Symplectic and multi-symplectic wavelet collocation methodsmath.njnu.edu.cn/meeting/meeting20121215/article/3.pdf · Symplectic and multi-symplectic wavelet collocation methods Background

Symplectic and multi-symplectic wavelet collocation methods

Symplectic and multi-symplectic waveletcollocation methods

ytÚIÆEâÆ§â

Üöµ/ ý§Áu§æµ2012 c 12 £H®Æ¤


Outline

1 Background

2 Symplectic and multi-symplectic methods based on wavelets(1) Symplectic wavelet collocation methods(2) Multi-symplectic wavelet collocation methods(3) Symplectic and multi-symplectic wavelet spectral methods(4) Generalize the methods to solve high dimensional PDEs

3 Conclusions and some unsolved problems


Background

Wavelet-based methods

Wavelets display good localization properties both in spaceand frequency. Wavelet-based methods have superior insolving singular problems.

WaveletsµDaubechies wavelets, Second generation wavelets.

Wavelet-based methodsµwavelet-Galerkin, waveletcollocation.

We use wavelet collocation method based on autocorrelationfunction of Daubechies wavelets.


Symplectic and multi-symplectic methods based on wavelets

(1) Symplectic wavelet collocation methods

Autocorrelation function of Daubechies wavelets

Define the autocorrelation function:

θ(x) = (φ ∗ φ(−·))(x) =

∫φ(x)φ(t − x)dt.

The function θ(x) has nice properties as follows:

Compactly supported:supp(θ(x)) = [−M + 1,M − 1]

Interpolation property:θ(l) =∫φ(x)φ(x − l)dx = δ0,l , l ∈ Z

Derivative property: the odd-order derivative is an oddfunction, and the even-order derivative is an even function.




Autocorrelation function of Daubechies wavelets

Figure: Daubechies scaling function D8 and its autocorrelation function.

Define an interpolation operator on VJ(space step h = 2−J)

uJ(x , t) =N−1∑m=0

u(xm, t)θ(2Jx −m), xm =m

2J, N = 2J , (1)

where VJ is the linear span of θJ,k(x) = 2J/2θ(2Jx − k), k ∈ Z.




Wavelet collocation method

NLW equation ut = v ,

vt = uxx − F ′(u),(2)

with Hamiltonian and momentum

H(u, v) =1

2

∫[v2 + ux

2 + 2F (u)]dx , M = −∫

utuxdx (3)

Making k-times differential:

∂kuJ(x , t)

∂xk|xm =

N−1∑m′=0

u(xm′ , t) · dkθ(2Jx −m′)

dxk|xm = (BkUJ)m,

UJ = (u0, u1, · · · , uN−1)T , Bk is the differentiation matrix.




Bk for D4

In particular for the autocorrelation function of D4, the matrix Bkcan be expressed as

Bk = 2kJ

b0 b−1 b−2 b−3 b3 b2 b1

b1 b0 b−1 b−2 b−3 b3 b2

b2 b1 b0 b−1 b−2 b−3 b3

b3 b2 b1 b0 b−1 b−2 b−3

. . .

b3 b2 b1 b0 b−1 b−2 b−3

b−3 b3 b2 b1 b0 b−1 b−2

b−2 b−3 b3 b2 b1 b0 b−1

b−1 b−2 b−3 b3 b2 b1 b0

N×N

where bl = θ(k)(l) and l is an integer in −3 ≤ l ≤ 3.




The properties of Bk

Theorem 1 For the autocorrelation function ADM, the space differentiationmatrix Bk has the following properties:(1) B2k is symmetric, and B2k+1 is antisymmetric.(2) Bk is a circulant matrix with bandwidth of 2M − 1, and B2kB2k′+1 is aantisymmetric circulant matrix with a bandwidth of 4M − 3. Recursively,B2kB2k′ and B2k+1B2k′+1 are symmetric circulant matrixes with bandwidth of4M − 3.(3) The eigenvalues of circulant matrix Bk are

λj = 2Jk θ(k)(ωj), ωj = −2π

Nj , j = 0, 1, · · · ,N − 1,

where θ(k)(ω) is the Fourier transform of θ(k)(x). And the following equalityholds

FBkF∗ = diag(θ(k)(ω0), θ(k)(ω1), · · · , θ(k)(ωN−1)),

where F ∗ is the Fourier matrix.

