A SIMPLE EXPLICIT OPERATOR-SPLITTING METHOD FOR EFFECTIVE

SIAM J. SCI. COMPUT. c© 2018 Society for Industrial and Applied MathematicsVol. 40, No. 1, pp. A484–A503

A SIMPLE EXPLICIT OPERATOR-SPLITTING METHODFOR EFFECTIVE HAMILTONIANS∗

ROLAND GLOWINSKI† , SHINGYU LEUNG‡ , AND JIANLIANG QIAN§

Abstract. Understanding effective Hamiltonians quantitatively is essential for the homogeniza-tion of Hamilton–Jacobi equations. We propose in this article a simple efficient operator-splittingmethod for computing effective Hamiltonians when the Hamiltonian is either convex or nonconvexin the gradient variable. To speed up our Lie scheme–based operator-splitting method, we furtherpropose a cascadic initialization strategy so that the steady state of the underlying time-dependentHamilton–Jacobi equation can be reached more rapidly. Extensive numerical examples demonstratethe efficiency and accuracy of the new algorithm.

Key words. Hamilton–Jacobi, homogenization, effective Hamiltonian, operator-splittingmethods

AMS subject classifications. 65N30, 65M60

DOI. 10.1137/17M1137322

1. Introduction. Because of the pioneering work of [22], the homogenization ofHamilton–Jacobi (HJ) equations has called for a better understanding of homogenizedequations which are characterized by the so-called effective Hamiltonians. Becauseof the nonlinearity of the HJ equations, explicit forms of effective Hamiltonians areseldom available so that it is hard to directly describe homogenized HJ equationsboth qualitatively and quantitatively. Inspired by [22], many mathematicians havemade a lot of effort to understand homogenization of HJ equations, with Evans re-alizing a long time ago the importance of computing effective Hamiltonians to guidetheoretical characterization of homogenized HJ equations. Observing that the effec-tive Hamiltonian is a kind of additive eigenvalue as characterized in [9], the thirdauthor of this article has designed a large-T and a small-δ method for computingthe effective Hamiltonian when the Hamiltonian is either convex or nonconvex in thegradient variable [28]. Almost at the same time, Gomes and Oberman have designedin [19] an optimization method to compute effective Hamiltonians by discretizinga min-max formula [10] when the Hamiltonian is convex in the gradient variable.When the Hamiltonian is nonconvex in the gradient variable, a recent work [30] hasdiscovered some min-max formulas for nonconvex effective Hamiltonians for certainclasses of nonconvex HJ equations by using the computational results from the large-T method as an inspiration. Since it is essential to compute effective Hamiltoniansin many applications, such as weak KAM theory [11], front propagation of turbulent

∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section July 5,2017; accepted for publication (in revised form) October 23, 2017; published electronically February20, 2018.

http://www.siam.org/journals/sisc/40-1/M113732.htmlFunding: The first author’s work was supported by the NSF grant DMS 1418308. The second

author’s work was supported by the Hong Kong RGC under grants 605612 and 16303114. The thirdauthor’s work was supported by NSF grants DMS 1522249 and 1614566.†Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204,

and Department of Mathematics, Hong Kong Baptist University, Hong Kong ([email protected]).‡Department of Mathematics, Hong Kong University of Science and Technology, Clear Water

Bay, Hong Kong ([email protected]).§Department of Mathematics, Michigan State University, East Lansing, MI 48824 (qian@

math.msu.edu).

A484

http://www.siam.org/journals/sisc/40-1/M113732.html

mailto:[email protected]




OPERATOR SPLITTING FOR EFFECTIVE HAMILTONIANS A485

reaction-diffusion equations [35], stationary mean field games [21], and dislocationdynamics [6], we propose in this article an operator-splitting method to compute ef-fective Hamiltonians for homogenizing HJ equations with either convex or nonconvexHamiltonians.

In the literature, the large-T method and the small-δ method [28] have been an-alyzed in [7], and it was shown that both methods yield an approximate effectiveHamiltonian with a convergence rate of half order in the L∞-norm when the Hamil-tonian is assumed to be convex. Furthermore, an estimate of the error between theexact and computed solutions of the homogenized HJ equation (defined by the effec-tive Hamiltonian) has been obtained in [1]. Based on a variational characterization ofeffective Hamiltonians when the Hamiltonian is convex [11], optimization techniqueshave been designed in [14] for computing the effective Hamiltonians. Since the so-lution of a certain class of convex HJ equations (so-called metric HJ equations) canbe related to the distance function in a certain metric, the authors of [26] designeda numerical method to recover the effective Hamiltonian from solving one auxiliaryeffective equation. Moreover, for a class of convex HJ equations in the kinetic form,[25] links the effective Hamiltonian to a suitable effective equation so that the entireHamiltonian can be recovered from solving one auxiliary equation. A recent workin [5] has designed a generalized Newton method for homogenization of HJ equa-tions, in which the two unknowns (the effective Hamiltonian and the correspondingviscosity solution) in a cell problem are computed simultaneously by solving a non-linear least-squares problem with a Newton-like method, this method being able tohandle both convex and nonconvex Hamiltonians. In comparison to the general-ized Newton method in [5] which requires the Hamiltonian being differentiable, ouroperator-splitting method does not require differentiability of the Hamiltonian so thatour method can be applicable to more complicated, nondifferentiable Hamiltonians.

