A uniformly convergent numerical scheme for singularly perturbed non linear eigenvalue problem under constraints

24 3.3.3 Matched asymptotic approximations for the first excited state 25 4 Numerical Methods for Singularly Perturbed Eigenvalue Problems 28 4.1 Gradient flow with discrete normalizatio

A UNIFORMLY CONVERGENT NUMERICAL SCHEME FOR SINGULARLY PERTURBED NON-LINEAR EIGENVALUE PROBLEM

UNDER CONSTRAINTS

CHAI MING HUANG

(B.Sc.(Hons.), NUS)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE

2006

First and foremost, I would like to thank my supervisor, Associate Professor BaoWeizhu for his patience, guidance and invaluable advice He had been extremelypatient with me throughout my studies and was always ready to guide me in myresearch

It is also my pleasure to express my appreciation and thanks to my fellow graduates Yang Li, Lim Fong Yin and especially my seniors Dr Wang Hanquan and

post-Dr Zhang Yanzhi They provided immense help especially with the Mathematicalderivation and programming This dissertation would not be completed smoothlywithout their kind assistance and heartfelt encouragement

Lastly, I would like to dedicate this work with love to my wife, my parents and

my family for being always there for me

Chai Ming Huang

Dec 2006

ii

1.1 Brief history of Bose-Einstein condensation 1

1.2 Review of existing numerical methods 3

1.3 The problem 3

1.4 The organization of the thesis 5

2 The Gross-Pitaevskii equation 7 2.1 The time-dependent GPE 7

2.2 Non-dimensionalization of GPE 8

2.3 Reduction of the GPE to lower dimensions 9

2.4 Stationary states of GPE 12

2.4.1 Ground state 13

2.4.2 Excited states 13

iii

Contents iv

3.1 The singularly perturbed nonlinear eigenvalue problem 14

3.1.1 For bounded domain Ω = [0, 1] d 15

3.1.2 For the whole space Ω = Rd 15

3.1.3 General formulation 16

3.2 Approximations in 1D box potential 17

3.2.1 Thomas-Fermi approximation for ground state 18

3.2.2 Matched asymptotic approximations for ground state 18

3.2.3 Matched asymptotic approximations for excited states 21

3.3 Approximations for 1D harmonic potential 22

3.3.1 Thomas-Fermi approximation for ground state 23

3.3.2 Thomas-Fermi approximation for the first excited state 24

3.3.3 Matched asymptotic approximations for the first excited state 25 4 Numerical Methods for Singularly Perturbed Eigenvalue Problems 28 4.1 Gradient flow with discrete normalization 28

4.2 Discretization with uniform mesh in 1D 29

4.3 Discretization with piecewise uniform mesh in 1D 32

4.3.1 The full discretization with piecewise uniform mesh 32

4.3.2 Piecewise uniform mesh for ground state with box potential 35 4.3.3 Piecewise uniform mesh for first excited state with box potential 35 4.3.4 Piecewise uniform mesh for first excited state with harmonic potential 36

4.4 Choice of initial data 37

4.5 Error analysis of uniform mesh 39

4.6 Error analysis of piecewise uniform mesh 47

4.7 Numerical comparisons 54

Contents v

5.1 Numerical results in 1D 65

5.1.1 Ground state and excited states with box potential 65

5.1.2 Ground state and excited states with harmonic potential in 1D 69 5.2 Numerical results in 2D for box potential 71

5.2.1 Choice of mesh 71

5.2.2 Choice of initial data 72

5.2.3 Results 72

5.3 Numerical results in 2D for harmonic potential 79

5.3.1 Choice of mesh 79

5.3.2 Choice of initial data 80

5.3.3 Results 80

5.4 Numerical results in 2D for harmonic plus optical lattice potential 85

5.4.1 Results 85

The time-independent Gross-Pitaevskii equation (GPE) in the semiclassical regime

is used to describe the equilibrium properties of Bose-Einstein Condensate at tremely low temperature In this regime, the GPE is a singular perturbed nonlineareigenvalue problem

ex-The aim of this thesis is to present a uniformly convergent numerical scheme tosolve the singularly perturbed nonlinear eigenvalue problem The adaptive numericalscheme proposed is based on a piecewise uniform mesh The scheme is found to beable to treat the interior layers or boundary layers inherent in solutions of singularlyperturbed nonlinear eigenvalue problems

A comparison of the new proposed scheme based on piecewise uniform mesh ismade against the classical numerical scheme based on uniform mesh We found thatthe numerical accuracy of the new numerical scheme proposed is greatly improvedover the classical numerical scheme

