk4uhn+1/2

)−(∇wh, 2Dk2∇uhn+1/2

)+(∆wh, D∆uhn+1/2

)= 0 (26)

where

∆tn = tn+1 − tn; JuhnK = uhn+1 − uhn; uhn+1/2 =1

2(uhn+1 + u

hn) (27)

We summarize the main properties of our time integration scheme in the following theorem.

Theorem 1 The fully-discrete variational formulation (26):

(1) Verifies the nonlinear stability condition

F(uhn) ≤ F(uhn−1) ∀n = 1, . . . , N

irrespectively of the time step.(2) Gives rise to a local truncation error τ that may be bounded as |τ(tn)| ≤ K∆t2n for all

tn ∈ [0, T ], where K is a constant independent of ∆tn.

Proof:

(1) Let f : [a, b] 7→ R be a sufficiently smooth function. We will make use of the followingquadrature formula:

∫ ba

f(x)dx =b− a

2(f(a) + f(b))− (b− a)

3

12f ′′(a)− (b− a)

4

24f ′′′(ξ); ξ ∈ (a, b) (28)

which was introduced in [18]. Let us apply the quadrature formula (28) to the right-handside of the identity ∫ uhn+1

uhn

Ψ′(z)dz =∫ uhn+1

uhn

µ(z)dz (29)

and rearrange the resulting equation. It follows that

JΨ(uhn)KJuhnK

+JuhnK324

µ′′′(uhn+ξ) =1

2

(µ(uhn) + µ(u

hn+1)

)− Ju

hnK2

12µ′′(uhn); ξ ∈ (0, 1) (30)

Taking wh = JuhnK in equation (26), applying equation (30), and making use of the identities(JuhnK, uhn+1/2

)=

1

2

∫ΩJ(uhn)2Kdx;

(∇JuhnK,∇uhn+1/2

)=

1

2

∫ΩJ|∇uhn|2Kdx (31)

7

(∆JuhnK, ∆uhn+1/2

)=

1

2

∫ΩJ(∆uhn)2Kdx (32)

we obtain the following relation

1

∆tn

∫ΩJuhnK2dx +

∫ΩJΨ(uhn)Kdx +

∫Ω

JuhnK412

µ′′′(uhn+ξ)dx

+∫Ω

Dk4

2J(uhn)2Kdx−

∫Ω

Dk2J|∇uhn|2Kdx +∫Ω

D

2J(∆uhn)2Kdx = 0 (33)

which may be rewritten as

JF(uhn)K = −1

∆tn

∫ΩJuhnK2dx−

∫Ω

JuhnK412

µ′′′(uhn+ξ)dx (34)

Since µ′′′(u) ≥ 0, it follows thatJF(uhn)K ≤ 0 (35)

which completes the proof.(2) We derive a bound on the local truncation error by replacing the time-continuous solution

uh(tn) into the algorithm (26). The time-continuous solution does not satisfy the algorithm,giving rise to the local truncation error τ , which is defined by the expression

(wh, τ(tn)

)=

(wh,

Juh(tn)K∆tn

)+(wh,

1

2

(µ(uh(tn+1)) + µ(u

h(tn))))

−(wh,

Juh(tn)K212

µ′′(uh(tn))

)+(wh, Dk4uh(tn+1/2)

)

−(∇wh, 2Dk2∇uh(tn+1/2)

)+(∆wh, D∆uh(tn+1/2)

)(36)

where tn+1/2 = (tn+1 + tn)/2. Assuming sufficient smoothness, Taylor series can be utilizedto prove that

Juh(tn)K∆tn

=∂uh

∂t(tn+1/2) +O(∆t2n) (37)

1

2

(µ(uh(tn+1)) + µ(u

h(tn)))

= µ(uh(tn+1/2)) +O(∆t2n) (38)

Juh(tn)K212

µ′′(uh(tn)) = O(∆t2n) (39)

where we have made use of the Landau notation. Equations (36)–(39), together with (12)lead to the identity (

wh, τ(tn))

=(wh,O(∆t2n)

)(40)

which indicates that |τ(tn)| ≤ K∆t2n where K is a constant independent of ∆tn.

�

8

4 Numerical examples

In this section we present some numerical examples for the Swift-Hohenberg equation. Ourcalculations also provide numerical corroboration of the theoretical results presented in theprevious sections. We also present a comparison of the performance of our new algorithm withother existing techniques. Finally, we present two examples related to the formation of rollpatterns in Rayleigh-Bénard convection both in square and circular domains.

