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Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic waveletcollocation methods
ytÚIÆEâƧâ
Üöµ/ ý§Áu§æµ2012 c 12 £H®Æ¤
Symplectic and multi-symplectic wavelet collocation methods
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
Symplectic and multi-symplectic wavelet collocation methods
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 wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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)
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(1) Symplectic wavelet collocation methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(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 .
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
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 .
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
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),
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
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)
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
Accuracy test of MSWCM for NLS equation
Table: Numerical errors and CPU times of MSWCM
Real ImaginaryMethods N
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
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
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
.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(2) Multi-symplectic wavelet collocation methods
Ito-type coupled KdV (ItcKdV) equation
Figure: Errors in global energy and momentum
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
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
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
ES-SWSM for NLS equation
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
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
ES-SWSM for NLS equation
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
Accuracy test for ES-SWSM
Table: Numerical errors and CPU time of ES-SWSM-ADM
Real ImaginaryMethods N
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
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(3) Symplectic and multi-symplectic wavelet spectral methods
ES-SWSM for NLS equation
Consider β = 2 · 72. Take τ = 0.000002 and N = 3840.
Figure: Seven soliton solutions of NLS equation.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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
MSWCM-AD8 32× 32 1.67E-08 7.43E-08 1.67E-08 7.43E-08 2.00 35.7564× 64 6.61E-11 2.93E-10 6.61E-11 2.93E-10 2.00 142.9216× 16 6.69E-06 1.57E-05 7.36E-06 1.57E-05 3.00 7.52
MSWCM-AD10 32× 32 1.59E-10 7.07E-10 1.59E-10 7.08E-10 2.00 37.1464× 64 1.27E-13 4.59E-13 1.28E-13 4.61E-13 1.79 163.9416× 16 8.28E-15 1.11E-14 7.38E-15 8.90E-15 1.00 1.91
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
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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-MSWSM-AD8 32× 32 1.67E-08 7.43E-08 1.67E-08 7.43E-08 9.1364× 64 6.78E-11 3.01E-10 6.78E-11 3.01E-10 37.6116× 16 1.56E-07 6.91E-07 1.56E-07 6.91E-07 2.41
ES-MSWSM-AD10 32× 32 1.62E-10 7.20E-10 1.62E-10 7.20E-10 9.4464× 64 2.44E-12 1.04E-11 2.44E-12 1.04E-11 38.0616× 16 1.39E-12 5.69E-12 1.39E-12 5.69E-12 2.39
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
Symplectic and multi-symplectic wavelet collocation methods
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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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
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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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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).
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Symplectic and multi-symplectic methods based on wavelets
(4) Generalize the methods to solve high dimensional PDEs
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.
Symplectic and multi-symplectic wavelet collocation methods
Conclusions and some unsolved problems
Conclusions
SWCM and MSWCM have some merits:
high accuracy,
less computation,
can capture singularities efficiently,
good invariant conserving properties
Symplectic and multi-symplectic wavelet collocation methods
Conclusions and some unsolved problems
Some unsolved problems
General boundary conditions
Use other wavelets
Combine with unstructured meshes and adaptive methods
Make more application in physics
Time discretizations
Symplectic and multi-symplectic wavelet collocation methods
Conclusions and some unsolved problems
Thank you!