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Gaussian Beam Methods for the Schr odinger

Equation in the Semi-classical Regime:Lagrangian and Eulerian Formulations

Shi Jin , Hao Wu , and Xu Yang

October 12, 2008

Abstract

The solution to the Schr odinger equation is highly oscillatory when

the rescaled Planck constant is small in the semiclassical regime. Adirect numerical simulation requires the mesh size to be O(). TheGaussian beam method is an efficient way to solve the high frequencywave equations asymptotically, outperforming geometric optics methodin that the Gaussian beam method is accurate even at caustics.

In this paper, we solve the Schr odinger equation using both theLagrangian and Eulerian formulations of the Gaussian beam methods.A new Eulerian Gaussian beam method is developed using the levelset method based only on solving the (complex-valued) homogeneousLiouville equations. A major contribution here is that we are ableto construct the Hessian matrices of the beams by using the level-setfunctions rst derivatives. This greatly reduces the computationalcost in computing the Hessian of the phase function in the Eulerian

framework, yielding an Eulerian Gaussian beam method with compu-tational complexity comparable to that of the geometric optics butwith a much better accuracy around caustics.

We verify through several numerical experiments that our Gaussianbeam solutions are good approximations to the Schr odinger solutionseven at caustics. We also numerically study the optimal relation be-tween the number of beams and the rescaled Planck constant in theGaussian beam summation.

This work was partially supported by NSF grant No. DMS-0608720, NSF FRG grantDMS-0757285, AFOSR grant No. FA9550-04-1-0143, NSFC Projects 10676017, the Na-tional Basic Research Program of China under the grant 2005CB321701. SJ was alsosupported by a Van Vleck Distinguished Research Prize from University of Wisconsin-Madison.

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA(jin@math.wisc.edu)

Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, China(hwu04@mails.tsinghua.edu.cn)

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA(xyang@math.wisc.edu)

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Key words. Schrodinger equation, Gaussian beam method, Liouvilleequation

AMS subject classications. 81Q20, 65M99

1 IntroductionWe are interested in the Gaussian beam methods for the numerical approx-imation of the Schrodinger equation:

i

t+

2

2 V (x ) = 0 , x R n , (1.1)

with the WKB initial data,

(0, x ) = A0(x )eiS 0 (x )/ . (1.2)

Here (x , t ) is the wave function, is the re-scaled Plank constant, andV (x ) is the smooth potential. The physical observables can be dened interms of (x , t ):

position density n = ||2 , (1.3)density ux J =

2i

( ) , (1.4)

kinetic energy E =2

2 ||2 . (1.5)

The wave function (x , t ) and the related physical observables becomeoscillatory of wave length O() when is small in the so-called semiclas-sical regime. A mesh size of O() is required when using the time-splittingspectral method [1] to simulate (1.1)-(1.2) directly. The mesh size (and thetime step as well) becomes even worse, since they need to be as small aso(), if nite difference methods are used [21, 22]. The mesh and time steprestrictions of these methods make the computation of (1.1)-(1.2) extremelyexpensive, especially in high dimensions.

One alternative efficient approach is to solve the equation asymptoticallyby the classical WKB method. Applying the ansatz

(t, x ) = A(t, x )eiS (t, x )/ (1.6)

to (1.1), one obtains the eikonal equation for the phase S (t, x ) and the

transport equation for the amplitude (t, x ) = |A(t, x )|2([34]):

S t +12 |S |

2 + V (x ) = 0 , (1.7)

t + (S ) = 0 . (1.8)

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Since the eikonal equation is of the Hamilton-Jacobi type, the solution be-comes singular after caustic formation. Beyond caustics, the correct semi-classical limit of the Schrodinger equation becomes multivalued. The mul-tivalued solution can be computed by the ray tracing methods [6, 2, 3],wave front methods [32, 9], moment methods [11, 14] and level set methods

[16, 5, 15, 17]. We also refer the readers to the review paper on computa-tional high frequency waves [8].

A problem with the WKB, or geometric optics based methods is thatthe asymptotic solution is invalid at caustics, since the amplitude (x , t )blows up there. None of the aforementioned methods could give accuratesolutions near caustics. But on the other hand, accurately computing thesolutions around the caustics is important in many applications, for example,in seismic imaging [12, 13].

The Gaussian beam method, developed for the high frequency linearwaves, is an efficient approximate method that allows accurate computationof the wave amplitude around the caustics [28]. Similar to the ray tracing

method, the Gaussian beam solution also has a WKB form of (1.6). The raydetermined by (1.7)-(1.8) is the center of the Gaussian beam. The differencelies in that the Gaussian beam allows the phase function S (x , t ) to be com-plex off its center, and the imaginary part of S (x , t ) is chosen so that thesolution decays exponentially away from the center. The Lagrangian formu-lation consists of the ray tracing equations determined by (1.7)-(1.8), whichdescribe the trajectory of the beam center, and the Riccati equation, whichdescribes the Hessian of the phase S (x , t ). The validity of this constructionat caustics is analyzed by Ralston in [29].

Lagrangian numerical methods of Gaussian beams were usually devel-oped based on the Taylor expansion and superposition principle. The accu-racy of the beam off the central ray is determined by the truncation error of Taylor expansion, and the approximate solution is given by a sum of all thebeams. See [4, 13, 25]. The accuracy of the Taylor expansion was studiedby Motamed and Runborg [24], and Tanushev [30] developed and analyzedhigher order Gaussian beams giving better accuracy of the approximations.

Compared to the Lagrangian methods based on ray tracing, which loseaccuracy when the ray diverges and need re-interpolation to maintain the nu-merical accuracy which can be very complicated, the Eulerian methods basedon solving PDEs on xed grids have the advantage of a uniform accuracy.Here we remark that the Lagrangian methods based on wave front construc-tions [32, 9, 25] do not suffer from the issue of the ray divergence. Recently,motivated by the work of Tanushev, Qian and Ralston [31], Leung, Qianand Burridge [19] designed an Eulerian Gaussian beam summation methodfor the Helmholtz equations. In their formulation for n space dimension,the Hessian matrix of the phase was solved by 2 n2 complex -valued inho-

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mogeneous Liouville equations. 1 They also introduced the semi-Lagrangianmethod to numerically evaluate the singular Eulerian beam summation in-tegral.