The article is organized as follows. We formulate the problem in section 2. Insection 3, we develop a Lie scheme–based operator-splitting method and propose acascadic strategy to speed up the convergence to the steady state of the underlyingHJ equation. In section 4, we use extensive numerical examples to demonstrate theefficiency and accuracy of our new algorithm.

2. Problem formulation. Consider the following homogenization problem forthe first-order nonlinear HJ equation:

uεt +H(∇uε, x

ε

)= 0 in Rd × (0,∞)(2.1)

uε = g in Rd × t = 0(2.2)

where H: Rd×Rd → R is smooth and periodic in the second variable with the periodY = [0, 1]d. Since ε > 0 characterizes the high-frequency oscillating perturbation,homogenization studies the response of a differential equation to such high-frequencyperturbation.

For linear differential equations, the homogenization has been well studied; see[4]. For the fully nonlinear equations under consideration, the homogenized equationis not obviously defined. Lions, Papanicolaou, and Varadhan [22] proved the firstexistence result of homogenized Hamiltonian (effective Hamiltonian) by defining a so-called cell problem; later, Evans [12] recounted their results for first-order nonlinearPDEs and made further development for second-order fully nonlinear PDEs by usinga “perturbed test function” method. The result roughly states that under suitableassumptions, for every ε > 0 the above equation has a unique viscosity solution uε;

A486 R. GLOWINSKI, S. LEUNG, AND J. QIAN

moreover, the sequence uεε>0 converges to the viscosity solution u of the averagedproblem

ut + H(∇u) = 0 in Rd × (0,∞)(2.3)

u = g in Rd × t = 0(2.4)

where the effective (averaged) Hamiltonian is defined through the cell problem: Foreach p ∈ Rd, there exists a unique H(p) such that

H(p+∇yv(p; y), y) = H(p)(2.5)

has a Y -periodic viscosity solution v(p; y).To see where this cell problem comes from, we give a formal derivation here.

Suppose that uε → u0 as ε→ 0 in the uniform topology and that uε has the asymptoticexpansion

uε(x, t) = u0(x, t) + εu1

(xε, t)

+O(ε2),(2.6)

where u1 is the first-order correction term to u0. Then by matching orders of ε in(2.1), we formally have

∂u0

∂t(εy, t) +H(∇xu0(εy, t) +∇yu1(y, t), y) = O(ε),(2.7)

where x = εy. Letting ε→ 0, we deduce that u1 satisfies

H(p+∇yu1(p, t; y), y) = H(p)(2.8)

with p = ∇xu0 and H(p, t) = −∂u0

∂t .

2.1. Small-δ and large-T methods. The work in [28] has developed two nu-merical approaches to determine the effective Hamiltonian. The first one is the so-called small-δ method, where one solves

δuδ +H(P +Duδ, x) = 0

for some small δ > 0. It has been shown in [7] that for δ small enough, the viscositysolution −δuδ gives a good approximation to the effective Hamiltonian H(P ).

A more widely used approach is the so-called large-T method. Established in [9],one could introduce the evolution problem (where Ω = Y = (0, 1)d)

∂u

∂t+H(P +∇u, ·) = 0 in Ω× (0,+∞),

u(t) periodic in Ω ∀t > 0,

u(0) = u0 (with u0 periodic in Ω).

Numerically, such an HJ equation can be solved using well-developed finite-differencemethods, such as the TVD Runge–Kutta (TVD-RK) stepping in time and weightedessentially nonoscillatory (WENO) finite-differencing in space [24, 34, 20, 28, 32]. Forthe eikonal Hamiltonian, one might also follow [13, 31] to use the semi-Lagrangianstrategy instead. Finally, the effective Hamiltonian can be approximated by

H(P ) = − limt→∞

u(x, t)

t.

Since the solution u(x, t) actually tends to infinity as t goes to infinity, the large-Tmethod looks for the linear growth rate of how fast u(x, t) diverges to infinity.


3. Our proposed method.

3.1. An explicit operator-splitting scheme. Our idea shares some similaritywith the large-T method [28]. But instead of solving for large T and then determiningthe growth in the average value of the solution, we enforce the uniqueness of thesolution by imposing the following condition to (2.5) [11]:∫

Ω

u(x)dx = 0.

With this constraint, we associate the following initial value problem:

(3.1)

∂u

∂t+H(P +∇u, ·) = λ in Ω× (0,+∞),∫

Ω

u(x, t)dx = 0 ∀t > 0,

u(t) periodic in Ω ∀t > 0,

u(0) = u0 (with u0 periodic in Ω and verifying

∫Ω

u0(x)dx = 0 ∀t > 0),

λ(t) ∈ R ∀t > 0 ,

where u(t) denotes the function x→ u(x, t). We first rewrite (3.1) as

(3.2)

∂u

∂t(t) +H(P +∇u, ·) + ∂I0(u(t)) 3 0 in Ω,

u(0) = u0,

u(t) periodic in Ω ,

where ∂I0(u(t)) is the subdifferential at u(t) of the indicator functional I0 of the space

L20(Ω) =

ϕ : ϕ ∈ L2(Ω),

∫Ω

ϕdx = 0

,

that is,

I0(ϕ) = 0 if ϕ ∈ L20(Ω),

I0(ϕ) = +∞ if ϕ ∈ L2(Ω)\L20(Ω).

In order to solve (3.2), we advocate the Lie scheme [15, 17]. Applying the Liescheme with ∆t > 0, tn = n∆t, we obtain for n ≥ 0: un → un+1/2 → un+1: Solvefirst

(3.3)

∂u

∂t(t) +H(P +∇u, ·) = 0 in Ω× (tn, tn+1),

u(tn) = un,

u(t) periodic in Ω.

