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SIAM J. SCI. COMPUT.
Vol. 35, No. 1, pp. A122–A149
c 2013 Society for Industrial and Applied Mathematics
ALTERNATING EVOLUTION SCHEMES FOR HAMILTON–JACOBI
EQUATIONS∗
HAILIANG LIU† , MICHAEL POLLACK† , AND HASEENA SARAN†
Abstract. In this work, we propose a high-resolution alternating evolution (AE) scheme to solve
Hamilton–Jacobi equations. The construction of the AE scheme is based on an alternating evolution
system of the Hamilton–Jacobi equation, following the idea previously developed for hyperbolic conservation laws. A semidiscrete scheme derives directly from a sampling of this system on alternating
grids. Higher order accuracy is achieved by a combination of high order nonoscillatory polynomial
reconstruction from the obtained grid values and a time discretization with matching accuracy. Local
AE schemes are made possible by choosing the scale parameter to reflect the local distribution of
waves. The AE schemes have the advantage of easy formulation and implementation and efficient
computation of the solution. For the first local AE scheme and the second order local AE scheme
with a limiter, we prove the numerical stability in the sense of satisfying the maximum principle.
Numerical experiments for a set of Hamilton–Jacobi equations are presented to demonstrate both
accuracy and capacity of these AE schemes.
Key words. alternating evolution, Hamilton–Jacobi equations, viscosity solution
AMS subject classifications. 65M06, 65M12, 35L02
DOI. 10.1137/120862806
1. Introduction. In this paper, we develop a new alternating evolution (AE)
method to solve time-dependent Hamilton–Jacobi (HJ) equations. We describe the
designing principle of our new AE scheme through the following form:
φt + H(x, ∇x φ) = 0, φ(x, 0) = φ0 (x),
x ∈ Rd , t > 0.
Here, d is the space dimension, the unknown φ is scalar, and H : Rd → R1 is a
nonlinear Hamiltonian. The HJ equation arises in many applications, ranging from
geometrical optics to differential games. These nonlinear equations typically develop
discontinuous derivatives even with smooth initial conditions, the solutions of which
are nonunique. In this paper, we are only interested in the viscosity solution [7, 6, 29],
which is the unique physically relevant solution in some important applications.
The difficulty encountered for the satisfactory approximation of the exact solutions of these equations lies in the presence of discontinuities in the solution derivatives. An important class of finite difference methods for computing the viscosity
solution is the class of monotone schemes introduced by Crandall and Lions [8]. Unfortunately, monotone schemes are at most first order accurate. The need for devising
more accurate numerical methods for HJ equations has prompted the abundant research in this area in the last two decades, including essentially nonoscillatory (ENO)
or weighted ENO (WENO) finite difference schemes (see, e.g., [24, 25, 13, 17, 28, 31])
and central or central-upwind finite difference schemes (see, e.g., [19, 16, 15, 4, 2, 3]),
as well as discontinuous Galerkin methods [12, 5, 30, 22].
∗ Submitted to the journal’s Methods and Algorithms for Scientific Computing section January 19,
2012; accepted for publication (in revised form) November 26, 2012; published electronically January
8, 2013. This research was partially supported by the National Science Foundation under grant DMS
09-07963.
http://www.siam.org/journals/sisc/35-1/86280.html
† Mathematics Department, Iowa State University, Ames, IA 50011 (hliu@iastate.edu, mpollack@
iastate.edu, haseena.saran@gmail.edu).
A122
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A123
The preliminary goal of this paper is to design high order finite difference schemes
for HJ equations by following the alternating evolution framework introduced in [21]
for hyperbolic conservation laws; see local AE schemes developed by Saran and Liu
[26]. The global AE scheme [21] shares similar features to central schemes by Y.
Liu [23] using overlapping cells. In such a framework, the general setting is to refine
the original PDE by an AE system which involves two representatives: {u, v}. In
the evolution of u, terms involving spatial derivatives are replaced by v’s derivatives
and augmented by an additional relaxation term (v − u)/, which serves to communicate the two representatives. The AE system for scalar hyperbolic conservation laws
was shown in [21] to be capable of capturing the exact solution when initially both
representatives are chosen as the given initial data. Such a feature allows for a sampling of two representatives over alternating grids. Using this alternating system as a
“building base,” we apply standard approximation techniques to the AE system: high
order accuracy is achieved by a combination of high order nonoscillatory polynomial
reconstruction in space and an ODE solver in time with matching accuracy, following
those implemented for hyperbolic conservation laws in [26]. Preliminary results on
AE schemes for HJ equations were reported in the third author’s thesis [1].
For Hamilton–Jacobi equations, we consider the following alternating evolution
system:
1
(v − u),
1
vt + H(x, ∇x u) = (u − v).
ut + H(x, ∇x v) =
Here, > 0 is a scale parameter of the user’s choice. At t = 0, we take the initial data
u0 (x) = v0 (x) = φ0 (x).
We sample the above system over alternating grids when performing the spatial discretization, leading to a class of high-resolution semidiscrete AE schemes. One of the
main differences between the formulation of AE schemes for HJ equations and that
for conservation laws is that here we use grid point values instead of cell averages.
The semidiscrete schemes thus obtained have the form
1
1
d
Φα + H(xα , ∇x pα [Φ](xα )) + Φα = pα [Φ](xα ),
dt
where pα [Φ](x) is the polynomial reconstruction based on grid values of Φ in the
domain centered at xα . The scale parameter and the time step size Δt are chosen
to stabilize the time discretization. In the one-dimensional case, the fully discrete AE
scheme of first order becomes
Φnk+1 − Φnk−1
Δt
Δt 1 n
n+1
n
n
(Φ
+ Φk−1 ) − H xk ,
Φk = 1 −
Φk +
, ≥ Δt,
2 k+1
2Δx
which when = Δt is taken reduces to the celebrated Lax–Friedrichs scheme
Φnk+1 − Φnk−1
1 n
n+1
n
.
Φk = (Φk+1 + Φk−1 ) − H xk ,
2
2Δx
Thus the class of AE schemes can be viewed as a higher order extension of the Lax–
Friedrichs scheme.
To put our study in the proper perspective, we recall that the success of existing
finite difference schemes for HJ equations is due to two factors: the local enforcement
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A124
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
of the equation in question and the nonoscillatory piecewise polynomial interpolation
from evolved grid values. The present AE scheme differs from the existing ones in the
local enforcement. The ENO/WENO type schemes [24, 25, 13, 17, 28, 31] are based
mainly on some local refinement of HJ equations by
−
ut + Ĥ(x, φ+
x , φx ) = 0,
where Ĥ is the numerical Hamiltonian which needs to be carefully chosen to ensure
the viscosity solution is captured when φx becomes discontinuous. Instead, the central type schemes [20, 19, 16, 15, 4, 2, 3] choose to evolve the constructed polynomials
in smooth regions such that the Taylor expansion may be used in the scheme derivation. Effort has been made to extend these ideas toward some discontinuous Galerkin
methods such as in [12, 5, 30, 22]; however, these extensions either are restricted or
involve some further local refinement.