In this paper we systematically study the Gaussian beam method forsolving the Schrodinger equation in the semiclassical regime using both the

Lagrangian and the Eulerian formulations. The Lagrangian formulationfollows the classical work for linear hyperbolic equations [29]. The maincontribution of this paper is a new Eulerian Gaussian beam method using thelevel set function. To compute the velocity and the Hessian of the phase, weonly use n complex valued homogeneous level-set Liouville equation, ratherthan 2 n2 inhomogeneous complex valued Liouville equations for the Hessianin addition to n real-valued homogeneous Liouville equations to the velocityas in [19]. The Hessian of the phase is evaluated from the rst derivativesof the level set functions. This new formulation signicantly reduces thenumber of Liouville equations to be used to construct the Hessian of thephase. As a matter of fact, the computational method for the (complex-

valued) phase and amplitude is not much different from the level set methodused for geometric optics computation as in [5, 16, 15]. In addition we alsoevaluate the Gaussian beam summation integral using the semi-Lagrangianmethod of [19] only near caustics, thus maintain largely the accuracy of the Eulerian method. This new method will be tested for its accuracy andefficiency by comparing with the solution of the Schr odinger equation (1.1)-(1.2).

Note that when there exists many caustics which make the velocity con-tours rather complicated, the implementation of this local semi-Lagrangianmethod is complex and hard. In these cases we refer to a discretized deltafunction method [33] for numerically computing the Gaussian beam sum-mation integral which will be further studied in a forthcoming work.

The paper is organized as follows. In Section 2, we introduce the La-grangian Gaussian beam formulation and summarize its properties. In Sec-tion 3, we give the detailed derivation of the new level set formulations forthe Eulerian Gaussian beam method; we also discuss how to implement thesummation of the Eulerian Gaussian beams, with an analysis of the compu-tational complexity of this new method in the end. The numerical examplesare given in Section 4 to test the accuracy and efficiency of the Gaussianbeam methods. We make some conclusive remarks in Section 5.

1 After we completed this manuscript, we received reference [18], where the EulerianGaussian beam method of [19] was extended to the semiclassical Schr odinger equationstill using the same formulations for the Hessian of the phase as in [19].

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2 The Lagrangian formulation

In this section, we adopt the Gaussian beam approximation to the Schr odingerequation (1.1). Let

la (t, x , y 0) = A(t, y )e

iT (t, x ,y )/ , (2.1)

where y = y (t, y 0) and T (t, x , y ) is given by the Taylor expansion

T (t, x , y ) = S (t, y )+ p(t, y )(x y )+12

(x y )M (t, y )(x y )+ O(|x y |3),(2.2)

in which (x y ) is the transpose of ( x y ). Here S R , p R n , AC ,M C n n . The imaginary part of M will be chosen so that (2.1) has aGaussian beam prole. We call (2.1) as the beam-shaped ansatz.The difference from the WKB ansatz is that, in (2.1)-(2.2), a free pa-

rameter y is used to control the domain where the WKB analysis is applied.Actually y plays the role of the beam center, chosen as

dydt

= p (t, y ), y (0) = y 0. (2.3)

2.1 Formulation for the beam-shaped ansatz

We rst derive the formulation for the beam-shaped ansatz (2.1). Plugging(2.1) into (1.1) and equating the rst two leading orders of , one obtainsthe equations for T and A

T t

+ yt y T +

12 |x T |

2 + V = 0 , (2.4)

At

+ yt y A +

12

(x T )A = 0 . (2.5)

Taking the rst and second derivatives with respect to x in (2.4) gives

(x T )t

+ yt yx T + x T

2x T + x V = 0 , (2.6)

(2x T )

t+

yt yxx T +

2x T

2 + x T 3x T + 2x V = 0 . (2.7)Using (2.2) and evaluating (2.4)-(2.7) at x = y yield (after ignoring theO(|x y |3) term)

S t

+ yt (y S p) +

12 | p |

2 + V = 0 , (2.8)

At +

yt y A +

12 Tr( M ) A = 0 , (2.9)

pt

+ yt (y p M ) + p M + y V = 0 , (2.10)M t

+ yt y M + M

2 + 2y V = 0 , (2.11)

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where Tr( M ) is the trace of the matrix M .Considering the y -trajectory given by (2.3), then (2.8)-(2.11) can be

written as a set of ODEs:

dydt

= p , (2.12)

d pdt

= y V, (2.13)dM dt

= M 2 2y V, (2.14)dS dt

=12 | p|

2 V, (2.15)dAdt

= 12

Tr( M ) A. (2.16)

Here y = y (t, y 0), p = p (t, y (t, y 0)) , V = V (y (t, y 0)) , M = M (t, y (t, y 0)) ,S = S (t, y (t, y 0)) , A = A(t, y (t, y 0)).

The equations (2.12)-(2.16) are the Lagrangian formulation of the Gaus-sian beam method. (2.12)-(2.13) are called the ray-tracing equations, (2.14)is a Riccati equation for the Hessian M , which will be solved by the dynamicrst order system of ray tracing equations (2.17).

We summarize some properties of these ODEs in Theorem 2.1. Theresults and proofs of Parts 1, 2 and 3 essentially follow those in [29].

Theorem 2.1 Let P (t, y (t, y 0)) and R(t, y (t, y 0)) be the (global) solutionsof the equations

dP dt

= R,dRdt

= (2y V )P, (2.17)with initial conditions

P (0, y 0) = I, R (0, y 0) = M (0, y 0), (2.18)

where matrix I is the identity matrix and Im (M (0, y 0)) is positive denite.Assume M (0, y 0) is symmetric, then for each initial position y 0, we havethe following results:

1. P (t, y (t, y 0)) is invertible for all t > 0.

2. The solution to equation (2.14) is given by

M (t, y (t, y 0)) = R(t, y (t, y 0))P 1(t, y (t, y 0)) (2.19)

3. M (t, y (t, y 0)) is symmetric and Im (M (t, y (t, y 0))) is positive denite for all t > 0.

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4. Not only is the Hamiltonian V +12 | p |

2 conserved along the y -trajectory,another quantity A2 det P is also conserved, which means A(t, y (t, y 0))can also be computed by

A(t, y (t, y0)) = (det P (t, y (t, y

0))) 1A2(0, y

0) 1/ 2 , (2.20)

where the square root is taken as the principle value.