(4) B4k+2 is negative semidefinite, and B4k is positive semidefinite.




SWCM (Symplectic wavelet collocation method)

Using B2, we obtain a finite-dimensional Hamiltonian system

Zt = J∇ZH(Z ), (4)

where Z = (UJ ,VJ)T , J =

[0 IN−IN 0

], the Hamiltonian

H(UJ ,VJ) =1

2〈VJ ,VJ〉+ 〈F (UJ), 1〉 − 1

2〈UJ ,B2UJ〉, (5)

where 〈·, ·〉 is the standard inner product.Integrating the semi-discrete system in time by the Euler-centeredscheme, we obtain a symplectic wavelet collocation method:

Un+1J = Un

J + τ ·V nJ + V n+1

J

2,

V n+1J = V n

J + τ · (B2 ·UnJ + Un+1

J

2− F ′(

UnJ + Un+1

J

2)).

(6)




Theoretical analysis

Theorem 2 Suppose u(x , t) ∈ Hs(a, b), s ≥ 52, ∀t ∈ [0,T ], u(x , t) ∈ C 4(a, b),

∀x ∈ [a, b]. Let F (u) be a smooth function. Let Un and UnJ be the exact

solution and numerical solution respectively, and en = Un − UnJ . Then the error

estimate of the SWCM at time T satisfies

‖eL‖ ≤ O(τ 2 + 2−J(s−2)), L =T

τ.

which means second order in time and (s-2) order in space. Here τ is time step.Theorem 3 Using the symplectic wavelet collocation method, we have

|hHLh − hH0

h | = O(τ 2), L =T

τ, h = 2−J .

which means error in discrete Hamiltonian is second order in time.Theorem 4 Assume the initial condition is symmetric, (Un

J ,VnJ ) is the

numerical approximation at tn of SWCM, we have

Mh(UnJ ,V

nJ ) = 〈V n

J ,B1UnJ 〉 = 0.

which means SWCM conserves the discrete momentum exactly.




SWCM for sine-Gordon equation

utt = uxx − sin(u)

with symmetric initial conditions u0(x) = 0, v0(x) = 4γsech(γx).Here sech(x) = 1.0/cosh(x) and γ = 20 is taken.

Figure: Numerical solution, error in Hamiltonian and momentum(τ = 0.0005, N = 3840, T = 200).




Accuracy test for nonlinear Schrodinger equation

iut + uxx + β|u|2u = 0 (7)

with one soliton solution u(x , t) = sech(x − 4t)exp(2i(cx − 32 t)),

where β = 2 and c = 1.Table 1: Comparison of the SWCM-ADM, SFPSM and SFDM (τ = 0.000001, T = 1)

Real ImaginaryMethods N

L∞ error L2 error L∞ error L2 errorCPU(s)

200 0.15 0.19 0.13 0.19 507.36SWCM-AD10400 3.72E-03 4.22E-03 3.37E-03 4.37E-03 1100.80800 2.76E-05 2.67E-05 2.21E-05 2.75E-05 2465.27200 1.47E-02 1.86E-02 1.38E-02 1.94E-02 1000.61SWCM-AD20400 1.59E-05 1.57E-05 1.49E-05 1.57E-05 2080.25800 1.38E-09 1.17E-09 1.36E-09 1.18E-09 4582.67200 4.08E-03 6.89E-03 4.47E-03 7.43E-03 1545.64SWCM-AD30400 1.17E-06 1.71E-06 1.57E-06 1.72E-06 3122.78800 4.42E-11 4.65E-11 4.05E-11 4.66E-11 6588.33200 2.68E-03 6.53E-03 2.56E-03 6.47E-03 5154.53SFPSM400 1.41E-08 3.01E-08 2.08E-08 3.03E-08 28074.52800 2.59E-10 2.58E-10 2.08E-10 2.40E-10 151417.86200 0.21 0.28 0.33 0.32 178.39SFDM400 1.73E-02 1.88E-02 1.52E-02 1.95E-02 359.66800 1.11E-03 1.21E-03 1.00E-03 1.26E-03 765.78

SFPSM: Symplectic Fourier pseudospectral method; SFDM: Symplectic finite difference method




SWCM for nonlinear Schrodinger equation

When β = 2 · K 2, it will produce a bound state of K solitons.Here β = 2 · 52 is taken.