The scheme we design here is different in that it is based on alternatively sampling
the AE system with spatial accuracy enhanced by interlaced local interpolations.
The procedure discussed in this paper opens a new door to deriving robust finite
difference schemes for HJ equations. Also the semidiscrete scheme thus obtained
offers an economic approach for achieving a matching accuracy in time. Another
attractive feature of the AE scheme is the amount of the leverage in the choice of
. Indeed, different choices of the scale parameter in such a procedure yield different
AE schemes such as local and global AE schemes. For the polynomial construction,
we use standard polynomial interpolation with ENO choice of stencils or appropriate
limiters to ensure the computed solution is the viscosity solution. Extension of the
AE idea to a discontinuous Galerkin setting is under investigation.
It should be noted that even though our AE schemes are derived based on sampling the AE system, we do not solve the system directly. The AE system simply
provides a systematic way for developing numerical schemes of both semidiscrete and
fully discrete form for the original problem, instead of as an approximation system at
the continuous level.
The article is organized as follows. In section 2, we formulate the AE method
for one-dimensional HJ equations. We then give a rigorous proof of L∞ stability
by the AE method of both first and second order in section 3. Extensions to the
multidimensional case are given in section 4, including stability for the AE method
of first and second order. In section 5, we show numerical results, which include both
one- and two-dimensional problems. The results illustrate accuracy, efficiency, and
high resolution near kinks. Section 6 ends this paper with our concluding remarks.
2. Alternating evolution methods. Our numerical schemes for HJ equations
consist of a semidiscrete formulation based on sampling of the AE system on alternating grids and a fully discrete version by further using an appropriate Runge–Kutta
solver.
To illustrate, we start with the one-dimensional HJ equation of the form
φt + H(x, φx ) = 0.
The “building base” is the following AE system:
(2.1)
(2.2)
1
(v − u),
1
vt + H(x, ux ) = (u − v).
ut + H(x, vx ) =
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A125
2.1. Semidiscrete formulation. Consider a uniform discretization {xk , k ∈ Z}
with grid size Δx. We sample the AE approximate system (2.1)–(2.2) at the even
and odd grid points, respectively, to obtain
1
1
d
u2i (t) = − u2i + L[v](x2i , t),
dt
d
1
1
v2i+1 (t) = − v2i+1 + L[u](x2i+1 , t),
dt
where
L[Φ](x, t) = Φ − H(x, Φx ).
We assume that we have computed the solution at t, denoted by
uk ,
k = 2i,
Φk =
k = 2i + 1.
vk ,
To update each grid value Φk with a high order of accuracy, we construct a nonoscillatory polynomial approximation pk [Φ](x) using grid point values {xk±i } for some
i = 1, 2, . . . . The numerical approximation for L is then realized by L[pk [Φ]](xk , t),
which leads to the following semidiscrete scheme:
d
1
1
Φk (t) = − Φk + Lk [Φ](t),
dt
(2.3)
where
Lk [Φ](t) = pk [Φ](xk ) − H (xk , ∂x pk [Φ](xk )) .
The fully discrete scheme follows from applying an appropriate Runge–Kutta solver
to (2.3). For a computational domain [a, b] with x0 = a, xN = b, and Δx = (b − a)/N ,
we summarize the algorithm as follows.
Algorithm 2.1.
1. Initialization: at any node xk , compute the initial data as Φ0k = φ0 (xk ),
k = 0, 1, . . . , N .
2. Polynomial construction: from {Φk±i } construct (r + 1)th order nonoscillatory polynomial pk [Φ](x) and rth order polynomial ∂x pk [Φ](x) on Ik , k =
1, 2, . . . , N , then sample at xk to get
Lk [Φ] = pk [Φ](xk ) − H (xk , ∂x pk [Φ](xk )) ,
k = 1, 2, . . . , N.
3. Evolution: obtain Φn+1 from Φn by the following TVD Runge–Kutta type
procedure [25]:
(l)
Φk =
(0)
Φk
=
l−1
(i)
αli Φk −
i=0
(n)
Φk ,
βli (i)
Φk + Lk [Φ(i) ] , l = 1, . . . , r,
(r)
Φn+1
= Φk .
k
In the AE schemes up to the third order, is chosen such that the stability
condition,
Δt ≤ ≤ Q
Δx
,
max |Hp (x, p)|
is satisfied. The choice of Q depends on the order of the scheme; see (3.1) and (3.5)
in section 3.
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A126
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
2.2. Fully discrete AE schemes. We now present AE schemes of the first,
second, and third order for HJ equations and then discuss advantages of using a local
parameter = nj .
2.2.1. First order scheme. We start with a linear interpolant at nodes {xk−1 ,
xk+1 },
p1k [Φ](x) = Φk−1 + sk (x − xk−1 ), sk =
Φk+1 − Φk−1
,
2Δx
and sample at xk ,
p1k [Φ](xk ) =
Φk+1 + Φk−1
2
and ∂x p1k [Φ](xk ) = sk =
Φk+1 − Φk−1
.
2Δx
This, when combined with a forward Euler in time discretization, gives the first order
numerical scheme
Φnk+1 − Φnk−1
1 n
n
n
(Φ
=
(1
−
κ)Φ
+
κ
+
Φ
)
−
H
x
,
(2.4) Φn+1
,
k
k
k−1
k
2 k+1
2Δx
Δx
where κ := Δt
< 1. When ≤ max |Hp (·)| , this forms a class of monotone schemes,
with the celebrated Lax–Friedrichs scheme being a special case of κ = 1.
2.2.2. Second order scheme. A second order continuous, piecewise quadratic
reconstruction gives
p2k [Φ](x) = Φk−1 + sk (x − xk−1 ) +
sk
(x − xk−1 )(x − xk+1 ),
2
where sk is the first numerical derivative defined as before and sk is an approximation
to the exact second derivative φxx . We then obtain
p2k [Φ](xk ) =
s
Φk+1 + Φk−1
− k (Δx)2
2
2
and ∂x p2k [Φ](xk ) = sk =
Φk+1 − Φk−1
,
2Δx
which when combined with the second order Runge–Kutta method (Heun’s method)
gives
Φ∗k = (1 − κ)Φnk + κLk [Φn ],
1−κ
1 n
κ
n+1
Φk = Φk +
Φ∗k + Lk [Φ∗ ],
2
2
2
(2.5)
(2.6)
where
(2.7)
Lk [Φ] =
1 n
(Δx)2 Φk+1 − Φk−1
(Φk−1 + Φnk+1 ) −
sk − H xk ,
.