Proof: Since y (t, y 0) is not involved in the proof, we drop it for conve-nience and simply write M (t, y (t, y 0)) , A(t, y (t, y 0)) , P (t, y (t, y 0)) andR(t, y (t, y 0)) as M (t), A(t), P (t) and R(t) respectively.(1) For any vector

C n , by (2.17), z 1 = P (t) and z 2 = R(t) satisfy

dz 1dt

= z 2 ,dz 2dt

= (2y V )z 1. (2.21)Dene

(P,R, ) = z 1 z 2 z 1 z 2 . (2.22)Note that

2y V is real and symmetric, then by differentiating (2.22) with

respect to t and using (2.21), one has

ddt

(P,R, ) =dz 1dt z 2 + z 1

dz 2dt

dz 1dt z 2 z 1

dz 2dt

= z 2 z 2 + z 1 (2y V )z 1 z 2 z 2 z 1 (2y V )z 1= 0 .

Assume that P (t) is singular at t = t0, then there exists a non-zerovector l

C n , such that P (t0)l = 0. So we have

0 = P (t0)l R(t0)l P (t0)l R(t0)l= (P (t0), R(t0), l ) = (P (0), R(0), l )= P (0) l R(0) l P (0) l R(0) l= l M (0) l l M (0) l = 2 il Im[M (0)]l ,

which is a contradiction since Im[ M (0)] is positive denite and l 0 is non-zero.In the last identity we used the symmetry of M (0).

This proves the invertibility of P (t).(2) Let M = RP 1. Using (2.17), one obtains

dM dt

+ M 2 + 2y V = dRdt

P 1 + R dP 1

dt+ RP 1RP 1 +

2y V

= (2y V )P P 1 RP 1dP dt

P 1 + RP 1RP 1 + 2y V

= RP 1RP 1 + RP 1RP 1 = 0 .

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Thus M satises (2.14).(3) First, since both M (t) and its transpose M (t) satisfy the same equation(2.14), the uniqueness (for example Theorem 1 in [7]) implies M (t) = M (t)for all t > 0 when the initial condition M (0) is symmetric.

Since we already proved that P (t) is invertible, l

C n there exists an

l that satises l = P (t)l . Then

2il Im[M (t)]l = 2 iP (t)l Im[M (t)]P (t)l= P (t)l M (t)P (t)l P (t)l M (t)P (t)l= P (t)l R(t)l P (t)l R(t)l= (P (t), R(t), l ) = (P (0), R(0), l )= P (0) l R(0) l P (0) l R(0) l= P (0) l M (0)P (0) l P (0) l M (0)P (0) l= 2 il Im[M (0)]l .

The second and last identities are got by the symmetries of M (t) and M (0).Because Im[M (0)] is positive denite, Im[ M (t)] is also positive denite.(4) Along the y -trajectory,

ddt

A2 det P = 2 AdAdt

det P + A2d(det P )

dt

= 2A12

Tr( M ) A det P + A2Tr P 1dP dt

det P

= A2Tr RP 1 det P + A2Tr P 1R det P = 0 .Hence A2 det P is conserved.

2.2 The Lagrangian Gaussian beam summation

In this subsection, we introduce the Lagrangian Gaussian beam summationformula (for example [12]). The approximation of the WKB initial data(1.2) is shown in the next theorem proved by Tanushev in [30]. Note thatthe regularities of the initial conditions are modied to be specic here.

Theorem 2.2 Let A0C 1(R n ) L2(R n ) and S 0C 3(R n ), dene

0(x ) = A0(x )eiS 0 (x )/ ,

0(x , y 0) = A0(y 0)e

iT 0 (x ,y 0 )/ ,

where

T 0(x , y 0) = T 0(y 0) + T 0 (x y 0) +12

(x y 0)T 0(x y 0),T 0(y 0) = S 0(y 0) , T 0(y 0) = x S 0(y 0) , T 0(y 0) =

2x S 0(y 0) + iI .

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Then

R n 12n2

r (x y 0)0(x , y 0)dy 0 0(x )l2

C12 ,

where r C 0 (R

n), r 0 is a truncation function with r 1 in a ball of radius > 0 about the origin and C is a constant related to .

By Theorem 2.2 we specify the initial data for (2.12)-(2.16) as

y (0, y 0) = y 0, (2.23) p(0, y 0) = x S 0(y 0), (2.24)

M (0, y 0) = 2x S 0(y 0) + iI, (2.25)

S (0, y 0) = S 0(y 0), (2.26)A(0, y 0) = A0(y 0). (2.27)

Then the Gaussian beam summation solution which approximates the Schr odingerequation (1.1) is constructed as:

la (t, x ) = R n 12n2

r (x y (t, y 0))la (t, x , y 0)dy 0 . (2.28)

The discrete form of (2.28) in a bounded domain is given by

la (t, x ) =N y 0

j =1

12

n2

r (x y (t, y j0))

la (t, x , y

j0) y 0, (2.29)

where y j

0s are the equidistant mesh points, and N y 0 is the number of thebeams initially centered at y j0 .Note that r works as a cut-off function, and the cut-off error becomes

large when the truncation parameter is taken too small. On the otherhand, a big for wide beams makes the Taylor expansion error large. Asfar as we know, it is still an open question that how large the should bechosen when the beams spread. However, for narrow beams one could takefairly large which makes the cut-off error almost zero. For example, aone-dimensional constant solution could be approximated by

1 =

R

1

2

exp (x y0)22

dy0 j

y0

2

exp (x y j0)2

2,

in which r 1.