Then we set un+1/2 = u(tn+1) and compute

(3.4) un+1 = un+1/2 −∫

Ω

un+1/2(x)dx.


Once a steady-state solution, say, u∞(P ) ≡ u∞(P ;x), has been obtained, we computeH(P ) via

(3.5) H(P ) =

∫Ω

H(∇u∞(P ;x))dx.

Remark 3.1. In applications when implicit schemes are more favorable, we ad-vocate some related operator-splitting methods based on the Marchuk–Yanenko ap-proach given as

un+ 12 − un

∆t+H

(∇un+ 1

2 , ·)

= λn in Ω,

un+1 = un+ 12 −

∫Ω

un+ 12 dx,

λn+1 =

∫Ω

H(∇un+1, ·

)dx,

or the relaxed version,

un+ 12 − un

∆t+H

(∇un+ 1

2 , ·)

= λn in Ω,

un+1 = un+ 12 −

∫Ω

un+ 12 dx,

γλn+1 − λn

∆t+ λn+1 =

∫Ω

H(∇un+1, ·

)dx,

for some relaxation parameter γ > 0. Possible choices for γ are γ = ∆t,√

∆t, orsimply O(1). To guarantee the existence of solution for the above implicit nonlinearsystem, see [8].

Since the solution un+1 is actually invariant with respect to the term λn on theright-hand side of the first equation in both numerical schemes, we can replace it byany constant, or simply by zero, in the implementation. To show this, let us introducea periodic function v(λ) ≡ v(λ;x) solution of

v(λ)− λ∆t+ ∆tH (∇v(λ), ·) = un.

Since H(∇v, x) = H(∇(v+c), x) for any constant c, we have v(λ) = v(0)+λ∆t. Thisimplies that

un+1 = v(λ)−∫

Ω

v(λ) dx = v(0) + λ∆t−∫

Ω

[v(0) + λ∆t] dx = v(0)−∫

Ω

v(0) dx.

For the current application, however, we found that the explicit scheme convergesrelatively fast even though we have to choose a time-marching step ∆t = O(∆x).Therefore, we did not further explore the alternative possibility of an implicit method.

Remark 3.2. In applications where regularization is necessary, we can considerthe Douglas–Rachford scheme defined by un → un+1, λn+1 → un+1 as

1

∆t

[un+1 − un − α∇2

(un+1 − un

)]+H(P +∇un, ·) = λn+1 in Ω,∫

Ω

un+1dx = 0 , λn+1 ∈ R , un+1 periodic over Ω,(3.6)


followed by

1

∆t

[un+1 − un − α∇2

(un+1 − un

)]+H(P +∇un+1, ·) = λn+1 in Ω,

un+1 periodic over Ω.

The linear problem (3.6) belongs to the following family of linearly constrained ellipticproblems:

u− α∇2u = f + γλ in Ω,∫Ω

u dx = 0 , λ ∈ R,

u periodic over Ω,

with γ > 0. Let us define the function ϕ : R→ R by ϕ(µ) =∫

Ωuµ(x)dx, where uµ is

the solution of

uµ − α∇2uµ = f + γµ in Ω,

uµ periodic over Ω.

Solving this problem numerically is routine nowadays. Now, since ϕ is an affinefunction of µ, we clearly have

ϕ(µ) =ϕ(M)− ϕ(0)

Mµ+ ϕ(0) ∀µ ∈ R,

which implies in turn that λ = Mϕ(0)/[ϕ(0) − ϕ(M)]. While we could simply useM = 1, one might also consider M 1 for numerical stability.

On the other hand, unlike the Lie or Strang splitting scheme, the Douglas–Rachford splitting scheme produces exact steady-state solutions, while the steady-state solutions produced by the Lie or Strang splitting scheme are splitting errorpolluted. However, as shown in [16], the Douglas–Rachford splitting is significantlyless robust than the Lie or Strang splitting, explaining why the Douglas–Rachford wasnot tested numerically considering that several of our test problems were nonsmoothones.

3.2. Numerical implementations. In this section, we briefly discuss someimplementation details of the proposed numerical algorithm for the effective Hamilto-nian. Some components of the method are well developed, and we will refer interestedreaders to the corresponding references and those therein. For simplicity, we consideronly the one-dimensional case, although two-dimensional test problems will be consid-ered in section 4. Actually, high-dimensional generalization is rather straightforward.

3.2.1. HJ equation. We first consider the numerical discretization of the HJequation (3.3). This time-dependent HJ equation is in a standard form and cantherefore be solved using the TVD-RK-WENO with the Lax–Friedrichs Hamiltonian[27, 24, 34, 20, 32]. We will just briefly state the discretization procedure here andrefer readers to the above references and those therein for detailed discussion.

We consider the one-dimension HJ equation given by

∂u

∂t+H

(P +

∂u

∂x, x

)= 0.