4.1 Accuracy test

This example provides numerical evidence for our time integration scheme being second-orderaccurate. The setup of this accuracy test is based on that presented in [20]. We solve theone-dimensional Swift-Hohenberg equation on the domain Ω = [0, 32]. The parameters of theSwift-Hohenberg equation are D = k = 1, � = 0.025, and g = 0. The initial condition is definedas:

u (x) = 0.07− 0.02 cos(

2π(x− 12)32

)+ 0.0171 cos2

(2π(x + 10)

32

)− 0.0085 sin2

(4πx

32

)(41)

We computed a reference solution at time t = 1 using a spatial mesh composed of 256 C1quadratic elements, and a time step ∆t = 7.8125 · 10−3. We assume that this space-timediscretization is fine enough as to suppose that the reference solution is exact. Then, we repeatedthe computation using larger time steps, and studied how the L2([0, 32]) spatial error normevolved as a function of ∆t. The results are presented on a doubly logarithmic scale in Figure 1.The data defines a straight line with slope 2.67, which confirms the results proven in Theorem1.

4.2 Comparison with other methods

In this section, we compare the performance of our new numerical scheme with other well-established techniques. The performance of our spatial discretization, that is, NURBS functionsin a variational formulation, has been shown superior to standard finite elements on a per-degree-of-freedom basis in several publications [42,48,50,51]. For this reason, we will focus onthe time discretization algorithm. We take a spatial mesh which is sufficiently fine as to supposethat all the error can be attributed to time integration. Then, we compare the performanceof our new time integration method with the following algorithms: (1) a semi-implicit methodthat treats the nonlinear terms explicitly and the linear terms implicitly; (2) the explicit first-order accurate exponential time integrator presented in [57]; (3) the convex-splitting methodproposed in [20] that has been shown to be unconditionally stable for the Swift-Hohenbergequation; (4) the midpoint rule.

To perform the comparison we solve the Swift-Hohenberg equation on the domain Ω = [0, 40]2.The parameters are D = k = 1, � = 2, and g = 0. The initial condition is a constant state(u = −1) in which we embed a curvy vertical stripe with the phase variable taking the valueu = 1. The initial pattern evolves developing horizontal fingers that might bifurcate. Figure 2shows the initial condition and solution at times t = 20 and t = 40 on a sufficiently fine space-

9

10−2

10−1

100

10−7

10−6

10−5

10−4

10−3

10−2

10−1

∆t

||e||2 2.67

1.00

Fig. 1. Accuracy test. L2([0, 32]) spatial error norm with respect to the time step ∆t. This plot confirmsthat our algorithm is second-order time-accurate.

(a) t = 0 (b) t = 20 (c) t = 40

Fig. 2. Comparison with other methods. Initial condition and reference solution at different times.The computational domain is Ω = [0, 40]2. This solution has been computed on a sufficiently smallspace-time mesh.

time mesh. The color scale ranges from −1.75 (blue) to +1.75 (red), and will be maintained forall the examples in this section. We consider the solution in Figure 2 as our reference solution.We will compare the results produced by all the above-mentioned methods at time t = 40.

Figure 3 shows the results produced by the semi-implicit scheme at time t = 40 using differenttime steps. For ∆t = 0.25 the solution is totally incorrect. Qualitative agreement is obtainedfor ∆t = 0.125, ∆t = 0.0625, and ∆t = 0.03125. For smaller time steps, the solution is almostindistinguishable from the reference solution.

10

(a) ∆t = 0.25 (b) ∆t = 0.125 (c) ∆t = 0.0625

(d) ∆t = 0.03125 (e) ∆t = 0.015625 (f) ∆t = 0.0078125

Fig. 3. Comparison with other methods. Numerical solution at t = 40 with the semi-implicit timeintegrator using different time steps.

Figure 4 presents the free energy evolution for the semi-implicit method using different timesteps. For ∆t = 0.25, the free energy is oscillating, which indicates that the computed solu-tion is incorrect. For smaller time steps, the energy does decrease in time, but the dissipa-tion rate is somewhat underestimated. For the two smallest time steps (∆t = 0.015625 and∆t = 0.0078125), the energy curves are one on top of each other, which is a sign of numericalconvergence.

Figure 5 shows the solution at time t = 40 using the exponential time integrator and differenttime steps. This method permits taking time steps larger than those used for the semi-implicitalgorithm. The solution for ∆t = 0.5 (Figure 5(a)) is incorrect. This is also reflected in theenergy plot (Figure 6), which shows an oscillating evolution. For smaller time steps, the solutionlooks qualitatively correct, but quantitative match is only achieved for the two smallest timesteps (∆t = 0.03125 and ∆t = 0.015625). This can also be observed in the energy plot (Figure6) in which the curves corresponding to those time steps are superposed.