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3 The Eulerian formulation

In the last few years, the level set method has been developed to computethe multi-valued solution of (1.7)-(1.8) which gives the correct semiclassicallimit of the Schrodinger solution [16, 5, 15] away from the caustics . The idea

is to build the velocity u = y S into the intersection of zero level sets of phase-space functions j (t, y , ), j = 1 , , n , i.e.

j (t, y , ) = 0 , at = u (t, y ), j = 1 , , n. (3.1)If we dene = ( 1 , , n ), then by differentiating (3.1) with respect toy for each j , the Hessian y u satises

2y S = y u = y ( ) 1. (3.2)

Comparing (3.2) with (2.19), we conjecture that

R = y , P = . (3.3)Note that this conjecture does not violate the symmetry of the Hessian

y uby the second and third parts of Theorem 2.1. Moreover, it also implies thedivergence-free condition

R + y P = 0 , (3.4)

which actually holds initially (2.18).In this section, we rst review the level set formulations developed in

[15, 16, 19] for geometrical optics, and then prove that the conjecture (3.3)is true under an appropriate initial condition for . We then describe thelevel set algorithm for the Eulerian Gaussian beam method and the con-struction of the wave function for the Schr odinger equation. Although ournew formulations are consistent with the Eulerian Gaussian beam methodconstructed in [19] for the Helmholtz equations, by making use of the obser-vation (3.3), we introduce a much simpler and efficient numerical methodthan [19].

3.1 The verication of (3.3)

Dene the linear Liouville operator as

L= t + y y V .As shown in [15, 16], the level set equations for the velocity, phase and

amplitude are given by

L = 0 , (3.5)LS =

12| |

2 V, (3.6)

LA =12

Tr ( ) 1y A. (3.7)

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If one introduces the new quantity ([15])

f (t, y , ) = A2(t, y , )det( ), (3.8)

then f (t, y , ) satises the Liouville equation

Lf = 0 . (3.9)For the Gaussian beam method in [19], two more inhomogeneous Liou-ville equations, which are the Eulerian formula of (2.17) for P and R, wereintroduced to construct the Hessian matrix

LR = (2y V )P, (3.10)LP = R. (3.11)

Note that the equations (3.5)-(3.9) are real , while (3.10)-(3.11) are complex and consist of 2 n2 equations.

By taking the gradient of the equation (3.5) with respect to y and separately, we have

L(y ) = 2y V , (3.12)L( ) = y , (3.13)

Compare (3.10)-(3.11) with (3.12)-(3.13), we observe that y and satisfy the same equations as R and P . Since the Liouville equations arelinear, the conjecture (3.3) is true by letting be complex and y , have the same initial conditions as R and P respectively.

From (2.18) and (2.25), this suggests the following initial condition for :

0(y , ) = iy + ( y S 0). (3.14)With this observation now we can solve (3.5) for complex , with initial data(3.14). Then the Hessian M is constructed by

M = y ( ) 1 (3.15)where velocity u = y S is given by the intersection of the zero-levelcontours of the real part of , i.e. for each component j ,

Re[ j (t, y , )] = 0 , at = u (t, y ) = y S. (3.16)

Note that to compute u , S and M we just need to solve n complex-valuedhomogeneous Liouville equation (3.5), while in the formulation of [19], oneneeds to solve n real-valued homogeneous Liouville equation (3.5) and 2 n2

complex-valued inhomogeneous Liouville equations (3.10) and (3.11). Ournew formulation has a computational complexity comparable to that of thelevel set methods for geometrical optics [15, 17]. The only difference hereis that (3.5) is complex-valued while in the geometrical optics computationone solves real part.

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3.2 The level set algorithm

In this section, we give the implementation details of the new level setmethod. We will also prove its validity at caustics.

Step 1. Solve (3.5) for complex , with initial condition (3.14), then ob-tain the velocity u by the intersection of the zero-level sets of Re j , j =1, , n .

Step 2. Use y and to construct the Hessian matrix by (3.15)(note these quantities actually are already available from the rst stepwhen one discretizes the Liouville equation for ).

Step 3. One can integrate the velocity u along the zero-level sets([10, 17]) to get the phase S . The idea is to do numerical integra-tion following each branch of the velocity. The integration constantsare obtained by both the boundary condition and the fact that themultivalued phase is continuous when it passes from one branch tothe other. For example, if we consider a bounded domain [ a, b] in onespace dimension, the phase function is given by

S (t, x ) = V (a)t 12 t0 u2(, a )d + xa u(t, s )ds + S (0, a). (3.17)

Because xa u(t, s )ds is the only term in (3.17) which affects the quadraticphysical observables for xed time t, one could take

S (t, x ) = xa u(t, s )ds (3.18)as the phase value in the numerical simulations of (1.3)-(1.5). Formore details and in higher dimensions, see [17]. (One could also solve(3.6) directly.)

Step 4. Solve (3.9) with the initial conditionf 0(y , ) = A20(y , ).

The amplitude A is given by

A = (det( ) 1f )1/ 2, (3.19)

in which the square root is taken as the principle value.

Remark 3.1 All of the functions in Steps 2-4 only need to be solved locally around the zero-level sets of Re j , j = 1 , , n . Namely, the entire algo-rithm can be implemented using the local level set methods [26, 27], thus thecomputational cost for mesh size y is O(( y) n ln( y) 1), about the sameas the local level set methods for geometrical optics computation [15, 17].

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The well-denedness of (3.15) and (3.19) is justied by the followingtheorem, which is the Eulerian version of Theorem 2.1.

Theorem 3.2 Let = (t, y , ) C be the solution of (3.5) with initial

data (3.14). Then we have the following properties:

1. is non-degenerate for all t > 0.

2. Im y ( ) 1 is positive denite for all t > 0, y , R n .Proof: (1) For

C n , by using (3.12)-(3.13) we have

L ( ) (y ) ( ) (y )= L ( ) (y ) + ( ) L(y )

L( ) (y ) ( ) L (y )= (y ) (y ) + ( ) (2y V )

+(y ) (y ) ( ) (2y V )= 0 .

The last equality is true because 2y V is symmetric and real.

If is singular at ( t2, y 2, 2), there exists a non-zero lC n such that

( )l |(t2 ,y 2 , 2 ) = 0. Then we have( )l (y )l ( )l (y )l (t 2 ,y 2 , 2 ) = 0 .