2
2
2Δx
The nonoscillatory nature of the scheme is realized by the choice of sk . We follow the
ENO interpolation technique [11, 25] to select the “smoother” polynomial candidates.
For a second order approximation, we choose from two quadratic interpolants which
use {xk−3 , xk−1 , xk+1 } and {xk−1 , xk+1 , xk+3 }, respectively, following
sk+2 − sk sk − sk−2
,
sk = M
(2.8)
,
2Δx
2Δx
where M(a, b) = a if |a| ≤ |b| and b if |b| < |a|.
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A127
2.2.3. Third order scheme. We formulate the third order scheme by using a
cubic polynomial interpolant with the ENO selection. For x∗ = xk−3 or x∗ = xk+3
determined through the ENO selection (2.8), a cubic polynomial interpolant is given
as
p3k [Φ](x) = Φk−1 + sk (x − xk−1 ) +
+
sk
(x − xk−1 )(x − xk+1 )
2
sk
(x − xk−1 )(x − xk+1 )(x − x∗ ),
6
where the ENO selection requires that
sk+2 − sk sk − sk−2
,
sk = M
(2.9)
.
2Δx
2Δx
The obtained polynomial and its derivative when evaluated at xk give
1
s
s
(Φk+1 + Φk−1 ) − k (Δx)2 − k (Δx)2 (xk − x∗ ),
2
2
6
s
−
Φ
Φ
k+1
k−1
− k (Δx)2 ,
∂x p3k [Φ](xk ) =
2Δx
6
p3k [Φ](xk ) =
which when combined with the third order Runge–Kutta method gives
(1)
Φk = (1 − κ)Φnk + κLk [Φn ],
3
1
(2)
(1)
Φk = Φnk + (1 − κ) Φk +
4
4
1
2
(2)
Φn+1
= Φnk + (1 − κ) Φk +
k
3
3
1
κLk [Φ(1) ],
4
2
κLk [Φ(2) ]
3
with
Lk [Φ] =
1 n
s
s
(Φk−1 + Φnk+1 ) − k (Δx)2 − k (Δx)2 (xk − x∗ )
2
2
6
s
− H xk , sk − k (Δx)2 .
6
Remark 2.2. The above ENO selection procedure can be applied hierarchically
to any higher order polynomial approximation. Also, to make stencils more compact, mixed grids {Φk±i } instead of even or odd grids {Φk+2i−1 } may be used in the
polynomial construction.
2.3. Local AE schemes. In practice, it is preferred to use local AE schemes,
in which local speeds instead of global speeds are explored in the scheme formulation
for . The notation Mk is used to denote the range
Mk = min ∂x pk [Φ](x), max ∂x pk [Φ](x) ,
Ik
Ik
where Ik = [xk−1 , xk+1 ]. Depending on the way k is defined, we can formulate two
local AE schemes:
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A128
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
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1. Local AE1 scheme. We choose k such that
k ≤ Q
(2.10)
Δx
.
max |Hp (x, p)|
p∈Mk ,x∈Ik
2. Local AE2 scheme. We choose k such that
k ≤ Q
(2.11)
maxx∈Ik
1
|Mk |
Δx
|Hp (x, p)| dp
.
Mk
In both cases we want the time step
Δt < min k ,
k
since the stability conditions require that κ = Δt
k < 1. Q is a factor that is dependent
on the order of the scheme and the stability conditions to be presented in section 3
provide the range of values Q can take.
3. Stability analysis. In this section we show that the first and second order
schemes are nonoscillatory in the sense of satisfying the maximum principle. Let Φn
be a computed solution and |Φn |∞ = maxk {|Φnk |} define the standard l∞ norm. For
simplicity we present the stability results only for the case H(x, p) = H(p).
We first prove stability for global AE schemes. The proofs for local schemes are
similar; only needs to be considered as defined locally instead of globally.
Theorem 3.1. Let Φ be computed from the first order AE scheme (2.4) for the
HJ equation φt + H(φx ) = 0. If
max |H (·)| ≤ 1
Δx
(3.1)
and
Δt < ,
hold, then
min Φnk ≤ Φn+1
− H(0)Δt ≤ max Φnk ,
k
(3.2)
k
k
n ∈ N.
Proof. The scheme (2.4) can be written as
Φn+1
k
When
have
= (1 −
Δx
κ)Φnk
+ κLk [Φ],
Φn + Φnk+1
− H
Lk [Φ] = k−1
2
Φnk+1 − Φnk−1
2Δx
.
max |H | ≤ 1, Lk [Φ] becomes nondecreasing in both Φk−1 and Φk+1 , we
min Φk − H(0) = Lk [min Φk ] ≤ Lk [Φ] ≤ Lk [max Φk ] = max Φk − H(0).
Note that for κ < 1, Φn+1
is a convex combination of Φnk and Lk [Φn ]; therefore the
k
claimed estimate (3.2) follows.
Without loss of generality we can assume H(0) = 0 (this can be made so by a
transform of φ − H(0)t in the original equation), so that the estimate (3.2) becomes
|Φn+1 |∞ ≤ |Φn |∞ .
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A129
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ENO from (2.8)
MM from (3.3)
MM from (3.4)
0.55
0.5
0.45
0.4
0.35
0.3
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Fig. 1. Comparison of three different approximations for sk , ENO from (2.8) and MM from
(3.3) and (3.4), when second order scheme is applied to Example 5.3 with T = 1, N = 80, Δt = 0.9.
Such an estimate for second order AE schemes still holds if some stricter limiter on
numerical derivatives sk is taken. As is known, an alternative limiter for the second
order scheme is the minmod limiter
sk+1 − sk sk − sk−1
sk = MM
(3.3)
,
.
Δx
Δx
Here M M denotes the minmod nonlinear limiter
⎧
⎨ minj {xj }
maxj {xj }
MM{x1 , x2 , . . . } =
⎩
0
if xj > 0 ∀j,
if xj < 0 ∀j,
otherwise.
We modify this limiter to obtain
(3.4)
sk =
1
(MM{sk+1 , sk } − MM{sk , sk−1 }) .
Δx
A comparison of the ENO selection and the minmod limiters from (3.3) and (3.4)
is shown in Figure 1 for Example 5.3. As one can see, the ENO approximation provides
the most accurate results for the reference solution.
In order to establish the stability result, sk will be chosen as the stricter limiter
from (3.4).