An extension of the new numerical scheme is made to two dimensions Thescheme is then applied to solve the singular perturbed nonlinear eigenvalue problem

in two dimensions

vi

Chapter 1

Introduction

In 1925, Indian physicist Satyendra Nath Bose published a paper devoted to thestatistical description of the quanta of light Based on Bose’s results, Albert Einstein[13] predicted that a phase transition in a gas of noninteracting atoms could occurdue to quantum statistical effects During this phase of transition period, a Bose-Einstein Condensate (BEC) will be formed when a macroscopic number of non-interacting bosons simultaneously occupy the single quantum state of the lowestenergy [31]

For many years, there was no practical application of BEC In 1938, after fluditiy was discovered in liquid helium, F London theorized that the superfluiditycould be a manifestation of BEC However in 1955, experiments on superfluid heliumshowed that only a small fraction of condensate is found In the 1970s, experimentalstudies on dilute atomic gases were developed The first of these studies focused onspin-polarized hydrogen This gas was chosen as it has a very light mass and is thuslikely to achieve BEC After numerous attempts, BEC was almost achieved but itwas not pure [44]

super-1

1.1 Brief history of Bose-Einstein condensation 2

In the 1980s, there was remarkable progress made in the application of based cooling techniques and magneto-optical trapping In 1995, a historical mile-stone was achieved when the experimental teams of Cornell and Wieman at Boulder

laser-of JILA and laser-of Ketterle at MIT succeeded in reaching the ultra low temperatureand densities required to observe BEC in vapors of 87Rb [2] and 23Na [22] Later inthe same year, occurrence of BEC in vapors of 7Li was also reported [15] For theirachievement, the Nobel Prize of Physics was awarded to the first three researcherswho created this fifth state of matter in the laboratory After realizing BEC in dilutebosonic atomic gases, BEC was also reached in other atomic matter, including thespin-polarized hydrogen, metastable 4He and 41K [28]

Since all the particles occupy the same state in the BEC at ultra low temperature,

the condensate is characterized by a complex-valued wave function ψ(~x, t), whose

time evolution is governed by the time-dependent Gross-Pitaevskii equation (GPE)[20, 37] It is impossible to solve the GPE analytically except for the simplest cases

of GPE Various numerical methods are used to solve the GPE instead When theproblems involve the static properties of the condensate, the numerical solutions ofthe time-independent GPE are of interest

Over the last several years, there were extensive progress made towards oping innovative approaches and algorithms in solving both time-dependent andtime-independent GPE We will survey some of the more important recent researchpapers written in the field, with more emphasis of the numerical methodology insolving the time-independent GPE, which is the main subject of interest in thisdissertation

devel-1.2 Review of existing numerical methods 3

The earliest attempts to solve the GPE might be started by Edwards and Burnetts[27] They developed a Runge-Kutta method based on finite-difference to solve thetime-independent GPE for spherical condensates Edwards [26] also designed a ba-sis set approach to solve GPE For the solving of time-independent GPE in groundstate and the vortex states in anisotropic traps, a finite-difference based imaginarytime method was developed by Dalfovo and Stringari [21] Adhikari [1] used a finite-difference based approach to solve the two-dimensional time-independent GPE Cer-imele, together with his coworkers [17], developed a finite-difference and imaginary-time approach for solving the time-independent GPE Schneider and Feder [48] used

a discrete variable representation that is coupled with a Gaussian quadrature tion scheme, to attain the ground and the excited states of GPE in three dimensions.Recently, Bao and Tang [11] used a different approach for obtaining the ground state

integra-of GPE They did this by directly minimizing the corresponding energy functionalwith a finite element discretization Utilizing the harmonic oscillator as the basisset, Dion and Canc´es [23] proposed a Gauss-Hermite quadrature integration scheme

to solve both the time-dependent and time-independent GPE More recently, Baoand Du [4] applied the gradient flow method with discrete normalization to find theground state of the GPE This numerical method is perhaps one of the most efficientways to solve the time-independent GPE [4, 5, 9, 17, 19, 21]

However, there are numerical difficulties when the time-independent GPE is in asemiclassical regime, i.e BEC is a strong repulsively interacting condensate Insuch a regime, the GPE is reduced into a singularly perturbed non-linear eigenvalue