First, we discuss the spatial discretization of the equation. We approximate the Hamil-tonian by the Lax–Friedrichs numerical Hamiltonian given by

HLxF

(P +

∂u

∂x

∣∣∣∣x=xi

, xi

)= H

(P +

p+i + p−i

2, xi

)− σx

(p+i − p

−i

2

),

where the derivatives p± are approximated by the WENO strategy. The viscosityparameter σx is chosen so that for any xi the following monotonicity requirement issatisfied:

σx ≥ maxu,ux

∣∣∣∣∣∂HLxF(P + ux)

∂ux

∣∣∣∣∣.In this work, we applied the third-order WENO (i.e., WENO3) given by

p−i = (1− ω−)

(ui+1 − ui−1

2∆x

)+ ω−

(3ui − 4ui−1 + ui−2

2∆x

),

p+i = (1− ω+)

(ui+1 − ui−1

2∆x

)+ ω+

(−3ui + 4ui+1 − ui+2

2∆x

),

with ω± =(1 + 2γ2

±)−1

and

γ− =δ2 + (ui − 2ui−1 + ui−2)2

δ2 + (ui+1 − 2ui + ui−1)2and γ+ =

δ2 + (ui − 2ui+1 + ui+2)2

δ2 + (ui+1 − 2ui + ui−1)2,

where, to avoid dividing by zero, we introduce δ, a small positive constant.Concerning the temporal derivative, we apply the TVD-RK2 method based on

the method of line strategy. In particular, at each grid point, we update the solutionusing

un+1i = uni −∆t HLxF

(P +

∂un

∂x

∣∣∣∣x=xi

, xi

),

un+2i = un+1

i −∆t HLxF

(P +

∂un+1

∂x

∣∣∣∣x=xi

, xi

),

un+1/2i =

1

2

(uni + un+2

i

).

Because of the CFL stability condition, the time-marching step is chosen to be ∆t =O(∆x). Indeed, higher-order methods including WENO5 and RK3 are available.However, since operator splitting has already introduced an error of order ∆t, it doesnot seem to be necessary to spend extra time further reducing the error from thenumerical discretization of the HJ equation.

3.2.2. Effective Hamiltonian. At any intermediate step for a particular vectorP , we compute an approximation to the effective Hamiltonian by integrating thenumerical Hamiltonian over the domain Ω using the simple trapezoidal rule, i.e.,

λn+1 =

∫Ω

H(∇un+1 + P, x

)dx

=

∫Ω

H

(P +

∂un+1

∂x, x

)dx

≈ ∆x∑i

HLxF

(P +

∂un+1

∂x

∣∣∣∣x=xi

, xi

).(3.7)


We assume that we have reached the steady-state solution u∞ when∣∣λn+1 − λn

∣∣ ≤ε for some tolerance ε > 0. The trapezoidal rule has an error of O(∆x2), and it hasenough accuracy to maintain the overall convergence of the numerical algorithm.

3.2.3. Initial condition. The initial condition u0 can be chosen by using severaldifferent approaches. If there is no prior information about the solution, one cansimply choose the zero initial condition. We found that such a simple initial conditionworks for all one-dimensional cases presented in section 4. Indeed, an initial conditionappearing to be closer to the steady-state solution will reduce the number of time-marching steps. One might, therefore, use the steady-state solution u∞(P ) for a givenP as an initial condition for a slightly different P .

In this work, we are using a cascadic strategy to improve the computationalefficiency. We first determine the steady-state solutions for a set of P ’s on a rathercoarse mesh. The computational cost of this step is very cheap even though we expectto have a rather large number of iterations until the steady-state solution is reached.Since the viscosity solution is Lipschitz, we interpolate the coarse mesh solution ontoa finer grid and use the resulting interpolant as the initial condition on the finer grid.

Algorithm 1. Effective Hamiltonians with cascadic initialization strategy.

1: Discretize P and Ω. Initialize the mesh in Ω at the coarsest level, and denote thenumber of mesh points in each direction by Mx. Set u0 = 0 on this mesh.

2: for each P do3: while Mx is not at the finest level do4: Obtain un by solving equation (3.3) followed by projection (3.4).5: Compute H(P ) from (3.5).6: Mx = 2Mx and interpolate u∞ onto the finer mesh and use it as u0.7: end while8: end for

3.3. Computational complexity. Let MP and Mx be the number of gridpoints for each direction in P and x. Further, assume that the number of itera-tions required to reach the steady state is N . The overall computational complexityis O(Md

PMdxN).

4. Numerical examples. In the following examples, unless specified otherwise,we will follow the cascadic strategy to determine the initial condition, and the resultingalgorithm will be compared with the algorithm using the zero initial condition u0 = 0.The stability condition for all time-marching steps is chosen to be ∆t = O(∆x).

4.1. One-dimensional examples. In this section, we first consider some one-dimensional examples where the effective Hamiltonians can be determined analyti-cally. When the cascadic strategy is used to initialize the iteration, we start with amesh of 32 points and then double the number of mesh points gradually. The stoppingcriterion for the iterations is chosen to be |λn+1 − λn| < 10−10.

4.1.1. Eikonal equation. We consider the eikonal equation with the Hamilto-nian given by

H(p, y) =1

2|p|2 − V (y),


where V is a Y -periodic potential. Here the Hamiltonian is convex in the p vari-able, and the corresponding effective Hamiltonian in the one-dimensional case can beobtained analytically [22] via

H(P ) =

−minV if |P | ≤ Pc,

α s.t. |P | =∫ 1

0

√2(V (y) + α)dy if |P | > Pc,

where Pc =∫ 1

0

√2(V (y)−minV )dy. In this example, we follow [28] and choose the

potential function V (y) = sin 2πy, which implies H(P ) = 1 for P ≤ Pc = 4/π. Thetime step is chosen to be

∆t =βτ∆x

σx

for some CFL number βτ = 0.75, and the regularization parameter in the numericalHamiltonian is σx = 2.