Theorem 3.2. Let Φn be computed from the second order AE scheme (2.5) for
HJ equations with H(0) = 0 and with slopes sk defined as in (3.4). If
(3.5)
1
max |H (·)| ≤
Δx
2
and
Δt < hold, then
(3.6)
|Φn+1 |∞ ≤ |Φn |∞ ,
n ∈ N.
Proof. It suffices to show that
(3.7)
|L[Φn ]|∞ ≤ |Φn |∞ ,
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A130
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
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so that when κ < 1,
|Φ∗ |∞ ≤ (1 − κ)|Φn |∞ + κ|Φn |∞ ≤ |Φn |∞ .
Then, from (2.6), it follows that
|Φ
n+1
|∞
1
≤ |Φn |∞ +
2
1−κ
2
|Φ∗ |∞ +
κ
|L[Φ∗ ]|∞
2
from which the maximum principle as claimed in (3.6) follows.
We now prove (3.7). From (2.7) we have that
n
Φk+1 − Φnk−1
1
s
Lk [Φn ] = (Φnk+1 + Φnk−1 ) − k (Δx)2 − H
.
2
2
2Δx
Notice that H(0) = 0 and define
⎧
sk
⎪
⎨ −(Δx)2
Φnk+1 − Φnk−1
βkn :=
⎪
⎩
0
if Φnk+1 = Φnk−1 ,
if Φnk+1 = Φnk−1 .
Hence,
(3.8)
1 n
βn
(Φk+1 + Φnk−1 ) + k (Φnk+1 − Φnk−1 ) −
H (·)(Φnk+1 − Φnk−1 )
2
2
2Δx
1
1
1 + βkn −
H (·) Φnk+1 +
1 − βkn +
H (·) Φnk−1 ,
=
2
Δx
2
Δx
Lk [Φn ] =
where (·) is an unspecified intermediate value between 0 and sk . The minmod property
on the second numerical derivative sk leads to
sk ≤
1
|sk |
=
|Φn − Φnk−1 |,
Δx
2(Δx)2 k+1
max |H | ≤ 12 ensures that all
and hence βkn ≤ 12 . This together with the condition Δx
the coefficients on the right of (3.8) are nonnegative and we can take the maximum
norm to obtain
|L[Φn ]|∞ ≤ |Φn |∞ ,
as claimed.
4. The multidimensional case. By similar procedures we can construct AE
schemes for multidimensional HJ equations:
φt + H(x, ∇x φ) = 0,
x ∈ Rd .
We start with the AE formulation
(4.1)
1
1
ut + u = v − H(x, ∇x v).
Let {xα } be uniformly distributed grids in Rd . Consider Iα to be a hypercube centered
at xα with vertices at xα±1 where the number of vertices is 2d .
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A131
We consider two different constructions of AE schemes for multidimensions. For
the first type, given grid values {Φα }, we construct a continuous, piecewise polynomial
pα [Φ](x) ∈ Pr defined in Iα such that
pα [Φ](xα±1 ) = Φα±1 .
These polynomials are then put into the right-hand side of (4.1). Here Pr denotes a
linear space of all polynomials of degree at most r in all xi :
⎧
⎫
⎨
⎬
Pr := p | p(x) =
aβ xβ , 1 ≤ i ≤ d, aβ ∈ R .
⎩
⎭
0≤βi ≤r
Note dim(Pr ) = (r + 1)d . Sampling the AE system (4.1) at xα , which is the common
vertex of Iα±1 , we obtain the semidiscrete AE scheme
(4.2)
1
1
d
Φα + Φα = Lα [Φ],
dt
Lα [Φ] = pα [Φ](xα ) − H(xk , ∇x pα [Φ](xα )).
The second type will involve a dimension-by-dimension approach. For the twodimensional case, we construct a polynomial pj,k in the x-direction and qj,k in the
y-direction in a similar fashion to the one-dimensional case. The AE formulation for
two dimensions then becomes
(4.3)
1
1
ut + u = v − H(∂x pj,k [v], ∂y qj,k [v]).
An average of the p and q polynomials will be used on the v term so that the
semidiscrete AE scheme becomes
d
1
1
Φj,k + Φj,k = Lj,k [Φ],
dt
where
Lj,k =
pj,k [Φ](xj , yk ) + qj,k [Φ](xj , yk )
− H (∂x pj,k [Φ](xj , yk ), ∂y qj,k [Φ](xj , yk )) .
2
The strong stability-preserving Runge–Kutta method [10] can be used to achieve
a time discretization with matching accuracy.
4.1. Averaging construction. We will consider the first type of construction
for an AE scheme in multidimensions, which will be referred to as the averaging construction. For simplicity, we now present our AE schemes in the two-dimensional case
d = 2 and use x, y instead of x1 , x2 . For mesh sizes Δx, Δy, Δt, Φnj,k will denote the
numerical approximation to the viscosity solution φ(xj , yk , tn ) = φ(jΔx, kΔy, nΔt)
of
φt + H(φx , φy ) = 0.
We shall use the notation
Dx0 Φj,k =
Φj+1,k − Φj−1,k
,
2Δx
Dy0 Φj,k =
Φj,k+1 − Φj,k−1
2Δy
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A132
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
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for central differences and
Ax Φj,k =
Φj+1,k + Φj−1,k
,
2
Ay Φj,k =
Φj,k+1 + Φj,k−1
2
for averages.
We begin with a continuous linear interpolation over each rectangular Ij,k centered
at (xj , yk ) = (jΔx, kΔy) with vertices at (xj±1 , yk±1 ). In this case the polynomial is
uniquely determined by grid values at four vertices of Ij,k satisfying
pj,k [Φ](xj±1 , yk±1 ) = Φj±1,k±1 .
Such an interpolant is given by
p1j,k [Φ](x, y) = Ax Ay Φj,k + Dx0 Ay Φj,k (x − xj )
+ Dy0 Ax Φj,k (y − yk ) + Dx0 Dy0 Φj,k (x − xj )(y − yk ).
Substitution of this into the right-hand side of (4.2) and use of forward Euler timediscretization yields the first order AE scheme,
(4.4)
(4.5)
n
n
Φn+1
j,k = (1 − κ)Φj,k + κLj,k [Φ ],
Lj,k [Φ] = Ax Ay Φj,k − H(Dx0 Ay Φj,k , Dy0 Ax Φj,k ).
The scale parameter is chosen to be
max |H1 (·)| max |H2 (·)|
·
+
≤ 1.
Δx
Δy
Here, Hi (p, q) is the partial derivative of H with respect to the ith argument (dependence of H on x causes no difficulty). The scheme becomes a local AE scheme
with = nj,k when the maximum is taken over Ij,k for all φ = p1j,k [Φn ](x, y). Indeed,
under this CFL type restriction, Lj,k [Φ] is nondecreasing in terms of its arguments.