1.3 The problem 4problem under a constraint as shown

layers or interior layers [8] The classical numerical scheme based on uniform mesh

to discretize the gradient flow would be difficult to track these layers [24] In order

to obtain a reliable numerical solution for (1.1) when ε ¿ 1, it is desirable to use

an adaptive mesh that concentrates nodes in the boundary layers or interior layers.Ideally, the mesh should be generated by adapting it to the features of the computedsolution There has been a great deal of research done on the use of adaptive methodsfor steady and unsteady partial differential equations recently [16, 18, 29, 42, 41,

34, 33, 35, 45, 46] Among which, Shishkin [49] in 1990 proposed an upwind schemebased on a piecewise uniform mesh to solve the two-point boundary layer problems—fine in the boundary and coarse in the rest of the domain This scheme is useful

and has been demonstrated to be ε-uniform convergenct by Miller et al [42, 41].

It has also been shown that the scheme is uniformly convergent near the boundarylayer and it has been pointed out that uniform convergence cannot be obtained atall interior mesh points unless the mesh is specially tailored to the solution of theproblem

In this thesis, we aim to design a uniformly convergent numerical scheme based

on piecewise uniform mesh for discretizing the gradient flow so that we can treatproblems with complicated boundary layer or interior layers effectively

1.4 The organization of the thesis 5

This thesis is organized as follows

In Chapter 2, starting from the time-dependent GPE, we first rescale it to adimensionless form and then reduce the time-dependent GPE from three dimensionsinto lower dimensions We next describe how to obtain the stationary states of BECand the time-independent GPE in a semiclassical regime, i.e., the ground state andexcited states

In Chapter 3, we arrive at the singularly perturbed nonlinear eigenvalue lem under a constraint to be solved For the sake of comparison with numericalapproximation later, we present some analytical approximations for the ground andexcited states in BEC with box potential in one dimension (1D) We also presentsome analytical approximations for the first excited states in BEC with harmonicpotential in 1D We demonstrate that there are boundary layers or interior layers inthese solutions

prob-In Chapter 4, we describe the numerical methods for solving such singularly turbed nonlinear eigenvalue problem under a constraint We apply one of the mostefficient numerical technique–the gradient flow with discrete normalization to solvethe singularly perturbed and constrained nonlinear eigenvalue problem We firstshow a classical numerical scheme based on uniform mesh to discretize the gradientflow We then analyze the shortcomings of the scheme and introduce the detailedalgorithm of our newly proposed numerical scheme based on piecewise uniform mesh

per-to discretize the gradient flow per-to treat boundary layers or interior layers Finally weprovide numerical error analysis for both uniform mesh and piecewise uniform mesh.The limitations of uniform mesh are shown and the advantages from using piece-wise uniform mesh are presented Comparisons between solutions obtained by ourproposed piecewise uniform mesh and solutions generated with the classical uniform

1.4 The organization of the thesis 6

mesh are shown in more details

In Chapter 5, we apply our newly proposed scheme based on piecewise uniformmesh to calculate the ground state, first, third, and ninth excited states of BEC withbox potential in 1D and the first excited state of BEC with harmonic potential in 1D

We compare the numerical results with those asymptotic approximation shown inChapter 3 We then extend our numerical scheme based on piecewise uniform mesh

to find numerical solutions of the singularly perturbed and constrained nonlineareigenvalue problem in two dimensions (2D), for example, ground state and excitedstates of BEC in three different potentials, box potential, harmonic potential andharmonic plus optical potential This is to illustrate the capability of the proposedpiecewise uniform scheme in solving the time-independent GPE under different po-tentials and conditions, more specifically, to treat the boundary layers or interiorlayers in two dimensions

Finally in Chapter 6, some conclusions on our results are drawn and possiblefuture works are highlighted

Chapter 2

The Gross-Pitaevskii equation

In this chapter, we derive the independent GPE from the well-known dependent GPE As preparatory steps, we introduce the time-dependent GPE withtwo kinds of external potentials, i.e., the harmonic oscillator potential and thebox potential The GPE is then non-dimensionalized, rescaled and reduced intolower-dimensional formulations Finally the solutions of the time-independent GPE,ground state and excited states are summarized

At temperatures T much lower than the critical temperature T c, the BEC is well

described by the macroscopic wave function ψ = ψ(~x, t) The evolution of this wave

function is governed by a self-consistent nonlinear Schr¨odinger equation known asthe Gross-Pitaevskii equation [32, 43]

7

2.2 Non-dimensionalization of GPE 8

length (positive for repulsive interaction and negative for attractive interaction),

V (~x) is an external trapping potential.