Figure 7 shows the solution using the convex-splitting algorithm proposed in [20]. This methodhas been shown to be unconditionally stable for the Swift-Hohenberg equation. This propertyis certainly reflected in the energy plot (Figure 8), which shows time-decreasing energies forall time steps. However, it is know that algorithms based on the convex-splitting concept tendto be significantly inaccurate for large time steps [18]. This statement is consistent with theresults in Figure 7, which show inaccurate solutions at least for ∆t = 1 and ∆t = 0.5. As wereduce the time step, the solution becomes more accurate, exhibiting good agreement with the

11

0 5 10 15 20 25 30 35 40−800

−600

−400

−200

0

200

400

600

�

���������

��������������������������

���������������������������������

���������������������

Fig. 4. Comparison with other methods. Energy evolution for several time steps using the semi-implicitalgorithm.

reference solution for ∆t = 0.0625.

Now, we analyze the results produced by the midpoint rule. This algorithm is an standardsecond-order accurate method. For a linear problem it is known to have the lowest truncationerror of all second-order accurate A-stable linear multistep methods [59]. Another feature ofthe midpoint rule that holds for linear problems is that the algorithm preserves the highestfrequency of the numerical solution at each time step, which makes it difficult to damp outthe spurious modes of the approximate solution. This feature of the method may be observedin Figure 9(a), which represents the numerical solution at t = 40 using the time step ∆t = 8.This image clearly shows how the high frequencies contained in the initial condition remain inthe solution after several time steps, leading to a significantly inaccurate solution. For ∆t = 4,Figure 9(b), the solution looks still shaky, showing again the inability of the algorithm to handlethe high frequencies of the solution. For smaller time steps the solution is accurate and smooth,and appears almost indistinguishable from the reference solution.

Figure 10 shows the energy evolution produced by the midpoint rule for the analyzed timesteps. All the curves are monotonically decreasing, which indicates that, for this particularproblem and the selected time steps, the midpoint rule respects the stability property of theSwift-Hohenberg equation. However, this result does not necessarily hold for other problemsor time steps, because the midpoint rule does not achieve unconditional stability for the Swift-Hohenberg equation.

Finally, Figure 11 shows the solution using our new algorithm. We observe that our methodpermits using significantly larger time steps than the semi-implicit, the exponential and theconvex-splitting algorithms. The time steps employed are comparable to those utilized for themidpoint rule. It may be said that for small and intermediate time steps our method producesresults similar to those achieved by the midpoint rule. However, for very large time steps (∆t =

12

(a) ∆t = 0.5 (b) ∆t = 0.25 (c) ∆t = 0.125

(d) ∆t = 0.0625 (e) ∆t = 0.03125 (f) ∆t = 0.015625

Fig. 5. Comparison with other methods. Numerical solution at t = 40 with the exponential timeintegrator using different time steps.

8), although the solution is inaccurate, is still smooth and its associated energy decreases withtime. The numerical solution actually looks like the exact solution at an earlier time (compareFigure 2 with Figure 11(a)). Physically speaking, it may be said that the method defers thedynamics of the equation for large time steps. This seems to be a feature of unconditionallystable methods for nonlinear dynamics [18,20,58], and may be also observed in the resultsproduced by the convex splitting method (Figure 7). However, the convex-splitting schemeis significantly less accurate than our method. In all, we may conclude that our algorithmrepresents a good balance between accuracy and stability, with the additional guarantee ofenergy-decreasing solutions, as shown in Figure 12. This plot also shows that the dissipationrate is underestimated for large time steps, which is consistent with the statement that thedynamics of the equation is deferred for large time steps.

4.3 Swift-Hohenberg equation on a periodic square

Here we present the numerical solution to the Swift-Hohenberg equation on the domain Ω =[0, 1200]2. We assume periodic boundary conditions in both directions. We take the parametersD = k = 1, � = 0.1, g = 0. To define our initial condition, we set all control variables to zero,and then, randomly perturb their values with a pseudo-random number which is uniformlydistributed on [−0.005, 0.005]. We employ an uniform spatial mesh composed of 20482 C1-quadratic elements. The time step is ∆t = 20.

13

0 5 10 15 20 25 30 35 40−800

−750

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−550

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�

���������

����������

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Fig. 6. Comparison with other methods. Energy evolution for several time steps using the exponentialalgorithm.

Figure 13 shows snapshots of the time-history of the phase variable. We observe that the initialcondition induces an instability into the equation that leads to amplification of the solutionand to the emergence of spatial patterns. Those patterns are composed of stripped regionswith zero- and one-dimensional defects as observed in Figures 13(a)-(b). Global reorganizationof the patterns leads to larger defect-free regions (Figures 13(c)-(d)) which eventually fill upthe whole domain. The simulation results suggest that for this set of parameters the equationpresents a stationary solution corresponding to an ordered stripped pattern as shown in Figures13(e)-(f).