Therefore, for H(0, 0) = 0, we have a local bound for Lj,k [Φ]:
{Φi,l } ≤ Lj,k [Φ] ≤
min
|i−j|=1,|l−k|=1
max
|i−j|=1,|l−k|=1
{Φi,l }.
This, when combined with κ ≤ 1 leads to the local maximum principle
min
{Φi,l } ≤ Φn+1
j,k ≤
|i−j|=1,|l−k|=1
max
|i−j|=1,|l−k|=1
{Φi,l },
n ∈ N.
4.2. Stability analysis of second order AE schemes. We start again assuming that Φnj,k has already been computed. We extend the linear reconstruction p1j,k [Φ]
to a continuous, piecewise quadratic polynomial in Ij,k . Maintaining the interpolation
at four vertices of Ij,k such a polynomial is given by
p2j,k [Φ](x, y) = p1j,k [Φ](x, y) +
sj,k
sj,k
(x − xj−1 )(x − xj+1 ) +
(y − yk−1 )(y − yk+1 ),
2
2
where sj,k and sj,k are approximations to the corresponding exact derivatives φxx (xj ,
yk , tn ) and φyy (xj , yk , tn ), respectively. The polynomial is not unique, leaving room
to impose a nonlinear limiter on the numerical derivatives to ensure the nonoscillatory
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A133
property. In our two-dimensional numerical experiments, we use the θ-refined minmod
limiter, defined as follows:
sj,k = MM θAy (Dx0 )2 Φj+1,k , Ay (Dx0 )2 Φj,k , θAy (Dx0 )2 Φj−1,k ,
(4.6)
sj,k = MM θAx (Dy0 )2 Φj,k+1 , Ax (Dy0 )2 Φj,k , θAx (Dy0 )2 Φj,k−1 .
(4.7)
Here, θ ∈ [1, 2] is a parameter that controls numerical dissipation where θ = 1 is the
most dissipative and θ = 2 is the least dissipative (see, e.g., [18]). We now substitute
p2j,k [Φ](x, y) into Lj,k [Φ] to obtain the second order semidiscrete AE scheme:
d
1
1
Φj,k + Φj,k = Lj,k [Φ],
dt
(4.8)
sj,k
sj,k
(Δx)2 −
(Δy)2
2
2
− H(Dx0 Ay Φj,k , Dy0 Ax Φj,k ).
Lj,k [Φ] = Ax Ay Φj,k −
(4.9)
An application of the second order Runge–Kutta solver gives the fully discrete second
order AE scheme:
Φ∗j,k = (1 − κ)Φnj,k + κLj,k [Φn ],
1−κ
1 n
κ
Φ
=
+
Φn+1
Φ∗j,k + Lj,k [Φ∗ ].
j,k
2 j,k
2
2
(4.10)
(4.11)
As in the one-dimensional case, for the second order scheme to still satisfy the
maximum principle, we fix θ = 1 and modify the limiter as
sj,k = Dx0 MM{Ay Dx0 Φj+1,k , Ay Dx0 Φj,k , Ay Dx0 Φj−1,k } ,
(4.12)
(4.13)
sj,k = Dy0 MM{Ax Dy0 Φj,k+1 , Ax Dy0 Φj,k , Ax Dy0 Φj,k−1 } .
As illustrated in Figure 1 for the one-dimensional case, this modified limiter could be
inferior to (4.6), (4.7), yet it ensures the maximum-satisfying property of the second
order scheme.
Theorem 4.1. Let Φn be computed from the second order AE scheme (4.10),
(4.11), and (4.9) for HJ equations with H(0, 0) = 0 and slopes sj,k , sj,k defined in
(4.12)–(4.13). If
max |H1 (∇φ)| max |H2 (∇φ)|
1
+
and Δt < (4.14)
·
≤
Δx
Δy
2
hold, then
|Φn+1 |∞ ≤ |Φn |∞ ,
n ∈ N.
Proof. It suffices to show
|L[Φn ]|∞ ≤ |Φn |∞ ,
so that when κ < 1, we have that
|Φ
n+1
|∞
1
≤ |Φn |∞ +
2
1 κ
−
2 2
|Φ∗ |∞ +
κ
|L[Φ∗ ]|∞ ≤ |Φn |∞ .
2
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A134
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
We now prove (4.2) as follows. Set
⎧
sj,k
⎨
−(Δx)2 0
βj,k :=
Dx Ay Φj,k
⎩
0
if
Dx0 Ay Φj,k = 0,
if
Dx0 Ay Φj,k = 0
if
Dy0 Ax Φj,k = 0,
if
Dy0 Ax Φj,k = 0.
and
βj,k
:=
⎧
⎨
⎩
−(Δy)2
sj,k
Dy0 Ax Φj,k
0
Notice that H(0, 0) = 0, and by taking the Taylor expansion of the Hamiltonian we
have
βj,k
βj,k
0
− H1 (·) Dx Ay Φj,k +
− H2 (·) Dy0 Ax Φj,k ,
Lj,k [Φ] = Ax Ay Φj,k +
2
2
where (·) denotes some unspecified intermediate values. Regrouping terms we obtain
a2
a2
1
1
a1
a1
+
−
Lj,k [Φ] =
1+
Φj+1,k+1 +
1+
Φj+1,k−1
4
Δx Δy
4
Δx Δy
(4.15)
a2
a2
1
1
a1
a1
+
−
+
1−
Φj−1,k+1 +
1−
Φj−1,k−1 ,
4
Δx Δy
4
Δx Δy
where
a1 =
βj,k
− H1 (·),
2
a2 =
βj,k
− H2 (·).
2
Due to the minmod limiters defined in (4.12), (4.13), we claim that
(4.16)
|sj,k | ≤
|Ay Dx0 Φj,k |
,
2Δx
|sj,k | ≤
|Ax Dy0 Φj,k |
.
2Δy
Δx
,
2
|βj,k
|≤
Δy
,
2
Thereby,
|βj,k
|≤
which when combined with the condition (4.14) yields
1 1 1
|a1 | |a2 |
+
≤ + + = 1.
Δx
Δy
4 4 2
Thus all the coefficients on the right of (4.15) are nonnegative, and Lj,k [Φ] is a convex
combination of four values Φj±1,k±1 . This implies
|L[Φn ]|∞ ≤ |Φn |∞ .
Finally we show claim (4.16). For the first inequality, we fix k and set bj := Ay Dx0 Φj,k
and Bj = M M {bj+1 , bj , bj−1 } so that
sj,k =
1
(Bj+1 − Bj−1 ).
2Δx
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A135
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By the definition of M M , we have
1
min{|Bj−1 |, |Bj+1 |}
2Δx
1
min2l=−2 |bj+l |
≤
2Δx
1
|bj |.