Since L ( )l (y )l ( )l (y )l = 0, there exists (0 , y 1, 1)that connects ( t2, y 2, 2) by a characteristic such that

( )l (y )l ( )l (y )l (0,y 1 , 1 )= ( )l (y )l ( )l (y )l (t 2 ,y 2 , 2 )= 0 .

This implies, by taking account of the initial condition (3.14),

2il l = 0 .This is a contradiction, which proves is not degenerate for all t > 0.

(2) Since we already proved that is non-degenerate, l

C n there

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exists an l that satises l = ( )l . Note that y ( ) 1 is symmet-ric from (3.2), then we have2il Im y ( ) 1 (t2 ,y 2 , 2 ) l

= 2 i (

)l

Im

y (

) 1 (

)l(t2 ,y 2 , 2 )

= ( )l (y )l ( )l (y )l (t2 ,y 2 , 2 )= ( )l (y )l ( )l (y )l (0,y 1 , 1 )= 2 il l ,

which implies that Im y ( ) 1 is positive denite.Remark 3.3 Although det Re[ ] = 0 at caustics, the complexied makes non-degenerate, and the amplitude A, dened in (3.19), doesnot blow-up at caustics!

So far we have got the phase S , the velocity u , the Hessian M andthe amplitude A. We show how to construct the wave function from thesequantities in the next subsection.

3.3 The Eulerian Gaussian beam summation

Dene

eu (t, x , y , ) = A(t, y , )e

iT (t, x ,y , )/ , (3.20)

where

T (t, x , y , ) = S (t, y , ) + (x y ) +12

(x y )M (t, y , )(x y ),then the wave function is constructed via the following Eulerian Gaussianbeam summation formula:

eu (t, x ) = R n R n 12n2

r (x y )eu (t, x , y , )n j =1 (Re[ j ])d dy ,(3.21)

in which r C 0 (R n ), r 0 is a truncation function with r 1 in a

ball of radius > 0 about the origin and is the Dirac delta function. Thechoice of r is the same as the one in the Lagrangian formulation.We show in the Appendix that (3.21) is consistent to the Lagrangian

summation formula (2.28). (3.21) could be evaluated as a single integral

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about y . Since the velocity becomes multivalued after caustic formation, wedenote u k , k = 1 , , K as the k-th branch of the velocity and write

eu (t, x ) = R n 12n2

r (x y )k

eu (t, x , y , u k )

|det(Re[ ] = u k )|dy . (3.22)

Since det Re[ ] = 0 at caustics, a direct numerical integration of (3.22) loses accuracy around singularities (see Example 3 in Section 5 forthe detailed numerical demonstrations). To get a better accuracy, we split(3.22) into two parts

I 1 =k L 1 12

n2

r (x y )

eu (t, x , y , u k )

|det(Re[ ] = u k )|dy , (3.23)

I 2 =k L 2 12

n2

r (x y )

eu (t, x , y , u k )

|det(Re[ ] = u k )|dy , (3.24)

where

L1 = y det(Re[p ](t, y , p j )) ,L2 = y det(Re[p ](t, y , p j )) < ,

with being a small parameter. In the numerical simulation one needs toconsider two facts about choosing the parameter :

1. to reduce the computational complexity one needs to make as smallas possible to minimize the cost of computing (3.24).

2. for convenience one needs to evaluate (3.23) easily by some quadraturerule using the mesh grid points.

See Figure 7 (bottom) for example. The circles are the grid points we needto compute (3.23) and the other cross points are those we need to evaluate(3.24).

In our numerical simulations, I 1 is approximated by the trapezoid quadra-ture rule, while the singular integral I 2 is evaluated by the semi-Lagrangianmethod introduced in [19]. For convenience we summarize the semi-Lagrangianmethod here. Suppose we take a number of discrete beams centered aty j , j = 1 , , M y with the velocity u

jk on the contour, the idea is to trace

each individual ( y j , u jk) back to the initial position ( y j

0, u j

k, 0) using (2.12)-

(2.13) with t t , then determine the weight function (y j0) for it. For

example, in one space dimension, if the two adjacent points y j 10 and y j 20 of

y j0 satisfy y j 10 < y

j0 < y

j 20 , then (y

j0) = ( y

j 20 y j 10 )/ 2 (see Page 68 in [19] for

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details). In this process one gets rid of the singular term by noticing that

dy 0 =1

|det(Re[ ] = u k )|dy . The discrete form of (3.24) reads as

I 2=

M y

j =1 k

1

2

n2

r (x

y j )

eu(t,

x,y j

,u j

k)

(y j0).

(3.25)

We remark here that the semi-Lagrangian method (3.25) can be usedto evaluate (3.22), as was done in [19]. However, when the rays divergebackward in time, (y j0) becomes large. The local semi-Lagrangian methodhelps to reduce such errors.

Remark 3.4 When the velocity contours are complicated due to large num-bers of caustics, the implementation of this local semi-Lagrangian method is complex and hard. In these cases we refer to a discritized delta function method [33] for numerically computing (3.21) which does not contain the

singular denominator. In this method one needs to compute (3.6) for solv-ing the phase function since all the function values near the support domain of the delta functions (Re[ j ]) are needed to evaluate (3.21) . This will be further studied in a forthcoming work.

3.4 Estimates on computational complexity

Since the errors of the Gaussian beam method were already discussed in[24, 30], we only focus on the analysis of the computational complexityhere. There are two steps for computing the Gaussian beam solution.

1. Solving the PDEs (3.5) and (3.9).

2. Constructing the asymptotic solution using (3.22).

The numerical cost of the rst step is related to the mesh size and thetime step. We will show later that the optimal mesh size is y = O(

12 )

in the numerical simulations. We denote the numerical error of solving thePDEs as E num , which introduces the error of E num / when one constructs

eu in (3.20). For this error to be minor, we require the time step taken

as t12 in a fourth order time discretization. Then the computational

complexity of the rst step is O(n +1

2 ln 12 ) if the local level set method

is used.The numerical cost of the second step is related to the number of nodes

N x in the x -mesh when we nally construct the solution by (3.22). To getthe whole wave eld , one needs to resolve the wave length, which requiresN x

n . Due to the truncation function r (x y ) and the fact that theimaginary part of the Hessian matrix M is positive denite by Theorem3.2, the computational cost of (3.23) is O(1) for each x . The computational

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cost of each backtrace is O(12 ) in the semi-Lagrangian method for solving

(3.24). However, since (3.24) is an integration on a small local domain ( issmall), the computational cost of (3.25) is negligible compared to the rest.If the discretized delta function method [33] is used to evaluate (3.21), thecomputational cost for each x is still O(1) because of the truncation function

r (x y ) and the fact that each beam decays exponentially. So the totalcomputational complexity of the second step is O( n ) for computing thewhole wave eld .