Figure: Five soliton solutions of NLS equation (τ = 0.00002, N = 960).




SWCM for nonlinear Schrodinger equation

Long time simulation:

Figure: The evolution of the five soliton solutions over t ∈ [0, 11.8] andthe corresponding contour picture.



(2) Multi-symplectic wavelet collocation methods

Multi-symplectic PDEs

Many PDEs can be written as a multi-symplectic system (Bridges,2000)

Mzt + Kzx = ∇zS(z), z ∈ Rd , (x , t) ∈ R2, (8)

where M and K are two skew-symmetric matrices and S : Rd → Ris a scalar-valued smooth function. The above system has amulti-symplectic conservation law

∂tω + ∂xκ = 0, (9)

where ω and κ are the pre-symplectic forms,

ω =1

2dz ∧Mdz , κ =

1

2dz ∧ Kdz .




MSWCM (Multi-symplectic wavelet collocation method)

Applying the wavelet collocation method for spatial discretizationand the implicit midpoint scheme for time integration, we obtainMSWCM for multi-symplectic PDEs:

Mzn+1l − znl

τ+ K

l+(M−1)∑m=l−(M−1)

(B1)lmzn+1/2m = ∇Sz(z

n+1/2l ). (10)

Theorem 5. The MSWCM has N full-discrete multi-symplectic conservationlaws

ωn+1l − ωn

l

τ+

l+(M−1)∑m=l−(M−1)

(B1)lmκn+1/2lm = 0, l = 0, 1, · · · ,N − 1,

where N = L · 2J , ωnl = 1

2(dznl ∧Mdznl ), κ

n+1/2lm = dz

n+1/2l ∧ Kdz

n+1/2m .




MSWCM for NLS equation

Consider NLS equation:

iψt + ψxx + a|ψ|2ψ = 0 (11)

By introducing a pair of conjugate momenta v = px ,w = qx , weobtain the following multi-symplecitc PDEs,

qt − vx = a(p2 + q2)p,

− pt − wx = a(p2 + q2)q,

px = v ,

qx = w ,

(12)

with state variable z = (p, q, v ,w)T and Hamiltonian function

S(z) =1

2(v2 + w2 +

a

2(p2 + q2)2),




MSWCM for NLS equation

Applying MSWCM for the NLS equation (11), we obtain

Qn+1 − Qn

τ− B1V

1/2 − a((P1/2)2 + (Q1/2)2) • P1/2 = 0,

Pn+1 − Pn

τ+ B1W

1/2 + a((Q1/2)2 + (P1/2)2) • Q1/2 = 0,

B1P1/2 = V 1/2,

B1Q1/2 = W 1/2.

By eliminating the values V and W , the scheme is equivalent toPn+1 = Pn − τ(B2

1Q1/2 + a((P1/2)2 + (Q1/2)2) • Q1/2),

Qn+1 = Qn + τ(B21P

1/2 + a((Q1/2)2 + (P1/2)2) • P1/2).(13)




Accuracy test of MSWCM for NLS equation

Table: Numerical errors and CPU times of MSWCM


L∞error L2error L∞error L2errorCPU(s)

200 0.20 0.27 0.16 0.25 775.02400 1.11E-03 1.16E-03 9.38E-04 1.18E-03 1621.55AD10

800 1.99E-06 1.72E-06 1.53E-06 1.75E-06 3405.66200 4.68E-02 9.79E-02 5.39E-02 9.84E-02 1654.77400 1.85E-05 5.40E-05 2.09E-05 5.41E-05 3818.88AD20

800 6.86E-10 5.80E-10 6.69E-10 5.84E-10 7765.75200 2.11E-02 5.87E-02 2.35E-02 5.97E-02 2595.00400 6.17E-06 2.59E-05 6.29E-06 2.59E-05 5947.86AD30

800 4.44E-11 5.17E-11 4.82E-11 5.20E-11 12075.13




MSWCM for Camas-Holm equation

ut − uxxt + 3uux − 2uxuxx − uuxxx = 0 (14)

with the initial condition u(x , 0) = φ1(x) + φ2(x)

φi (x) =

ci

cosh(a/2)cosh(x − xi ), |x − x0| ≤ a/2,

ci

cosh(a/2)cosh(a − (x − xi )), |x − x0| > a/2,

i = 1, 2. (15)

Figure: Interaction of two peaked traveling waves, c1 = 3, c2 = 1,x1 = 4.5, x2 = 12.5, a = 25 (τ = 0.0001, N = 1600).