≤
2Δx
|sj,k | ≤
The second inequality in (4.16) can be shown in same fashion.
Using this averaging method, construction of schemes higher than second order
will become cumbersome. We therefore adopt a dimension-by-dimension approach
that can be easily derived for higher order schemes.
4.3. Dimension-by-dimension approach. We now consider the second construction method of the two-dimensional AE scheme which will be referred to as the
dimension-by-dimension approach. We consider interpolated polynomials, pj,k and
qj,k in the x- and y-directions, satisfying
pj,k [Φ](xj±1 , yk ) = Φj±1,k ,
qj,k [Φ](xj , yk±1 ) = Φj,y±1
so that
(4.17)
Lj,k =
pj,k [Φ](xj , yk ) + qj,k [Φ](xj , yk )
2
− H (∂x pj,k [Φ](xj , yk ), ∂y qj,k [Φ](xj , yk )) .
4.3.1. First order scheme. For the first order scheme, such an interpolant is
given by
p1j,k [Φ](x, y) = Φj−1,k + sxj,k (x − xj−1 ),
1
[Φ](x, y) = Φj,k−1 + syj,k (y − yk−1 ),
qj,k
where
sxj,k =
Φj+1,k − Φj−1,k
,
2Δx
syj,k =
Φj,k+1 − Φj,k−1
.
2Δy
Evaluating at the polynomials and their partial derivatives at (xj , yk ) yields
Φj+1,k + Φj−1,k
,
2
Φj,k+1 + Φj,k−1
1
qj,k
,
[Φ](xj , yk ) =
2
p1j,k [Φ](xj , yk ) =
∂x p1j,k [Φ](xj , yk ) = sxj,k ,
1
∂y qj,k
[Φ](xj , yk ) = syj,k .
Substituting this into (4.17) yields
(4.18)
Lj,k [Φ] = Ax Ay Φj,k − H(sxj,k , syj,k )
so that
n
n
Φn+1
j,k = (1 − κ)Φj,k + κLj,k [Φ ].
Note that the only difference in first order schemes for (4.5) and (4.18) is in the
computation of φx and φy for H(φx , φy ).
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A136
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
4.3.2. Second order scheme. For the second order scheme, the continuous,
piecewise interpolating polynomial is given by
(sxj,k )
(x − xj−1 )(x − xj+1 ),
2
(syj,k )
2
(y − yk−1 )(y − yk+1 ),
[Φ](x, y) = Φj,k−1 + syj,k (y − yj−1 ) +
qj,k
2
p2j,k [Φ](x, y) = Φj−1,k + sxj,k (x − xj−1 ) +
where the approximations to the second derivative are approximated using the ENO
interpolation technique,
x
sj+2,k − sxj,k sxj,k − sxj−2,k
(sxj,k ) = M
,
,
2Δx
2Δx
y
sj,k+2 − syj,k syj,k − syj,k−2
y ,
.
(sj,k ) = M
2Δy
2Δy
We then obtain
(sxj,k )
Φj+1,k + Φj−1,k
−
(Δx)2 ,
2
2
(syj,k )
Φj,k+1 + Φj,k−1
2
−
(Δy)2 ,
[Φ](xj , yk ) =
qj,k
2
2
p2j,k [Φ](xj , yk ) =
∂x p2j,k [Φ](xj , yk ) = sxj,k ,
2
∂y qj,k
[Φ](xj , yk ) = syj,k ,
so that
Lj,k [Φ] = Ax Ay Φj,k −
(syj,k )
(sxj,k )
(Δx)2 −
(Δy)2 − H(sxj,k , syj,k ).
2
2
Combined with a second order Runge–Kutta method this gives
Φ∗j,k = (1 − κ)Φnj,k + κLj,k [Φn ],
1 n
1−κ ∗
κ
Φn+1
Φj,k + Lj,k [Φ∗ ].
j,k = Φj,k +
2
2
2
Again, the only difference in second order schemes for the two types of twodimensional AE schemes is in the computation of φx and φy for H(φx , φy ).
4.3.3. Third order scheme. The third order scheme formulation is based on
the cubic polynomials
(sxj,k )
(x − xj−1 )(x − xj+1 )(x − x∗ ),
6
(syj,k )
3
2
[Φ](x, y) = qj,k
[Φ](x, y) +
qj,k
(y − yk−1 )(y − yk+1 )(y − y ∗ ),
6
p3j,k [Φ](x, y) = p2j,k [Φ](x, y) +
where
(sxj,k ) = M
(syj,k )
=M
(sxj+2,k ) − (sxj,k ) (sxj,k ) − (sxj−2,k )
,
2Δx
2Δx
(syj,k+2 ) − (syj,k ) (syj,k ) − (syj,k−2 )
,
2Δy
2Δy
,
.
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A137
Here, x∗ is chosen as the grid point value used in the ENO procedure for (sxj,k ) so
that x∗ = xj−3 or x∗ = xj+3 . The value y ∗ is chosen in a similar way. This gives
(sxj,k )
(sxj,k )
Φj+1,k + Φj−1,k
−
(Δx)2 −
(Δx)2 (x − x∗ ),
2
2
6
(sxj,k )
(Δx)2 ,
∂x p3j,k = sxj,k −
6
(syj,k )
(syj,k )
Φj,k+1 + Φj,k−1
3
2
−
(Δy) −
(Δy)2 (y − y ∗ ),
qj,k [Φ](xj , yk ) =
2
2
6
(syj,k )
3
∂y qj,k
(Δy)2 .
= syj,k −
6
p3j,k [Φ](xj , yk ) =
When combined with the third order Runge–Kutta method, this gives
(1)
Φj,k = (1 − κ)Φnj,k + κLj,k [Φn ],
3
1
1
(2)
(1)
Φj,k = Φnj,k + (1 − κ)Φj,k + κLj,k [Φ(1) ],
4
4
4
1 n
2
2
(2)
(2)
Φn+1
],
j,k = Φj,k + (1 − κ)Φj,k + κLj,k [Φ
3
3
3
where
(sj,k )
(sxj,k )
(Δx)2 −
(Δy)2
2
2
(syj,k )
(sxj,k )
2
∗
(Δx) (x − x ) −
(Δy)2 (y − y ∗ )
−
6
6
y x )
(s
)
(s
j,k
j,k
y
(Δx)2 , sj,k −
(Δy)2 .
− H sxj,k −
6
6
y
Lj,k [Φ] = Ax Ay Φj,k −
Since this scheme can be easily derived for higher orders using an ENO procedure for derivative approximations, it will be used for all convergence studies in the
numerical results.