Two important invariants of (2.1) are the normalization of the wave function

There are two typical external potentials V (~x) considered in this dissertation:

1 The box potential:

2.3 Reduction of the GPE to lower dimensions 9

and the dimensionless energy functional E(ψ) is defined as follows:

E(ψ) =

Z

R 3

·1

where the interaction parameter β = 4πa s N

x s The choices used for the scaling

param-eters, t s and x s for the two different dimensionless potential V (~x) are:

1 The box potential:

In order to illustrate dimension reduction of the GPE in 3D to two dimensions (2D)

or one dimension (1D), we first consider the dimensionless GPE with the harmonicpotential The dimensionless GPE with its normalization is given by:

In a disk-shaped condensation with parameters ω x ≈ ω y and ω z À ω x (⇐⇒

γ y ≈ 1 and γ z À 1), the three-dimensional GPE (2.14) can be reduced to a

two-dimensional GPE by assuming that the time evolution does not cause excitations

2.3 Reduction of the GPE to lower dimensions 10

along the z-axis, since the excitations along the z-axis have large energy (of order

~ω z ) compared to that along the x- and y-axis with energies of order ~ω x Thus

we may assume that the condensation wave function along the z-axis is always well

described by the ground state wave function and set

¡

x2+ γ2

y y2+ C¢ψ2+ β2|ψ2|2ψ2, (2.16)where

Since this GPE is time-transverse invariant, we can replace ψ2 → ψe −iCt/2 which

drops the constant C in the trapping potential and obtain:

In a cigar-shaped condensation where the energies along x-axis is much smaller than energies along y- and z-axis, i.e ω y À ω x and ω z À ω x, and there is almost no

excitation along the y- and z-axis as time evolves, we can obtain a one-dimensional GPE In fact, for any fixed β ≥ 0 and when γ y À 1 and γ z À 1, we set

2.3 Reduction of the GPE to lower dimensions 11

Substituting (2.19) into (2.8), multiplying both sides by ¯φho

23(y, z) (the conjugate of

Since (2.21) is time-transverse invariant, we let ψ1 → ψe −i Ct

2 This will remove theterm containing the constant C and we obtain the GPE in 1D as:

i ∂

∂t ψ(x, t) = −

12

2.4 Stationary states of GPE 12

Hence, a general d-dimensional (d=1,2,3) GPE will be as follows:

i∂ t ψ(~x, t) = −1

2∇

2ψ(~x, t) + V d (~x)ψ(~x, t) + β d |ψ(~x, t)|2ψ(~x, t), ~x ∈ Ω, (2.26) ψ(~x, t) = 0, ~x ∈ ∂Ω,

where Ω is a bounded domain in Rd Two important invariants of (2.26) are thenormalization of the wave function

In order to find stationary state of (2.26), we let

where φ(~x) is a function independent of time t and µ is the chemical potential of

the condensate Substitute (2.29) into (2.26), we get

2.4 Stationary states of GPE 13

This is a nonlinear eigenvalue problem with a constraint and the eigenvalue µ can

be calculated from the corresponding eignfunction φ(~x) by

µ = µ β (φ) =

Z

Ω

·1

Any eigenfunction φ(~x) of (2.30) under the constraint (2.32) whose energy E β (φ) >

E β (φ g) is usually called as an excited state in the physics literature

Suppose the eigenfunctions of the eigenvalue problem (2.30) under the constraint(2.32) are

±φ g (~x), ±φ1(~x), ±φ2(~x), · · · , (2.35)whose energies satisfy

E β (φ g ) < E β (φ1) < E β (φ2) < · · · (2.36)

Then φ j (~x), j = 1, 2, 3, · · · , is called as the j-th excited state solution.

Chapter 3

The singularly perturbed nonlinear

eigenvalue problem

In this chapter, we derive the singularly perturbed nonlinear eigenvalue problem

from the time-independent GPE (2.30) When β d À 1, the time-independent GPE,

in the bounded domain or whole space, is then rescaled and reduced into sical formulations We finally obtain the singularly perturbed nonlinear eigenvalueproblem under a constraint in a general form

problem

When β d À 1, i.e the time-independent GPE (2.30) is in a strongly repulsive

interacting condensation or in the semiclassical regime, we need another scaling forthe GPE