Figure 1 shows the computed effective Hamiltonian H(P ) for P ∈ [0, 12/π] usingthe proposed scheme with βτ = 0.75. We discretize the interval [0, 12/π] into 10subintervals and determine the effective Hamiltonian for P = 6i

5π with i = 0, 1, · · · , 10.We plot the solutions computed on different sets of mesh points ranging from Mx =128 (i.e., ∆x = 1/128) to Mx = 512 (i.e., ∆x = 1/512). The initial condition fora given P ∈ [0, 12/π] is chosen according to the cascadic strategy we discussed insection 3.2. The initial condition for the case where Mx = 32 is chosen to be the zeroinitial condition u0 = 0 for all P .

To consider the efficiency of the numerical approach, we show in Table 1(a) thenumber of iterations until the time marching reaches the steady state for differentMx’s. For the finest mesh we have tested (Mx = 512), starting from the zero initialcondition takes more than 3000 iterations to reach the steady-state solution for thiscase, P = 6

5π < Pc. If the cascadic strategy is used, the number of iterations reducesto approximately 1000, while the accuracy in the numerical solutions is almost thesame, as shown in Table 1(b). More importantly, these numbers suggest that thenumber of time-marching steps required to reach the steady state is of the sameorder of the number of grid points in each physical dimension, i.e., N = O(Mx).The overall computational complexity will be given by O(Md

PMdxN) = O(M2d+1) if

M = O(MP ) = O(Mx) = O(N).

(a) P

0 1 2 3

H(P

)

0

1

2

3

4

5

6

7

8 N=128 N=256 N=512

(b) P

0 0.5 1 1.5

H(P

)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4 N=128 N=256 N=512

Fig. 1. (Example 4.1.1 with β = 0.75) The effective Hamiltonian H(P ) and the zoom-in of thesolution.


Table 1(Example 4.1.1, the explicit time-splitting scheme using two different initial conditions)

(a) Number of iterations necessary to reach the steady-state solution when P = 65π

with βτ = 0.1.

(b) The computed H( 65π

). The exact solution is H( 65π

) = 1. (c) The convergence rate to H( 65π

).

(d) The estimated H( 65π

) by applying the Richardson extrapolation to the data in (b).

(a)

Mx 128 256 512

u0 = 0 946 1982 3871u0: cascadic 369 638 1059

(b)128 256 512

0.9849 0.9908 0.99600.9849 0.9908 0.9960

(c)

Mx 128–256 256–512

u0 = 0 0.7075 1.1868u0: cascadic 0.7075 1.1868

(d)128–256 256–5120.9966 1.00110.9966 1.0011

Since P < Pc, the exact effective Hamiltonian is known and is given by the con-stant one. The convergence rates are approximately first order, as shown in Table 1(c).In Table 1(d), we apply the Richardson extrapolation method to obtain a more accu-rate solution by assuming that the order of convergence is indeed first order.

One might expect to obtain high-order convergence using higher-order splittingmethods, such as Strang’s splitting scheme given by un → un+1/2 → u∗ → un+1:Solve first

∂u

∂t(t) +H(P +∇u, ·) = 0 in Ω× (tn, tn+1/2),

u(tn) = un,

u(t) periodic in Ω.

Then we set un+1/2 = u(tn+1/2) and compute u∗ = un+1/2−∫

Ωun+1/2(x)dx. Finally,

we repeat the first step for another half time interval on (tn+1/2, tn+1) with the initialcondition u(tn+1/2) = u∗. We have implemented this scheme but were unable toobserve the expected results.

There are two major issues of Strang’s splitting-based approach. To have second-order accurate solutions in the temporal direction, one requires high-order accuratesolutions for each splitting step. For the first stage where one solves the HJ equation,it is relatively easy to obtain high-order solution (we have already implemented theRK2-WENO3). The challenge lies, surprisingly, in the other splitting step and alsothe final step of evaluating the effective Hamiltonian. Note that the solution un tothe HJ solution is a viscosity solution and in general has kinks. Therefore, in theprojection step when we compute the integral of un+1/2, the trapezoidal quadraturerule (even higher-order quadratures including Simpson’s rule) will yield at most first-order accurate solution. This is not too serious though since the projection step isused to guarantee the uniqueness of the overall problem. We eventually need onlythe derivative of the function u. The main issue, however, is in the integral of theHamiltonian (3.5). Since the derivative of u is discontinuous at a kink, we cannotexpect that this derivative will have pointwise convergence in general. This impliesthat the numerical Hamiltonian itself might have an O(1) error near a kink of u. Sothe integral of this numerical Hamiltonian can be of first-order accurate at most.

Moreover, even if one is able to derive a high-order projection scheme and also ahigh-order quadrature for the numerical Hamiltonian, there is actually no guaranteethat Strang’s splitting scheme can always produce second-order accurate solutionfor nonlinear problems. For example, [18] has recently observed a convergence ofonly O(∆t4/3) for an integral-differential problem rather than a typical second-orderconvergence as in a linear problem.