5. Numerical tests. In this section, we use some model problems to numerically
test the first, second, and third order global and local AE schemes. If φ is the exact
solution and Φ is the computed solution, then the numerical L1 and L∞ errors in the
one-dimensional case are calculated as
L1 error =
|φk − Φk |Δx, L∞ error = max |φk − Φk |.
k
k
We call Q in (2.10) and (2.11) the CFL type number for our numerical tests. In
calculating the numerical derivatives in the approximation, ENO constructions are
used in all one-dimensional tests. The θ-refined minmod limiters (4.6), (4.7) from
the averaging construction scheme are used in two-dimensional tests of the second
order scheme while all convergence studies use the dimension-by-dimension approach.
A performance comparison of different limiters is illustrated in Figure 1. In what
follows we use N to denote the number of intervals the domain is divided into, i.e.,
Δx = (b − a)/N for domain [a, b]. The same notation will be used in the twodimensional case as long as the same partition number is used in both x- and ydirections.
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A138
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
Table 1
The L1 and L∞ errors for convex Hamiltonian, Example 5.1, using N equally spaced cells for
global and local AE first order schemes at T = 0.5
when the solution is smooth.
π2
N
10
20
40
80
160
320
640
1280
Scheme
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
L1 error
2.427554E-01
1.658496E-01
6.191588E-02
1.238305E-01
6.336444E-02
3.617763E-02
6.009927E-02
2.439781E-02
1.737636E-02
2.964582E-02
1.042672E-02
8.684065E-03
1.467102E-02
4.711692E-03
4.278979E-03
7.293956E-03
2.228255E-03
2.120360E-03
3.634536E-03
1.080074E-03
1.053280E-03
1.814853E-03
5.323055E-04
5.256089E-04
L1 order
1.0410
1.4880
0.8310
1.0805
1.4265
1.0961
1.0379
1.2486
1.0187
1.0240
1.1563
1.0303
1.0127
1.0852
1.0175
1.0072
1.0471
1.0117
1.0030
1.0220
1.0040
L∞ error
1.866756E-01
1.380253E-01
6.012093E-02
1.180291E-01
6.304142E-02
3.248349E-02
6.612020E-02
2.764345E-02
1.794381E-02
3.556733E-02
1.289366E-02
1.033219E-02
1.848597E-02
6.139405E-03
5.490336E-03
9.463572E-03
2.992475E-03
2.824118E-03
4.788741E-03
1.473178E-03
1.430170E-03
2.409586E-03
7.316975E-04
7.208656E-04
L∞ order
0.7090
1.2119
0.9521
0.8661
1.2322
0.8871
0.9107
1.1201
0.8107
0.9526
1.0801
0.9204
0.9703
1.0414
0.9634
0.9850
1.0247
0.9838
0.9920
1.0108
0.9895
Example 5.1. We first test the numerical accuracy of the designed schemes by
using the Hamilton–Jacobi equation with convex Hamiltonian and with smooth initial
data. The equation is
Φt +
(Φx + 1)2
= 0,
2
−1 ≤ x ≤ 1,
with initial data
Φ(x, 0) = − cos πx.
For numerical accuracy results, the reference solution is computed using a fifth order
WENO [14, 27], fourth order TVD Runge–Kutta [10, 9], and N = 10240 with periodic
boundary conditions. In Tables 1–3, the numerical accuracy results for the first,
second, and third order global and local schemes are presented when the solution is
smooth (time T = 0.5
π 2 ). All the proposed methods give the desired order of accuracy.
The solution becomes discontinuous at time π12 , and in Figures 2–4 we plot the
solutions at time T = 1.5
π 2 . We can see that local schemes give better numerical
resolution than the global schemes since they have smaller numerical viscosity. For
the numerical experiments the CFL number is taken to be 0.9 and Δt = 0.9.
Example 5.2. The following example is with a nonconvex Hamiltonian:
1
Φt + (Φ2x − 1)(Φ2x − 4) = 0, −1 ≤ x ≤ 1,
4
Φ(x, 0) = −2|x|.
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
A139
Table 2
The L1 error for convex Hamiltonian, Example 5.1, using N equally spaced cells for global and
local AE second order schemes at T = 0.5
when the solution is smooth.
π2
N
10
20
40
80
160
320
640
1280
Scheme
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
L1 error
1.808044E-01
1.605897E-01
7.158793E-02
5.035859E-02
3.793765E-02
2.226361E-02
1.227800E-02
8.356335E-03
6.406106E-03
3.046794E-03
1.931040E-03
1.698534E-03
7.525114E-04
4.586449E-04
4.309275E-04
1.870988E-04
1.119356E-04
1.085638E-04
4.662925E-05
2.763822E-05
2.722453E-05
1.164294E-05
6.869442E-06
6.818169E-06
L1 order
1.9768
2.2314
1.8063
2.1095
2.2613
1.8619
2.0469
2.1516
1.9497
2.0357
2.0926
1.9966
2.0170
2.0439
1.9979
2.0090
2.0225
2.0001
2.0040
2.0107
1.9997
L∞ error
1.230779E-01
1.205706E-01
7.280369E-02
4.249643E-02
3.704884E-02
2.548232E-02
1.152047E-02
1.039131E-02
8.868757E-03
2.759160E-03
2.648734E-03
2.546340E-03
7.102240E-04
6.910419E-04
6.860096E-04
1.829021E-04
1.815522E-04
1.810838E-04
4.765022E-05
4.714968E-05
4.711239E-05
1.224287E-05
1.212996E-05
1.212727E-05
L∞ order
1.6445
1.8249
1.6235
1.9510
1.9001
1.5775
2.0991
2.0075
1.8328
1.9755
1.9559
1.9092
1.9660
1.9371
1.9302
1.9449
1.9494
1.9469
1.9628
1.9609
1.9601
Table 3
The L1 error for convex Hamiltonian, Example 5.1, using N equally spaced cells for global and
0.5
local AE third order schemes at T = π2 when the solution is smooth.
N
10
20
40
80
160
320
640
1280
Scheme
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
AE
AE1
AE2
L1 error
9.217579E-02
8.879066E-02
3.882247E-02
1.452062E-02
1.200976E-02
5.699280E-03
1.997324E-03
1.374387E-03
7.962369E-04
2.501019E-04
1.454294E-04
1.064920E-04
3.087791E-05
1.586493E-05
1.342152E-05
3.820356E-06
1.827268E-06
1.675592E-06
4.747847E-07
2.183270E-07
2.089154E-07
5.916589E-08
2.665971E-08
2.607256E-08
L1 order
2.8581
3.0938
2.9672
2.9650
3.2400
2.9418
3.0515
3.2988
2.9548
3.0451
3.2252
3.0151
3.0284
3.1321
3.0153
3.0151
3.0720
3.0105
3.0078
3.0372
3.0057
L∞ error
9.436562E-02
9.366517E-02
3.899054E-02
2.009461E-02
1.889613E-02
7.722367E-03
3.885110E-03
2.847301E-03
1.256969E-03
5.805090E-04
3.220735E-04
1.787705E-04
7.616378E-05
3.398078E-05
2.507739E-05
9.628130E-06
3.763566E-06
3.215076E-06
1.206421E-06
4.391123E-07
4.054263E-07
1.509158E-07
5.295197E-08
5.086366E-08
L∞ order
2.3920
2.4756
2.5041
2.4562
2.8288
2.7134
2.7920
3.2008
2.8645
2.9566
3.2738
2.8592
2.9972
3.1889
2.9768
3.0033
3.1064
2.9941
3.0023
3.0553
2.9981
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A140
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
(a) AE scheme.