14

3.1 The singularly perturbed nonlinear eigenvalue problem 15

3.1.1 For bounded domain Ω = [0, 1]d

When Ω = [0, 1] d, the GPE (2.30) with box potential is

µφ(~x) = − ε2

2∇

2φ(~x) + |φ(~x)|2φ(~x), ~x ∈ [0, 1] d , (3.3)and the normalization as

When Ω = Rd is the whole space, the time-independent GPE (2.30) with the monic potential is as follows,

har-µφ(~x) = −1

2∇

2φ(~x) + V d (~x)φ(~x) + β d |φ(~x)|2φ(~x), ~x ∈ Ω = R d , (3.5)

3.1 The singularly perturbed nonlinear eigenvalue problem 16

In order to rescale the GPE, We let

~x = ε 1/2e~x, φ = ε d/4 φ,e µ = εe µ, ε = β −d/d+2

Substituting the above scaling parameters into (2.30), and rearranging the variables,

we have the singularly perturbed nonlinear eigenvalue problem

In conclusion, we have the following singularly perturbed nonlinear eigenvalue

prob-lem whatever the potentials V d (~x) as

3.2 Approximations in 1D box potential 17

The chemical potential µ in (3.9) can be computed from its corresponding

where Ekin, Epot and Eint are the kinetic energy, potential energy and interactionenergy respectively They are defined as

In addition, the chemical potential µ can also be given by

µ ε (φ) = Ekin(φ) + Epot(φ) + 2Eint(φ). (3.17)The equation (3.9) with the constraint (3.11) is a singularly perturbed nonlineareigenvalue problem and its solutions are of main interest in this thesis In the nextsection, some approximated solutions for the problem in 1D, which have boundary

layer or interior layer for small ε, are summarized.

In this section, we present the matched asymptotic approximations for the ground

state and excited states of BEC confined in a 1D box potential, i.e., V1(x) = 0, for

3.2 Approximations in 1D box potential 18

0 ≤ x ≤ 1; V1(x) = ∞, otherwise We truncate the eigenvalue problem into [0, 1]

with homogeneous Dirichlet boundary condition in this case

We first consider (3.9) with box potential in 1D Since 0 < ε ¿ 1, we can drop the

first term on the right side and obtain the ground state approximation as:

near x = 0 and near x = 1 in the ground state of BEC with box potential when we

remove the diffusion term in (3.3)

Since the layers exist at the two boundaries x = 0 and x = 1 when 0 < ε ¿ 1, we solve (3.8) near x = 0 and x = 1, respectively Let us suppose the boundary layer

is of width δ (0 < δ < 1) We do a rescaling in the region of x ∈ [0, δ] and let

3.2 Approximations in 1D box potential 19

We substitute (3.21) into (3.8) and obtain

µΦ(X) = − ε2

2δ2ΦXX (X) + φ2

sΦ3(X), X ∈ (0, 1), (3.22)

In order to solve the above equation, we need to rescale all the terms to O(1) We

choose δ = ε/ √ µ and φ s =√ µ in (3.21), the above equation reduces to

Since µ ≈ µTF = 1 for the ground state, we can conclude that the width of boundary

layer near x = 0 is O(ε) Thus finally we have

3.2 Approximations in 1D box potential 20Using the normalization condition (3.11), we find

htanh

³q

µMA

g x/ε

´+ tanh

³cosh(

hsech2³q

3.2 Approximations in 1D box potential 21and

For the BEC in 1D box potential, when 0 < ε ¿ 1, the kth (k ∈ N) excited state not only has boundary layers near x = 0 and x = 1, but also has k interior layers at

Using the matched asymptotic method described in the previous subsection, we can

obtain an approximation for φMA

k , i.e., the kth (k ∈ N) excited states as

where [τ ] takes the integer part of the real number τ and the constant C k= 1 when

k is odd and C k = 0 when k is even Plugging equation (3.37) into the normalizationcondition (3.11), we have

3.3 Approximations for 1D harmonic potential 22

1 Boundary layers are observed at x = 0 and x = 1 for all ground state and excited states when 0 < ε ¿ 1 The width of these layers are of O(ε).