Ito-type coupled KdV (ItcKdV) equation

ut + αuux + βvvx + γuxxx = 0,

vt + β(uv)x = 0,(16)

It can be written as a multi-symplectic Hamiltonian PDE with

z = [ϕ,ψ, u, v ,w , p, q]T, S(z) = −α6u3− β

2uv2− pu− qv − γ

2w2,

M =

0 0 − 12

0 0 0 0

0 0 0 − 12

0 0 012

0 0 0 0 0 0

0 12

0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

, K =

0 0 0 0 0 1 00 0 0 0 0 0 10 0 0 0 γ 0 00 0 0 0 0 0 00 0 −γ 0 0 0 0−1 0 0 0 0 0 00 −1 0 0 0 0 0

.




MSWCM for the ItcKdV equation

The initial condition is u(x , 0) = cos(x), v(x , 0) = cos(x).

Figure: MSWCM for the ItcKdV equation (τ = 0.0005, N = 400).




Ito-type coupled KdV (ItcKdV) equation

Figure: Errors in global energy and momentum



(3) Symplectic and multi-symplectic wavelet spectral methods

symplectic and multi-symplectic wavelet spectral methods

Using Fourier matrix, the wavelet collocation differentiation matrixcan be transformed to diagonal matrix, which is called waveletspectral matrix. The matrix is similar with the Fourier spectralmatrix. The spatial discretization based on the matrix is calledwavelet spectral methods.Combining with splitting scheme, we construct explicit splittingsymplectic and multi-symplectic wavelet spectral methods(ES-SWSM and ES-MSWSM). FFT can be used to reduce CPUcosts.




Properties of wavelet spectral matrix

The kth order differential matrix of wavelet collocation method Bk

can be transformed to diagonal form Θk

F−1BkF = diag(θ(k)1 , θ

(k)2 , · · · , θ(k)

N ) = Θk ,

where F is the Fourier matrix, θ(k)l = 2Jk θ(k)(ωl) are the

eigenvalues of the matrix Bk .Theorem 6. The wavelet spectral matrix Θk has following properties:(1) λN−j = λj , j = 1, 2, · · · , N

2;

(2) when k is an odd, λ0 = λ N2

= 0, λN−j = −λj , j = 1, 2, · · · , N2− 1, λj is

pure imaginary;(3) when k = 4m + 2, λ0 = 0, λ N

2< 0; λN−j = λj , j = 1, 2, · · · , N

2− 1, λj is

real and λj < 0;(4) when k = 4m, λ0 = 0, λ N

2> 0; λN−j = λj , j = 1, 2, · · · , N

2− 1, λj is real

and λj > 0.




Comparison with Fourier spectral matrix

Consider computational area [0, 1] and grid number N = 64, theelements of the wavelet spectral matrix approximate that of theFourier spectral matrix, as shown in the following figure

Figure: Elements of the wavelet spectral matrix and the Fourier spectralmatrix




ES-SWSM for NLS equation

We split the NLS equation into the linear subproblem andnonlinear subproblem:

iψt = Lψ = −ψxx ,

iψt = Nψ = −β|ψ|2ψ.(17)

Both of the above subproblems can be written as Hamiltoniansystems and multi-symplectic systems.





For the nonlinear subproblem, we first discrete it in space and get afinite-dimensional Hamiltonian system

idψl

dt= −β|ψl |2ψl , l = 1, 2, · · · ,N,

which can be solved exactly and the solution is ψn+1l = e iβ|ψ

nl |

2τψnl .