In summary, the total computational cost is O(n +1

2 ln 12 + n ). We

remark that the computational cost mainly comes from solving the Liouvilleequations, which is O(

n +12 ln

12 ), if only some selected points in the wave

eld are needed to be computed. This is better than the complexity of directly solving the Schr odinger equation (1.1)-(1.2) using the Time SplittingSpectral Method [1], which is O( n 1/ 2 ln n ).

4 The numerical examples

In this section, we present several numerical examples using both the La-grangian and the Eulerian formulations to show the accuracy of the Gaussianbeam solutions and the numerical efficiency. We compute the solution of theSchr odinger equation by the Strang splitting spectral method [1]. The ex-act Schr odinger solution is obtained by using a very ne mesh and avery small time step with an appropriately large domain so that the peri-odic boundary condition does not introduce a signicant error to the initialvalue problem. Since all of the Gaussian beams are observed to be narrowin the numerical examples, the truncation parameter in (2.29) and (3.22)is chosen large enough so that the cut-off error is almost zero. We gain

convergence rates higher than O() in the numerical simulation becauseof the error cancelations in the summation of Gaussian beams (see [24]).4.1 The Lagrangian Gaussian beam examples

We study the Lagrangian formulation numerically in this subsection. Thenumerical examples are chosen such that the fourth order Runger-Kuttascheme with the time step t

1/ 2 is good to be used for solving (2.12)-(2.16). We analyze the asymptotic expansion error and the initial conditionerror in Example 1. The numerical integration errors and the errors of theODE solvers are discussed in Example 2.

Example 1: A free particle with zero potential V (x) = 0. The initialconditions for the Schrodinger equation are given by

A0(x) = e 25(x 0.5)2, S 0(x) = x2.

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In this example, the Gaussian beam solution (2.28) can be solved analytically

la (t, x ) =e 6.25

x 22 (5 t 2 4 t +1)

+ i(5 t 2) x 2

2(5 t 2 4 t +1)+ B

24A

2A (2 + i)t + 1

,

where A and B are

A = 25 +(1 2t)2

2(5t2 4t + 1) (1 2t)t

2(5t2 4t + 1)i,

B = 25 +2(1 2t)x

2(5t2 4t + 1) xt

(5t2 4t + 1)i.

In this example the initial velocity contour is a straight line with theslope of 1. As time increases the slope steepens and becomes innityat t = 0 .5 which generates an instantaneous caustic. After t = 0 .5 theslope becomes positive and the caustic disappears. This type of caustic is

different from the cusp caustic which will be studied in Example 2 and 3.As a consequence, we observed a higher convergence rate in l1 norm than inthe other examples.

We plot the solutions at time t = 0 .5 when the caustic is generated, seeFigure 1. We compare the wave amplitudes between the Gaussian beamsolution la (t, x ) and the Schrodinger solution

(t, x ) for different s. Theerror comes mostly from the asymptotic expansion and the initial data ap-proximation.

The l1, l2 and l errors between la and for different s are plotted

in Figure 2. One can see that the Gaussian beam solution la converges in with an order of about 1 .5 in the l1 norm, one in the l2 norm, and onehalf in the l norm.

Example 2: Consider the potential V (x) = 2 x3, with the initial conditions

A0(x) = e 25x2, S 0(x) =

1

cos(x ).

This example deals with a variable potential case. Since the potentialV (x) = 2 x3 makes the force one-directional , the wave function of (1.1)-(1.2)is asymmetric . Two caustics form at time t = 0 .5.

We use the fourth order Runge-Kutta method to solve the ODE system(2.12)-(2.16) with the initial data (2.23)-(2.27) in the interval [ 1.5, 1.5] withperiodic boundary conditions. Note that the Hessian M and the amplitudeA are solved via the dynamic ray tracing system (2.17), (2.19) and (2.20).In Table 1, we present the l1, l2 and l errors between and usingdifferent ts and N y0 = 200. We draw the conclusion that the numericalerrors are negligible compared to the other types of errors when t 0.02for = 14096 , n = 1.

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0.2 0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

x

||

|

la |

0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 0.008 0.01

0

5

10

15

20

25

x

||

|

la|

(a) = 11024

0.2 0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

x

||

|

la|

0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 0.008 0.01

0

5

10

15

20

25

x

||

|

la |

(b) = 14096

0.2 0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

x

||

|

la|

0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 0.008 0.01

0

5

10

15

20

25

x

||

|

la|

(c) = 116384

Figure 1: Example 1, the Schr odinger solution

|

|versus the Gaussian

beam solution |la | at different s. The left gures are the comparisons att = 0 (the initial time); the right gures are the comparisons at time t = 0 .5.

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106

105

104

103

108

107

106

105

104

103

102

101

100

l1

error

l2

error

Maximum error

3/2

1

1/2

Figure 2: Example 1, the convergence rate in of the l1, l2 and l errors of

the wave function.

Next we study on the error coming from evaluating the integral (2.28)numerically. Evidently, the more nodes of y0 are used, the more accuratethe numerical integration will be; however, each node of y0 corresponds toan ODE system, thus a large number of nodes will result in a high compu-tational cost. In this example, we investigate the accuracy versus N y0 for1. We also study the optimal relation between N y0 and which ensuresa good approximation and low computational cost.

We plot the solutions at time t = 0 .5 in Figure 3. The wave amplitudesare compared between the Schr odinger solution and the Gaussian beam

solution

using different numbers of beams N y0 at =1

4096 . The l1, l

2

and l errors between and are given in Table 2. We can see thatthe Gaussian beam solution converges quickly with the increasing number of beams. In Figure 4 (left), we show that N y0

12 is pretty much enough,and larger N y0 will not reduce the error further since the other errors startto dominate.