4.1.2. A nonconvex Hamiltonian. Next we consider a slightly more complexnonconvex Hamiltonian given by

H(p, y) =1

2(|p|2 − 1)2 − V (y),

where V is a Y -periodic potential. Here the Hamiltonian is nonconvex in the pvariable. Assuming that V is periodic with minV = 0, we have the following formulafor the effective Hamiltonian [28, 30]:

Case 1. maxR1 V = 12 − δ for some δ > 0. The effective Hamiltonian H(P ) is a

nonconvex function having three flat pieces, namely,

H(P ) =

δ if |P | ≤∫ 1

0

√1−

√2δ + 2V (x)dx,

|P | =∫ 1

0

√1−

√2H(P ) + 2V (x)dx if |P | ≤

∫ 1

0

√1−

√2V (x)dx,

0 if

∫ 1

0

√1−

√2V (x)dx ≤ |P | ≤

∫ 1

0

√1 +

√2V (x)dx,

|P | =∫ 1

0

√1 +

√2H(P ) + 2V (x)dx if |P | ≥

∫ 1

0

√1 +

√2V (x)dx.

Case 2. maxR1 V ≥ 12 . The effective Hamiltonian H(P ) is a convex function,

namely,

H(P ) =

0 when |P | ≤

∫ 1

0

√1 +

√2V (x)dx,

|P | =∫ 1

0

√1 +

√2H(P ) + 2V (x)dx when |P | ≥

∫ 1

0

√1 +

√2V (x)dx.

To validate our computational methodology, we assume further that the potentialfunction is chosen as

V (y) =Vmax

2(sin 2πy + 1),

where Vmax is a scaling parameter to control the amplitude of the potential. Wehave chosen two different values of Vmax given by 0.25 and 0.75 so that the numericalsolutions illustrate the two cases as shown above. When solving the time evolutionequations, the time step is chosen to be

∆t =β∆x

σ3x

for some CFL number βτ = 0.75 and σx = 2.Figure 2 shows the effective Hamiltonian computed using the proposed operator-

splitting scheme with P = 0.2i for i = 0, 1, · · · , 10. To check the convergence ofthe numerical solutions, we follow the previous example and collect the number ofiterations necessary to reach the steady-state solution at the first nontrivial P . Inthis example, we consider P = 0.2 which gives the effective Hamiltonian 0.25 and 0for the cases Vmax = 0.25 and 0.75, respectively. Table 2 shows the studies for thecase with Vmax = 0.25. In Table 2(a), we collect the number of time-marching stepsnecessary to reach the steady state using the initial condition u0 = 0 and also the


(a) P

0 0.5 1 1.5 2

H(P

)

-1

0

1

2

3

4

5 N=128 N=256 N=512

(b) P

0 0.2 0.4 0.6 0.8 1 1.2

H(P

)

-0.2

-0.1

0

0.1

0.2

0.3

0.4 N=128 N=256 N=512

(c) P

0 0.5 1 1.5 2

H(P

)

-1

0

1

2

3

4

5 N=128 N=256 N=512

(d) P

0 0.2 0.4 0.6 0.8 1 1.2

H(P

)

-0.2

-0.1

0

0.1

0.2

0.3

0.4 N=128 N=256 N=512

Fig. 2. (Example 4.1.2) The effective Hamiltonian H(P ) and the zoom-in of the solution with(a)–(b) Vmax = 0.25 and (c)–(d) Vmax = 0.75.

Table 2(Example 4.1.2, the explicit time-splitting scheme with Vmax = 0.25 using two different initial

conditions) (a) Number of iterations necessary to reach the steady-state solution when P = 0.2.(b) The computed H(0.2). The exact solution is H(0.2) = 0.25. (c) The convergence rate to H(0.2).(d) The estimated H(0.2) by applying the Richardson extrapolation to the data in (b).

(a)

Mx 128 256 512

u0 = 0 7047 13373 25436u0: cascadic 476 769 250

(b)128 256 512

0.2524 0.2513 0.25060.2524 0.2513 0.2506

(c)

Mx 128–256 256–512

u0 = 0 0.8726 1.0303u0: cascadic 0.8757 1.0156

(d)128–256 256–5120.2502 0.25000.2502 0.2500

one obtained by the cascadic strategy. The algorithm using the initial condition frominterpolating the coarser mesh solution significantly out-performs the one using thezero initial condition. Using Mx = 512 as an example, the number of marchingsteps using the interpolated solution is around 1600 while the one using the zeroinitial condition is 49000. Note that the accuracy of the final solutions by these twoinitializations are comparable, as shown in Table 2(b). Since the exact solution toboth cases of Vmax are known, we calculate the rate of convergence in Table 2(c). Thisshows that the numerical method is roughly first order accurate. Using the Richardsonextrapolation, we show in Table 2(d) the extrapolated solutions. We consider also thecase Vmax = 0.75 and the corresponding results are shown in Table 3.


Table 3(Example 4.1.2, the explicit time-splitting scheme with Vmax = 0.75 using two different initial

conditions) (a) Number of iterations necessary to reach the steady-state solution when P = 0.2.(b) The computed H(0.2). The exact solution is H(0.2) = 0. (c) The convergence rate to H(0.2).(d) The estimated H(0.2) by applying the Richardson extrapolation to the data in (b).

(a)

Mx 128 256 512

u0 = 0 17918 10590 19189u0: cascadic 1562 2942 5705

(b)128 256 512

-0.0134 -0.0073 -0.0069-0.0133 -0.0051 -0.0026

(c)

Mx 128–256 256–512

u0 = 0 0.8853 0.0796u0: cascadic 1.3745 0.9924

(d)128–256 256–512-0.0011 -0.00650.0030 -0.0000

4.2. Two-dimensional examples. In this section, we consider some two di-mensional examples. In this case, the computational efficiency of the numerical algo-rithm is particularly important since for each P , one has to solve a two dimensionalproblem up to the steady state. This implies the overall computational complexitywill be O(M5) if M = O(MP ) = O(Mx) = O(N). In this section, therefore, weconsider only the cascadic strategy to initialize the algorithm. In this case, we startwith a mesh of 64-by-64 grid points and then double the number of mesh points ineach direction gradually.