The linear problem is converted to Fourier modes with wavenumbers ψl , then we need to solve the following ODE:

id

dtψl = −θ(2)

l ψl , l = 1, 2, · · · ,N,

which can be solved exactly in O(Nlog2N) operations using the

FFTs and the solution is ψn+1l = e iθ

(2)l τ ψn

l





We choose the second-order Strang splitting method

ψ(x , t + τ) = e−iτN/2e−iτLe−iτN/2ψ(x , t), (18)

to compose the solutions of the subproblems and obtain an explicitsplitting symplectic wavelet collocation method (ES-SWSM) forthe NLS equation:

ψ(1)l = e iβ|ψ

nl |

2τ/2ψnl ,

ψl(1)

= FFT(ψ(1)l ), ψ

(2)l = e iθ

(2)l τ ψ

(1)l , ψ

(2)l = IFFT(ψl

(2))

ψn+1l = e iβ|ψ

(2)l |

2τ/2ψ(2)l ,

(19)Similarly, we can obtain explicit splitting multi-symplectic waveletspectral methods (ES-MSWSM).




Accuracy test for ES-SWSM

Table: Numerical errors and CPU time of ES-SWSM-ADM


L∞ error L2 error L∞ error L2 errorCPU(s)

200 0.15 0.19 0.13 0.19 176.83400 3.72E-03 4.22E-03 3.37E-03 4.37E-03 352.39AD10

800 2.76E-05 2.67E-05 2.21E-05 2.75E-05 705.61200 1.47E-02 1.86E-02 1.38E-02 1.94E-02 177.25400 1.59E-05 1.57E-05 1.49E-05 1.57E-05 352.61AD20

800 1.31E-09 1.17E-09 1.35E-09 1.18E-09 705.89200 4.08E-03 6.89E-03 4.47E-03 7.43E-03 177.80400 1.17E-06 1.71E-06 1.57E-06 1.72E-06 352.13AD30

800 1.12E-10 1.61E-10 9.43E-10 1.62E-10 706.19





Consider β = 2 · 72. Take τ = 0.000002 and N = 3840.

Figure: Seven soliton solutions of NLS equation.



(4) Generalize the methods to solve high dimensional PDEs

SWCM and MSWCM for high dimensional PDEs

The basis for two-dimensional calculations is constructed bymaking a tensor product of two one-dimensional bases

θj ,l ,l ′(x , y) = θjl(x)θjl ′(y),

where θjl(x) = 2j/2θ(2jx − l) and θjl ′(y) = 2j/2θ(2jy − l ′).Define an interpolation operator on Vj ⊗ Vj as

Iju(x , y) = 2−j∑l ,l ′

u(xj ,l ,l ′)θj ,l ,l ′(x , y) =∑l ,l ′

u(xj ,l ,l ′)θ(2jx−l)θ(2jy−l ′),

here xj ,l ,l ′ = (xj ,l , yj ,l ′) = (2−j l , 2−j l ′) and ⊗ means Kroneckerinner product.SWCM and MSWCM: For Hamiltonian system andmulti-symplectic PDEs, we use wavelet collocation method inspace and the implicit midpoint rule in time.




SWCM and MSWCM for 2D NLS equation

2D NLS equationµiut + uxx + uyy + β|u|2u = 0, i =√−1§

SWCMµDtP

n = −AQ1/2 − β((P1/2)2 + (Q1/2)2) • Q1/2,

DtQn = AP1/2 + β((P1/2)2 + (Q1/2)2) • P1/2,

(20)

where A = B2 ⊗ IN + IN ⊗ B2§DtPn = Pn+1−Pn

τ §τ is the time

step§P1/2 = 12 (Pn + Pn+1)§(P1/2)2 = P1/2 • P1/2§P • Q =

(p1,1q1,1, · · · , pN,1qN,1, · · · , p1,Nq1,N , · · · , pN,NqN,N)T.




Invariants preserving

Theorem 7.Under periodic boundary conditions§SWCM (20) hasthe discrete total norm conservation law

N n =N∑

k=1

N∑l=1

((pnk,l)2 + (qnk,l)

2) =N∑

k=1

N∑l=1

((p0k,l)

2 + (q0k,l)

2) = N 0.

Theorem 8. SWCM (20)The error in the discrete Hamiltonian attime T for the SWCM (20) is

(HS − H0)hxhy = O(τ3), S =T

τ.