Figure 4 (right) gives the l1, l2 and l errors between la and for

different s at time t = 0 .5, where we use N y0 = 800 and t = 0 .001. Theconvergence rate of the error in is about rst order in l1 and l2 norms, theorder of 0.7894 in the l norm. We also plot the comparisons of the positiondensity, the density ux and the kinetic energy for = 14096 , using N y0 =200, t = 0 .5 and t = 0 .01 in Figure 5.

4.2 The Eulerian Gaussian beam examples

In this subsection, we study the Gaussian beam method numerically usingthe Eulerian formulation. The numerical methods to the Liouville equation

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0.15 0.1 0.05 0 0.05 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

||

|

la|

0.15 0.1 0.05 0 0.05 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

|

la|

(a) N y0 = 50

0.15 0.1 0.05 0 0.05 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

||

|

la |

0.15 0.1 0.05 0 0.05 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

|

la|

(b) N y0 = 100

0.15 0.1 0.05 0 0.05 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

||

|

la |

0.15 0.1 0.05 0 0.05 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

|

la |

(c) N y0 = 200

Figure 3: Example 2, the Schr odinger solution

|

|versus the Gaussian

beams solution |la | at = 14096 using different N y0 s. The left gures arethe comparisons at t = 0 .5; the right gures plot the errors | la |.

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t 0.04 0.02 0.01 0.005la 1 7.88 10 2 4.18 10 4 4.18 10 4 4.18 10 4

la

23.51

10 1 1.68

10 3 1.68

10 3 1.68

10 3

la 2.58 100 1.24 10 2 1.24 10 2 1.24 10 2

Table 1: the l1, l2 and l errors for different ts for Example 2.

N y0 50 100 150 200la

1 2.72

10 2 6.92

10 4 4.18

10 4 4.18

10 4

la 2 1.09 10 1 2.69 10 3 1.68 10 3 1.68 10 3la 7.10 10 1 2.02 10 2 1.24 10 2 1.24 10 2

Table 2: the l1, l2 and l errors at = 14096 for different N y0 s for Example2.

104

103

101

102

103

104

105

Ny0

1

1/2

106

105

104

103

104

103

102

101

l1 error

l2 errorMaximum error 1

1/2

Figure 4: Example 2. Left: the plot of the optimal beam number N y0 foreach ; right: the convergence rate in of the l1, l2 and l errors of thewave function.

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0.15 0.1 0.05 0 0.05 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

position density of

position density of la

0.15 0.1 0.05 0 0.05 0.1

0

0.005

0.01

0.015

0.02

0.025

x

(a) The position density

0.15 0.1 0.05 0 0.05 0.1

0.8

0.6

0.4

0.2

0

0.2

0.4

x

density flux of

density flux of la

0.15 0.1 0.05 0 0.05 0.1

0

2

4

6

8

10

12x 10

3

x

(b) The density ux

0.15 0.1 0.05 0 0.05 0.1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x

kinetic energy of

kinetic energy of la

0.15 0.1 0.05 0 0.05 0.1

0

0.5

1

1.5

2

.5

3

x 103

x

(c) The kinetic energy

Figure 5: Example 2, = 14096 , N y0 = 200 , t = 0 .5 and t = 0 .01. The

left gures are the comparisons of the position density, the density ux andthe kinetic energy of the Schr odinger solution and the Gaussian beamssolution la at t = 0 .5; the right gures show the errors between them.

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have been very mature by now. It could be solved by either the nitedifference method [16, 17] or the semi-Lagrangian method [19, 20]. We donot address the issue of an optimal Liouville solver here.

We present two numerical examples to show the effectiveness of the Eu-lerian Gaussian beams method. In Example 3 (rst proposed in [16]), we

point out that aroundthe caustics one needs to take enough discrete beamsto resolve the velocity contour well to obtain a good accuracy. This is anissue not discussed in [19]. We also study a simple two-dimensional case inExample 4 which rst appeared in [26].

Example 3 (1D) : Free motion particles with zero potential V (x) = 0. Theinitial conditions for the Schr odinger equation are given by

A0(x) = e 25x2, S 0(x) =

15

log(2cosh(5x)) ,

which implies that the initial density and velocity are

0(x) = |A0(x)|2 = exp( 50x2), u0(x) = x S 0(x) = tanh(5 x).The solutions of the Liouville equations (3.5), (3.6) and (3.9) can be

solved by the method of characteristics analytically:

(t,y, ) = 0(y t, ),S (t ,y, ) = S 0(y t ) + tu 20(y t )/ 2,f (t ,y, ) = A20(y t, ).

We use y = = 0 .002 in the mesh for the Eulerian beam simulation,and take x =

1

65536, the same mesh size as the numerical solution to the

Schr odinger equation, to construct the wave function (3.22). Figure 6 showsthe comparison of the wave amplitude between the Schr odinger solution

and the Eulerian beams solution eu for = 1 10 4 and at t = 0 .5.From the left gures of Figure 6(a)-(c), one could see that summing upthe beams centered on the mesh could only approximate the Schr odingersolution accurately away from the caustics. In order to make good ap-proximations around the caustics, one has to sum up more beams whosecenters better represent the velocity contour curve. In other words, oneneeds to numerically resolve the singular integration (3.24) accurately bythe semi-Lagrangian method. This process could be easily implemented bythe Matlab subroutine contour. The results are shown in the right g-ures of Figure 6(a)-(c). To make some comparisons we list other types of asymptotic solutions in Figure 6(d)-(f) given by geometric optics [16] andthe corrected geometric optics with phase shift information (Keller-Maslovindex) [17]. Since both the approximations blow up at caustics, we makesome truncations when plot them. One could also notice that the Eulerian

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beam solution eu naturally carries the phase shift information even whenthe velocity contour is not well resolved. The error of eu is substantiallysmaller than GO and

P S around the caustics.