4.2.1. Eikonal equation. We consider the two dimensional eikonal equationwith Hamiltonian given by

H(px, py, x, y) =1

2

(p2x + p2

y

)− V (x, y),

where V is a periodic potential. In this example, we consider the following threedifferent potentials:

• V1(x, y) = cos(2πx) + cos(2πy) (for which we have an exact formula since theoverall Hamiltonian is separable in a dimension-by-dimension fashion);

• V2(x, y) = sin(2πx) sin(2πy);• V3(x, y) = cos(2πx) + cos(2πy) + cos[2π(x− y)].

Figures 3–5 show the computed effective Hamiltonian for different V ’s. For these

examples, we consider the domain for P ∈[− 4.4

π ,4.4π

]2with MP = 41. We compute

(a)

2

0

-2-2

0

4

3.5

3

2.5

22

(b) -2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

Fig. 3. (Example 4.2.1) (a) The effective Hamiltonian H(P ) with V = V1 with Mx = 256 and(b) the contour plot.


(a)

10

-1-1

0

1

1.5

1

2

(b) -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Fig. 4. (Example 4.2.1) (a) The effective Hamiltonian H(P ) with V = V2 and Mx = 256 and(b) the contour plot.

(a)

10

-1-1

0

1

1.5

3

2.5

2

(b) -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Fig. 5. (Example 4.2.1) (a) The effective Hamiltonian H(P ) with V = V3 with Mx = 256 and(b) the contour plot.

the effective Hamiltonian on the physical domain [0, 1]2 using various meshes rangingfrom the coarse mesh with Mx = 64 to the fine mesh with Mx = 256. We use thecascadic approach to determine the initial condition for each refinement of Mx. Forthe coarsest level, i.e., Mx = 64, we simply use the zero initial condition. Figure 3(a)shows the computed effective Hamiltonians on the finest mesh of Mx. The contourplot of the effective Hamiltonian is shown in Figure 3(b). Like the one-dimensionalexamples, we have also collected various convergence statistics for this particularHamiltonian under the potential V1 when we have the exact solution. These numbersare shown in Table 4. In particular, when P = (0, 0), we have the effective Hamiltoniangiven by H(0, 0) = 2. Similar to what we have demonstrated in the one-dimensionalcases, the method converges to the exact solution with an order of approximately one,as shown in Table 4(c). We have also plotted the effective Hamiltonians and theircontour plots for the cases with V = V2 and V3 in Figures 4 and 5.

4.2.2. A double pendulum. The double-pendulum Hamiltonian is a noninte-grable example for which the effective Hamiltonian is not known. The Hamiltonianfor the double pendulum is given by

H(px, py, x, y) =p2x − 2px py cos(2π(x− y)) + 2p2

y

2− cos2(2π(x− y))+ 2 cos 2πx+ cos 2πy .


Table 4(Example 4.2.1 with V = V1) (a) Number of iterations necessary to reach the steady-state

solution when P = (0, 0) and P = ( 4.4π, 4.4π

). (b) The computed H(0, 0) and H( 4.4π, 4.4π

). The

exact solution is H(0, 0) = 2. (c) The convergence rate to H(0, 0). (d) The estimated H(0, 0) andH( 4.4

π, 4.4π

) by applying the Richardson extrapolation to the data in (b).

(a)

Mx 64 128 256P = (0, 0) 521 577 98

P =(4.4π, 4.4π

)19471 623 912

(b)

Mx 64 128 256H(0, 0) 1.937 1.968 1.984

H(4.4π, 4.4π

)2.247 2.244 2.246

(c)Mx 64–128 128–256

Rate, P = (0, 0) 0.9985 0.9527(d)

Mx 64–128 128–256H(0, 0) 1.9999 1.9989

H(4.4π, 4.4π

)2.2405 2.2485

(a)

10

-1-1

0

1

7

6

5

4

3

(b) -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Fig. 6. (Example 4.2.2) (a) The effective Hamiltonian H(P ) for P ∈ [− 4.4π, 4.4π

]2 with Mx =256 and (b) the contour plot.

Figure 6 shows the effective Hamiltonian for P ∈[− 4.4

π ,4.4π

]2computed on a mesh

using Mx = 64 with a cascadic initialization strategy from Mx = 16. Our solution ofthe effective Hamiltonian on the computational domain seems to be smoother thanthe one obtained in the literature, such as [25].

4.2.3. A Lipschitz nonconvex Hamiltonian. We follow [30] and consider thefollowing nonconvex Hamiltonian H with the potential V :

H(p, x) = min|p− e1|, |p+ e1| − V (x),

where p = (p1, p2) and e1 = (1, 0). We consider two cases for the potential V givenby

• V1(x) = Vmax

4 [1 + sin(2πx1)] [1 + sin(2πx2)] and

• V2(x) = Vmax

2

[sin2(2πx1) + sin2(2πx2)

],

where the constant Vmax serves as the scaling parameter to increase or decrease theeffect of the potential.

Figure 7(a) shows the effective Hamiltonian for P ∈[− 4.4

π ,4.4π

]2computed on a

mesh using Mx = 256 with a cascadic initialization strategy from Mx = 64. We plotthe corresponding contour plot for the potential V1 in Figure 7(b). The correspondingplots for V2 are shown in Figure 8. See [30] for more details on properties of theseeffective Hamiltonians.