2 For k-th excited states, interior layers are also observed at x = j

k+1 , (j =

1, , k) when 0 < ε ¿ 1 The widths of these interior layers are twice the

size of widths at the boundary layers

Similarly, we can extend the above asymptotic approximations to ground stateand excited states of BEC with box potential in higher dimensions These ap-proximate results will be useful since they tell us the locations and width of theboundary and interior layers of the solutions These results also help us in choosingthe piecewise uniform mesh more effectively, which we will discuss in next chapter

In this section, we present some approximations for both ground state and the first

excited states of BEC with 1D harmonic potential, i.e., V1(x) = 1

2x2

3.3 Approximations for 1D harmonic potential 23

We first consider the Thomas-Fermi (TF) approximation for 1D harmonic oscillator

potential From (3.9), we drop the first term on the right side because 0 < ε ¿ 1

1 =Z

3.3 Approximations for 1D harmonic potential 24

¶2/3

3.3.2 Thomas-Fermi approximation for the first excited state

Similarly, using the same approach in the derivation of the TF approximation forthe ground state, we can obtain the TF approximation for the 1st excited state,

x2<2µTF 1

|φTF

1 (x)|2 dx =

Z √ 2µTF 1

− √

2µTF 1

¡

µTF 1

¢5/2

10

µ32

¶2/3

3.3 Approximations for 1D harmonic potential 25the interaction energy

¡

µTF 1

¢5/2

5

µ32

(3.56) and get

µΦ(X) = − ε2

2δ2ΦXX (X) + φ2

sΦ3(X), X ∈ (−1, 1). (3.57)The equation above is similar to the equation (3.22) obtained for the box potentialand the first excited state is an odd function In order to solve the above equation

3.3 Approximations for 1D harmonic potential 26

for 0 ≤ X < 1 , we need to rescale all the terms to O(1) By choosing δ = ε/ √ µ and φ s=√ µ, the equation becomes

From (3.60), we can conclude that the width of the interior layer near x = 0 is O(ε).

In fact the first excited solution of the equation from x = 0 can be approximated

ε x

¶+

1 can be determined by the normalization condition (3.11)

Based on the analytical results obtained, we make the following observations forthe ground state and first excited state of BEC with harmonic potential:

1 No boundary layer or interior layer is observed for ground state solutions

2 For the 1st excited states, an interior layer is observed at x = 0 The width of the interior layer is O(ε).

3.3 Approximations for 1D harmonic potential 27

Similar to the box potential, we can extend these observations accordingly tohigher dimensions These observations will be useful on how we choose the piecewiseuniform mesh when we numerically solve the 2D eigenvalue problem (3.8) in thesubsequent chapters

Chapter 4

Numerical Methods for Singularly

Perturbed Eigenvalue Problems

In this chapter, we apply the gradient flow with discrete normalization to solve thesingularly perturbed nonlinear eigenvalue problem (3.8) under the constraint (3.9).The efficiency and mathematical justification of this numerical method to solve theproblem can be found in [4] The ground state and excited states of BEC under

a box or harmonic potential are difficult to solve due to the presence of boundaryand interior layers In order to overcome this difficulty, we discretize the gradientflow with a new numerical scheme based on a piecewise uniform mesh also known

as ”Shishkin” mesh [49]

The gradient flow with discrete normalization (GFDN) is one of the most populartechniques for dealing with the normalization constraint (3.9) The key idea of themethod is as follows: (i) apply the steepest decent method to an unconstrained

minimization problem; (ii) project the solution back to the unit sphere S For

28

4.2 Discretization with uniform mesh in 1D 29

simplification of notation, we only consider the following GFDN in 1D as extension

of the method to higher dimension is straightforward:

− n+1)

In order to discretize the gradient flow equation (4.1), we divide the spatial interval

Ω = [a, b] into N sub-intervals Then, the mesh size h, time step k, spatial grid points x j and time grid points t n are given by

h = ∆x = b − a

x j = a + jh, j = 0, 1, 2, , N, (4.8)

4.2 Discretization with uniform mesh in 1D 30

ε2

2

φ ∗ j−1 − 2φ ∗

j + φ ∗ j+1

||Φ ∗ || , j = 1, 2, , N − 1,

||Φ ∗ || =

vuu

4.2 Discretization with uniform mesh in 1D 31

where Φn ∈ R N −1, Φ∗ ∈ R N −1 and A is a (N − 1) × (N − 1) symmetric tridiagonal

time t n+1, we can repeat the same procedure to calculate the solution for the nexttime step The energy functional is discretized as

¶2+

4.3 Discretization with piecewise uniform mesh in 1D 32

Here, we used the composite midpoint rule for the first term and the compositetrapezoidal quadrature rule for the second term Both quadrature rules are secondorder accuracy Similarly to energy, the chemical potential is discretized as

¶2+

Based on the new mesh, assuming that φ ∗

∆x j−1 (∆x j−1 + ∆x j) −

2φ ∗ j

∆x j−1 ∆x j +

2φ ∗ j+1

4.3 Discretization with piecewise uniform mesh in 1D 33with boundary conditions

4.3 Discretization with piecewise uniform mesh in 1D 34