We give the velocity contour and beam distribution around one causticin Figure 7. This gure shows that the contour of velocity around the caustic

(near x = 0.18) was not resolved well which causes large errors around thecaustic shown in Figure 6(c) (left). Figure 8 (left) shows the convergencerates of amplitude in are of order 0 .9082 in the l1 norm, order 0 .8799 in l2

norm and order 0 .7654 in the l norm. In Figure 8 (right), we numericallyshow that the optimal relationship between y and is y

1/ 2 whichensures a good approximation.

Example 4 (2D) : Take the potential V (x1, x2) = 10 and the initial condi-tions of the Schrodinger equation as

A0(x1, x2) = e 25(x21 + x

22 ) ,

S 0(x1, x2) = 15(log(2cosh(5 x1)) + log(2cosh(5 x2))) .

then the initial density and two components of the velocity are

0(x1, x2) = exp( 50(x21 + x22)) ,u0(x1, x2) = tanh (5 x1), v0(x1, x2) = tanh (5 x2).

This is an easily-implemented two-dimensional example, since the twocomponents are either x1-isotropic or x2-isotropic (see Figure 9). Morecomplicated examples could be dealt with similarly, but one needs moresophisticated interpolation technique to get the intersection of the zero-levelcontours. We do not address this issue here and refer to [23] for interestedreaders. We use y1 = y2 = 1 = 2 = 0 .004 for the computationof the Liouville equations (3.5) and (3.9) by the semi-Lagrangian method

[19, 20]. The mesh size is x1 = x2 =1

65536for the construction of the

Eulerian beam solution (3.22). We take = 0 .001 and compare the waveamplitudes of the Schr odinger solution and the Eulerian beam solutioneu in Figure 10 at time t = 0 .5. The error between them is shown in Figure11.

5 Conclusion

In this paper, we present the Lagrangian and Eulerian formulations of theGaussian beam method for solving the Schr odinger equation. A new Eu-lerian Gaussian beam method is developed using the level set method. Byusing the observation (3.3), we evaluate the Hessian matrix of the phasedirectly from the rst derivatives of the (complexied) level set function for

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0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

eu |

||

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

eu |

||

(a) Comparison of wave amplitude

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

eu |

||

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

eu |

||

(b) Comparison of undersampled wave amplitude

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

0.25

x

||

eu ||

||

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

0.25

x

||

eu ||

||

(c) Errors

Figure 6: Example 3, = 1 10 4. The gure shows the comparisonof the wave amplitudes between the Eulerian beam solution eu and theSchr odinger solution at t = 0 .5. The Eulerian beam solution on the leftis obtained by the summation of all the beams centered on the mesh, andthe one on the right is obtained by the summation of the beams which betterresolve the zero contour curve around caustics.

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0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

GO |

||

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

PS |

||

(d) Comparison of wave amplitude

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

GO |

||

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

|

PS |

||

(e) Comparison of undersampled wave amplitude

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

0.25

x

||

GO ||

||

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

0.25

x

||

PS ||

||

(f) Errors

Figure 6: (continued). Left: the comparison between the geometric optics

approximation GO and the Schrodinger solution

; right: the compari-son between the geometric optics corrected with phase shift (Keller-Maslov

index) P S and the Schrodinger solution .

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x

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

0.18 0.16 0.14 0.12 0.1 0.08

0.95

0.9

0.85

0.8

0.75

0.7

0.65

x

beams center on the mesh

beams resolve the contour

Figure 7: Example 3, t=0.5. The top gure shows the multivalued velocity;the bottom gure shows the beams centered on the mesh and the beamsneeded to resolve the zero contour around the caustic point. The former

case does not resolve the caustic around x = 0.18 well thus causes errorsshown in Figure 6(c) (left).

105

104

10310

5

104

103

102

101

l1 error

l2 errorMaximum error 1

1/2

105

104

10310

5

104

103

102

101

y 1

1/2

Figure 8: Example 3. Left: the convergence rates in of the l1, l2 and l

errors of the wave amplitude; right: the plot of the optimal y for each .

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Figure 9: Example 4, the two components of the multivalued velocity att = 0 .5.

Figure 10: Example 4, the comparison of the wave amplitude between the

Schr odinger solution

on the left and the Eulerian beams solution eu onthe right for = 0 .001 and at t = 0 .5.

Figure 11: Example 4, the error plot of ||| |eu ||.

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velocity, thus avoid using an extra 2 n2 (for n space dimension) complex-valued, inhomogeneous Liouville equations as was done in [19]. We verifythe accuracy of the Gaussian beam solutions by several numerical exampleswhich show small error around caustics. We also point out that one needsto resolve the velocity contour near caustics to obtain good approximations

in the Eulerian formulation. Moreover, we give a numerical criteria for theoptimal choice of the beam numbers in the Gaussian beam summation. Wewill extend this method to other problems in high frequency waves in thenear future.

Acknowledgement

Xu Yang would like to thank Prof. Bj orn Engquist and Prof. Richard Tsaifor the kind invitation to the workshop on Gaussian beams held in ICES,Austin, TX, which made him start to know and become interested in thissubject.

Appendix

The Lagrangian summation formulation (2.28) reads as

la (t, x ) = R n 12n2

r (x y (t, y 0))la (t, x , y 0)dy 0 , (A-1)which is equivalent to

eu (t, x ) =

R n

R n

1

2

n2

r (x

y )eu (t, x , y , p )(Re[ ](0, y 0, p0))d p0dy 0.

(A-2)based on Re[p 0 ](0, y 0, p 0) = I . Note that the integrated variables y 0, p 0are the initial positions of y , p . Changing variables by y , p , we get

eu (t, x ) = R n R n 12n2

r (x y )eu (t, x , y , p )(Re[](t, y , p ))d pdy .(A-3)

in which we need det J = 1 for

J = y 0 y p 0 y

y 0 p p 0 p.

First it is easy to see det J |t=0 = 1. Moreover,dJ dt

= UJ,

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with U = 0 I

2y V 0. This gives

ddt

(det J ) = tr(adj( J )dJ dt

) = tr(adj( J )UJ ) = tr( UJ adj( J )) = det( J )tr( U ) = 0 ,

where adj( J ) = (det J )J 1.

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