HS =1

2(Pn)TA(Pn)+

1

2(Qn)TA(Qn)+

β

4

N∑k=1

N∑l=1

((pnk,l)2+(qnk,l)

2)2.




SWCM and MSWCM for solving plane wave solution

Table: Numerical errors and CPU times of SWCM and MSWCM

Real ImaginaryMethods N × NL∞ error L2 error L∞ error L2 error

Nitera CPU(s)

16× 16 6.63E-03 2.95E-02 6.63E-03 2.95E-02 3.00 6.27SWCM-AD6 32× 32 4.29E-04 1.91E-03 4.29E-04 1.91E-03 2.00 16.39

64× 64 2.71E-05 1.20E-04 2.71E-05 1.20E-04 2.23 72.8416× 16 1.67E-04 7.42E-04 1.67E-04 7.42E-04 2.00 5.50

SWCM-AD8 32× 32 2.73E-06 1.21E-05 2.73E-06 1.21E-05 2.00 21.2864× 64 4.32E-08 1.92E-07 4.32E-08 1.92E-07 2.00 85.0616× 16 5.34E-06 2.30E-05 5.38E-06 2.30E-05 2.00 6.67

SWCM-AD10 32× 32 2.17E-08 9.65E-08 2.17E-08 9.65E-08 2.00 26.2264× 64 8.64E-11 3.84E-10 8.64E-11 3.84E-10 2.00 104.7516× 16 1.11E-04 4.94E-04 1.11E-04 4.94E-04 2.00 5.03

MSWCM-AD6 32× 32 1.78E-06 7.91E-06 1.78E-06 7.91E-06 2.00 25.1464× 64 2.80E-08 1.24E-07 2.80E-08 1.24E-07 2.00 101.5516× 16 4.13E-06 1.83E-05 4.13E-06 1.83E-05 2.00 5.02



SPSM 32× 32 1.02E-12 2.84E-12 1.34E-12 4.16E-12 2.00 12.6164× 64 3.31E-13 1.23E-12 3.29E-13 1.13E-12 2.00 55.86




ES-SWCM and ES-MSWCM for solving plane wavesolution

Table: Numerical errors and CPU times of ES-SWCM and ES-MSWCM

Real ImaginaryMethods N × NL∞ error L2 error L∞ error L2 error

CPU(s)

16× 16 6.63E-03 2.95E-02 6.63E-03 2.95E-02 2.34ES-SWSM-AD6 32× 32 4.29E-04 1.91E-03 4.29E-04 1.91E-03 9.22

64× 64 2.71E-05 1.20E-04 2.71E-05 1.20E-04 37.1916× 16 1.67E-04 7.42E-04 1.67E-04 7.42E-04 2.34

ES-SWSM-AD8 32× 32 2.73E-06 1.21E-05 2.73E-06 1.21E-05 9.0264× 64 4.32E-08 1.92E-07 4.32E-08 1.92E-07 36.5016× 16 5.24E-06 2.32E-05 5.24E-06 2.32E-05 2.33

ES-SWSM-AD10 32× 32 2.17E-08 9.65E-08 2.17E-08 9.65E-08 9.1364× 64 8.65E-11 3.84E-10 8.65E-11 3.84E-10 37.0316× 16 1.11E-04 4.94E-04 1.11E-04 4.94E-04 2.44

ES-MSWSM-AD6 32× 32 1.78E-06 7.91E-06 1.78E-06 7.91E-06 9.4564× 64 2.80E-08 1.24E-07 2.80E-08 1.24E-07 38.0916× 16 4.13E-06 1.83E-05 4.13E-06 1.83E-05 2.44



ES-SPSM 32× 32 3.44E-12 1.40E-11 3.91E-12 1.44E-11 9.3664× 64 2.57E-12 1.11E-11 2.57E-12 1.11E-11 38.53




SWCM and MSWCM for solving singular problems

Initial conditionµu(x , y , 0) = (1 + sin x)(2 + sin y).

Figure: Singular solution of 2D NLS equation (τ = 0.00001, N = 128).