Figure 14 shows the time evolution of the energy functional (1). We appended some snapshotsof the solution to the free-energy curve. Note that the time scale is logarithmic, because thedynamics of the equation becomes slower as time evolves. We observe that the energy functionaldiminishes at all times, which confirms our theoretical predictions.

4.4 Swift-Hohenberg equation on a disk

We present the numerical solution to the Swift-Hohenberg equation on a disk. This geometrybelongs to the class of conic sections that can be exactly reproduced by NURBS. To generatethis geometry we employ quadratic basis functions and the parametrization defined in [60]. Thisleads to a mapping with four singular points on the boundary, as shown in the mesh picturepresented in Figure 15(a). These singularities did not produce any issues in the calculations.The radius disk is 15, and the computational mesh is composed of 1282 C1 quadratic elements.On the boundary we set homogeneous Dirichlet boundary conditions. The time step is ∆t = 5.

The parameters of the Swift-Hohenberg equation are D = k = 1, � = 0.1, and g = 1. The initial

14

(a) ∆t = 1 (b) ∆t = 0.5 (c) ∆t = 0.25

(d) ∆t = 0.125 (e) ∆t = 0.0625 (f) ∆t = 0.03125

Fig. 7. Comparison with other methods. Numerical solution at t = 40 with the convex splittingalgorithm using different time steps.

condition was generated following the same process as in the last example. Figure 15(b)–(d)shows the time evolution of the phase variable u until the steady state is reached. We noticethat the choice of g = 1 leads to the formation of circular structures, rather than stripes.Note that the circular structures are distorted near the boundary, due to the effect of Dirichletboundary conditions.

In Figure 16 we plot the time evolution of the energy functional. We notice that the energydecreases at all times, which supports our theoretical result about the unconditional stabilityof the presented space-time discretization.

5 Conclusions

The Swift-Hohenberg equation is a higher-order nonlinear partial differential equation endowedwith a nonlinear stability property. This equation governs the formation and evolution of rollpatterns in Rayleigh-Bénard convection. We introduce a new space-time discretization thatinherits the nonlinear stability relationship of the continuous equation irrespectively of themesh and time step sizes, and that is second-order time-accurate. We present several numericalexamples dealing with fluid convection on square and circular domains. These examples supportour theoretical results and show the accuracy, efficiency and robustness of our new method. Wealso compare our method to other existing schemes, showing that is feasible alternative to the

15

0 5 10 15 20 25 30 35 40−800

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��

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Fig. 8. Comparison with other methods. Energy evolution for several time steps using the convexsplitting method.

available methods.

6 Acknowledgements

The authors were partially supported by Xunta de Galicia (grants # 09REM005118PR and#09MDS00718PR), Ministerio de Ciencia e Innovación (grant #DPI2009-14546-C02-01) cofi-nanced with FEDER funds, and Universidad de A Coruña.

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(a) ∆t = 8 (b) ∆t = 4 (c) ∆t = 2

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(a) ∆t = 8 (b) ∆t = 4 (c) ∆t = 2

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Fig. 12. Comparison with other methods. Energy evolution for several time steps using the proposedmethod. We observe that for ∆t = 1, ∆t = 0.5 and ∆t = 0.25 the curves are almost superposed,which indicates numerical convergence.

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20

(a) t = 800, full domain (b) t = 800, detail

(c) t = 4 · 104, full domain (d) t = 4 · 104, detail

(e) t = 2.2 · 106, full domain (f) t = 2.2 · 106, detail

Fig. 13. Swift-Hohenberg equation on a periodic square. Snapshots of the numerical approximation tothe phase variable u at different computational times. The parameters of the equation are D = k = 1,� = 0.1, and g = 0. The spatial mesh is composed of 20482 C1-quadratic elements. The time step is∆t = 20.

21

Fig. 14. Swift-Hohenberg equation on a periodic square. Time evolution of the energy functional.We appended some snapshots of the solution to the free-energy curve. We observe that the energyfunctional diminishes at all times, which confirms our theoretical predictions.

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22

(a) Computational mesh (b) Numerical solution at t = 100

(c) Numerical solution at t = 500 (d) Numerical solution at t = 6000

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23

0 500 1000 1500 2000−14

−12

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−6

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0

2

t

F

Fig. 16. Swift-Hohenberg equation on a disk. Time evolution of the energy functional. We appendedsome snapshots of the solution to the free-energy curve. These snapshots show that fast time-variationsof the free-energy correspond to abrupt variations in the spatial pattern. We observe that the energyfunctional diminishes at all times, which confirms our theoretical predictions.

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24

Highlights 

‐ We propose a new space‐time discretization algorithm for the Swift‐Hohenberg equation 

‐ We prove the method to be nonlinearly stable irrespectively of the discretization 

‐ We present computations that support the theory and show the efficiency of the method