(a)

10

-1-1

0

1

10.80.60.40.2

1.21.4

(b) -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

0.2

0.4

0.6

0.8

1

1.2

Fig. 7. (Example 4.2.3) (a) The effective Hamiltonian H(P ) for V = V1 with Vmax = 1 andMx = 256 and the contour plot.

(a)

10

-1-1

0

1

0.4

1

0.2

0.6

0.8

(b) -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Fig. 8. (Example 4.2.3) (a) The effective Hamiltonian H(P ) for V = V2 with Vmax = 1 andMx = 256 and the contour plot.

4.2.4. Vakonomic mechanics. In this example, we consider the following non-strictly convex Hamiltonian relating to the vakonomic mechanics [2, 19], where theHamiltonian is given by

H(p, x) =(f · p)2

2+

(g · p)2

2+ V (x),

where p = (p1, p2), f = (0, 1), g = (cos 2πx2, sin 2πx2), and a potential V (x) =cos 2πx1 + sin 2π(x1 − x2). Since the potential V is not symmetric in x1 and x2,the effective Hamiltonian cannot be determined analytically. Our numerical effectiveHamiltonian for P = [−π, π]2 on a mesh of Mx = 256 is shown in Figure 9.

4.2.5. Nonexistence of viscosity solutions. It is known that some HJ equa-tions do not admit viscosity solutions [3, 23, 19]. One particular example is thefollowing quasi-periodic Hamiltonian taken from [23]:

H(px, py, x, y) = |px + αpy|+ sinx+ sin y.

For an irrational number α, it can be shown that the HJ equation does not admita viscosity solution. In this example, we have chosen α = π numerically, and ourcomputed effective Hamiltonian is shown in Figure 10.


2

0

-2-2

0

2

8

2

4

6

-2 0 2-3

-2

-1

0

1

2

3

3

4

5

6

7

8

9

Fig. 9. (Example 4.2.4) The effective Hamiltonian H(P ) and the contour plot.

2

0

-2-2

0

6

2

8

4

2

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2

3

4

5

6

7

8

9

Fig. 10. (Example 4.2.5) The effective Hamiltonian H(P ) and the contour plot.

4.2.6. Finsler metrics. In this example, we consider the Hamiltonians formodeling the quasi-P and the quasi-SV waves described by the quadratic equation[29, 36, 33]:

c1p4x + c2p

2xp

2y + c3p

4y + c4p

2x + c5p

2y + 1 = 0,

where c1 = a11a44, c2 = a11a33 + a244 − (a13 + a44)2, c3 = a33a44, c4 = −(a11 + a44),

and c5 = −(a33 + a44) with some given elastic parameters aij ’s.The corresponding squared quasi-P wave Hamiltonian is given by

(4.1) − 1

2(c4p

2x + c5p

2y) +

√1

4(c4p2

x + c5p2y)2 − (c1p4

x + c2p2xp

2y + c3p4

y) .

The elastic parameters for this convex Hamiltonian is given by a11 = 15.0638, a33 =10.8373, a13 = 1.6381, and a44 = 3.1258.

The corresponding squared quasi-SV wave Hamiltonian is given by

−1

2(c4p

2x + c5p

2y)−

√1

4(c4p2

x + c5p2y)2 − (c1p4

x + c2p2xp

2y + c3p4

y) ,(4.2)


1

0

-1-1

0

5

0

20

15

10

1

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

2

4

6

8

10

12

14

16

18

Fig. 11. (Example 4.2.6) The effective Hamiltonian H(P ) for the quasi-P wave (4.1) withVmax = 2 and the contour plot.

(a) -1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(b) -1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(c) -1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(d) -1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 12. (Example 4.2.6) The effective Hamiltonian H(P ) for the quasi-SV wave (4.2) with(a) Vmax = 0, (b) Vmax = 0.25, (c) Vmax = 0.5, and (d) Vmax = 1.

which is nonconvex. The elastic parameters for this case are a11 = 15.9, a33 = 6.21,a13 = 4.82, and a44 = 4.00. We consider the potential

V (x) =Vmax

2

[sin2(2πx1) + sin2(2πx2)

],

where the constant Vmax serves as the scaling parameter to increase or decrease theeffect of the potential.


Figure 11 shows the effective Hamiltonian H(P ) for the convex squared quasi-Pwave Hamilontian with Vmax = 2 in the potential. The Hamiltonian is convex, and theplateau region near the origin is nicely captured. Figure 12 shows the correspondingeffective Hamiltonian for the quasi-SV wave Hamilontian for various Vmax’s. Forthe first case when Vmax = 0 as shown in Figure 12(a), we see that the effectiveHamiltonian reduces to the original Hamiltonian. The nonconvex structure in theHamiltonian is clearly shown. As we increase the magnitude of the potential fromzero to one by changing the coefficient Vmax, we see that the size of the flat regionnear the origin grows and that the effective Hamiltonian is becoming more convex.

5. Conclusion. We have developed a simple efficient operator-splitting methodfor computing effective Hamiltonians, and the method works for both convex and non-convex Hamiltonians. To speed up the Lie scheme–based operator-splitting method,we further propose a cascadic initialization strategy so that the steady state of theunderlying time-dependent HJ equation can be reached rapidly. Extensive numericalexamples demonstrate the effectiveness of the proposed algorithm.

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A SIMPLE EXPLICIT OPERATOR-SPLITTING METHOD FOR EFFECTIVE