SWCM and MSWCM for 2D TDLS equation

iψt +1

2ψxx +

1

2ψyy +

1√x2 + y2

ψ − ε(t)xψ = 0 (21)

with the laser field profile ε(t) = ε0f (t) cos(wt), where ε0 is thepeak amplitude of the laser field, ω is the frequency, 2π/ω is theoptical period of the laser field, and f (t) describes the temporalshape of the pulse,

f (t) =

sin(π

2· t

T0), 0 < t ≤ T0,

1, t > T0.(22)

Here, we choose parameters ω = 2, ε0 = 1, T0 = 4.




SWCM and MSWCM for 2D TDLS equation

The initial condition is set to be the following ground statewavefunction

ψ(x , y , 0) = 2

√2

π· e−2

√x2+y2

. (23)

Figure: The laser-atom interaction obtained by using SWCM(τ = 0.0005, N = 800).




MSWCM for 3D Maxwell’s equations

Consider 3D Maxwell’s equations∂E

∂t=

1

ε∇×H,

∂H

∂t= − 1

µ∇× E,

(24)

where E= (Ex ,Ey ,Ez)T is the electric field and H = (Hx ,Hy ,Hz)T

is the magnetic field.The Maxwell’s equations (24) have following two energyconservation lawsµ

Energy I :

∫Ω

(ε|E(x , t)|2 + µ|H(x , t)|2)dΩ = C1, (25)

Energy II :

∫Ω

(ε∣∣∣∂E(x , t)

∂t

∣∣∣2 + µ∣∣∣∂H(x , t)

∂t

∣∣∣2)dΩ = C2, (26)

where C1 and C2 are constants.




MSWCM for 3D Maxwell’s equations

MSWCM for 3D Maxwell’s equations:

Exn+1 − Ex

n

τ=

1

ε(A2Hz

n+1/2 − A3Hyn+1/2),

Eyn+1 − Ey

n

τ=

1

ε(A3Hx

n+1/2 − A1Hzn+1/2),

Ezn+1 − Ez

n

τ=

1

ε(A1Hy

n+1/2 − A2Hxn+1/2),

Hxn+1 −Hx

n

τ= − 1

µ(A2Ez

n+1/2 − A3Eyn+1/2),

Hyn+1 −Hy

n

τ= − 1

µ(A3Ex

n+1/2 − A1Ezn+1/2),

Hzn+1 −Hz

n

τ= − 1

µ(A1Ey

n+1/2 − A2Exn+1/2),

(27)

where τ is the time step, Exn+1/2 = 1

2(Ex

n+1 + Exn), Ey

n+1/2 = 12(Ey

n+1 + Eyn).

A1 = (Bx1 ⊗ INy ⊗ INz )§A2 = (INx ⊗ By

1 ⊗ INz )§A3 = (INx ⊗ INy ⊗ Bz1 ) are all

anti-symmetric matrices.




Stability, dispersion and energy-preserving property analysis

Theorem 9.Under periodic boundary conditions, the MSWCM (27) isunconditionally stable.Continuous dispersion relations:

ω1,2 = 0, ω3,4 =√

(k2x + k2

y + k2z )/εµ, ω5,6 = −

√(k2

x + k2y + k2

z )/εµ.

(28)the numerical dispersion relations of MSWCMµ

ω1,2 = 0, ω3,4 =tan−1(τ

√(|d1|2 + |d2|2 + |d3|2)/εµ)

τ,

ω5,6 = −tan−1(τ

√(|d1|2 + |d2|2 + |d3|2)/εµ)

τ.

(29)

Theorem 10.Under periodic boundary conditions, the MSWCM (27) conservesthe discrete total energy conservation laws (25) and (26), that is,

ε‖En‖2 + µ‖Hn‖2 = C3, (30)

ε‖DtEn‖2 + µ‖DtH

n‖2 = C4, (31)

where Dt is the difference operator, C1 and C2 are constants.




MSWCM for 2D Maxwell’s equation

Consider the 2D Maxwell’s equations(TE Case)

∂Ex

∂t=

1

ε

∂Hz

∂y,

∂Ey

∂t= −1

ε

∂Hz

∂x,

∂Hz

∂t=

1

µ(∂Ex

∂y− ∂Ey

∂x).

(32)

Ex =ky

ε√µω

cos(ωπt) cos(kxπx) sin(kyπy), Ey = − kxε√µω

cos(ωπt) sin(kxπx) cos(kyπy),Hz =

1µ

sin(ωπt) cos(kxπx) cos(kyπy), ω2 = 1µε

(k2x + k2

y ).

Figure: Ex at t = 10 and errors in energy I and II (τ = 0.0001, N = 64).




Accuracy test of MSWCM for 2D Maxwell’s equation

Consider the 2D Maxwell’s equations (TM case)

∂Ez

∂t=

1

ε(∂Hy

∂x− ∂Hx

∂y),

∂Hx

∂t= − 1

µ

∂Ez

∂y,

∂Hy

∂t=

1

µ

∂Ez

∂x.

(33)

HxHyEz

=

−βα1

exp(cos(αx + βy + t)), α = cos(0.3π), β = sin(0.3π). (34)

Figure: L∞ errors of Hx and CPU time.




MSWCM for 3D Maxwell’s equation

Consider the following periodic solution:Ex = cos(2π(x + y + z)− 2

√3πt), Hx =

√3Ex ,

Ey = −2Ex , Hy = 0,

Ez = Ex , Hz = −√

3Ex .

(35)

The computational domain is [0, 1]× [0, 1]× [0, 1] with periodic boundaryconditions.

Figure: L∞ errors of Ex and CPU time (t=1).




Our Paper

[1] Zhu Huajun, Tang Lingyan, Song Songhe, Wang Desheng, Tang Yifa.Symplectic wavelet collocation method for Hamiltonian wave equation[J], J.Comput. Phys., 2010, 229: 2550-2572.[2] Zhu Huajun, Song Songhe, Tang Yifa. Multi-symplectic wavelet collocationmethod for the nonlinear Schrodinger and Camassa-Holm equations[J],Comput. Phys. Comm., 2011, 182:616627.[3] Zhu Huajun, Chen Yaming, Song Songhe, Hu Huayu. Symplectic andmulti-symplectic wavelet collocation methods for two-dimensional Schrodingerequations[J], Appl. Numer. Math., 2011, 61:308321.[4] Zhu Huajun, Song Songhe, Chen Yaming, Multi-symplectic waveletcollocation method for Maxwell’s equations[J], Adv. Appl. Math. Mech., 2011,3:663-688.

[5] Chen Yaming, Song Songhe, Zhu Huajun, The multi-symplectic Fourier

pseudospectral method for solving two-dimensional Hamiltonian PDEs[J], J.

Comput. Appl. Math., 2011, 236:1354-1369.




Our Paper

[6] Chen Yaming, Zhu Huajun, Song Songhe, Multi-Symplectic splittingmethod for two-Dimensional nonlinear Schrodinger equation[J], Comm. Theor.Phys., 2011, 56:617-622.[7] Chen Yaming, Zhu Huajun, Song Songhe, Multi-symplectic splitting methodfor the coupled nonlinear Schrodinger equation[J], Comput. Phys. Comm.,2010, 181:1231-1241.[8] Chen Yaming, Song Songhe, Zhu Huajun, Multi-symplectic methods for theIto-type coupled KdV equation[J], Appl. Math. Comput., 2012, 218:5552-5561.[9] Qian Xu, Song Songhe, Gao Er, Li Weibin, Explicit multi-symplectic methodfor the Zakharov Kuznetsov equation[J], Chin. Phys. B, 2012, 21(7):070206.

[10] Qian Xu, Chen Yaming, Gao Er, Song Songhe, Multi-symplectic wavelet

splitting method for the strongly coupled Schrodinger system[J], Chin. Phys.

B, 2012, 21(12):120202.


Conclusions and some unsolved problems

Conclusions

SWCM and MSWCM have some merits:

high accuracy,

less computation,

can capture singularities efficiently,

good invariant conserving properties



Some unsolved problems

General boundary conditions

Use other wavelets

Combine with unstructured meshes and adaptive methods

Make more application in physics

Time discretizations



Thank you!

Documents

Symplectic and multi-symplectic wavelet collocation methodsmath.njnu.edu.cn/meeting/meeting20121215/article/3.pdf · Symplectic and multi-symplectic wavelet collocation methods Background