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(b) AE1 scheme.
0.8
0.8
0.6
Reference
AE
AE1
AE2
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(c) AE2 scheme.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(d) Comparison with all schemes.
Fig. 2. Comparison of plots for HJ equation with convex Hamiltonian, Example 5.1, at discontinuity on [−1, 1], T = 1.5/π 2 , N = 40, Δt = 0.8, first order scheme.
This test problem is used to show resolution of discontinuities when the initial data
has discontinuous derivatives. As in the previous problem, the reference solution is
computed using a fifth order WENO [14, 27], fourth order TVD Runge–Kutta [10, 9],
and N = 10240. In testing our AE methods, the CFL number used for all order
schemes was 0.9. Linear extension boundary conditions are used for the numerical
tests. In Figures 5–7, we can clearly see that the local AE2 scheme gives the best
numerical accuracy, followed by local AE1 and the global AE for first, second, and
third order methods.
Example 5.3. The one-dimensional Eikonal equation is
Φt + |Φx | = 0,
0 ≤ x ≤ 2π,
with initial data
Φ(x, 0) = sin x.
The exact solution is given as in [5]:
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A141
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
(a) AE scheme.
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(b) AE1 scheme.
0.8
0.8
0.6
Reference
AE
AE1
AE2
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
(c) AE2 scheme.
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(d) Comparison with all schemes.
Fig. 3. Comparison of plots for HJ equation with convex Hamiltonian, Example 5.1, at discontinuity on [−1, 1], T = 1.5/π 2 , N = 40, Δt = 0.8, second order scheme.
• If 0 ≤ t ≤ π/2,
⎧
sin(x − t)
⎪
⎪
⎪
⎨ sin(x + t)
Φ(x, t) =
−1
⎪
⎪
⎪
⎩ sin(x − t)
• If π/2 ≤ t ≤ π,
⎧
⎪
⎪
⎪
⎨
if
if
if
if
−1
sin(x − t)
Φ(x, t) =
sin(x
+ t)
⎪
⎪
⎪
⎩
−1
0 ≤ x ≤ π/2,
π/2 ≤ x ≤ 3π/2 − t,
3π/2 − t ≤ x ≤ 3π/2 + t,
3π/2 + t ≤ x ≤ 2π.
if
if
if
if
0 ≤ x ≤ t − π/2.
t − π/2 ≤ x ≤ π/2.
π/2 ≤ x ≤ 3π/2 − t.
3π/2 − t ≤ x ≤ 2π.
• If t ≥ π,
Φ(x, t) = −1.
Note that local and global schemes are all the same for this example since |Hp | = 1.
The numerical solution is compared for first, second, and third order schemes at time
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A142
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
(a) AE scheme.
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(b) AE1 scheme.
0.8
0.8
0.6
Reference
AE
AE1
AE2
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
(c) AE2 scheme.
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(d) Comparison with all schemes.
Fig. 4. Comparison of plots for HJ equation with convex Hamiltonian, Example 5.1, at discontinuity on [−1, 1], T = 1.5/π 2 , N = 40, Δt = 0.8, third order scheme.
T = 1 in Figure 8. For the viscosity solution, there is a shock wave in φx at x = π/2
and a rarefaction wave at x = 3π/2. The CFL number used on the numerical tests is
0.7.
Example 5.4. The two-dimensional linear HJ equation with variable coefficients
is
φt − yφx + xφy = 0
(5.1)
with initial condition
⎧
⎨
φ0 (x, y) =
0 if 0.3 ≤ r,
0.3 − r if 0.1 ≤ r ≤ 0.3,
⎩
0.2 if r ≤ 0.1,
where r = (x − 0.4)2 + (y − 0.4)2 . The computational domain is [−1, 1]2 and we
impose periodic boundary conditions. The exact solution can be expressed as
(5.2)
φ(x, y, t) = φ0 (x cos(t) + y sin(t), −x sin(t) + y cos(t)).
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A143
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
−1
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−1.8
−2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−2
−1
−0.8
−0.6
(a) AE scheme.
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(b) AE1 scheme.
Reference
AE
AE1
AE2
−1
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−2
−1
−1.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(c) AE2 scheme.
0.8
1
−2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(d) Comparison with all schemes.
Fig. 5. Comparison of plots for HJ equation with Riemann initial data, Example 5.2, at
discontinuity on [−1, 1], T = 1, N = 80, Δt = 0.8, first order scheme.
The results for the second order schemes at time t = 2π are shown in Figure 9.
We choose θ = 2 in the minmod computation for the approximations sj,k , sj,k and let
Δt = .95. Here, we look along the line x = y and compare the global AE scheme to
the local AE1 scheme for various mesh sizes. We can see that the local AE1 scheme
provides better resolution to the exact solution for the same number of grid points as
the global AE scheme.
Example 5.5. We solve the two-dimensional Burgers equation using first and
second order global schemes by the dimension-by-dimension approach:
φt +
(φx + φy + 1)2
=0
2
with initial data
φ(x, y, 0) = − cos(x + y).
The computational domain is [0, 2π]2 and we impose periodic boundary conditions.
The solution is still smooth at time t = 0.1 and we test the order of accuracy at this
time in Tables 4 and 5. For the first order and second order schemes, we get the
expected order in all norms. A convergence test for the third order scheme will be
applied to the problem with a nonconvex Hamiltonian (see Example 5.7).
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A144
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
−1
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−1.8
−2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−2
−1
−0.8
−0.6
(a) AE scheme.
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(b) AE1 scheme.
Reference
AE
AE1
AE2
−1
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−2
−1
−1.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−2
−1
(c) AE2 scheme.
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(d) Comparison with all schemes.
Fig. 6. Comparison of plots for HJ equation with Riemann initial data, Example 5.2, at
discontinuity on [−1, 1], T = 1, N = 80, Δt = 0.8, second order scheme.
Example 5.6.
φt +
φ2x + φ2y + 1 = 0
with smooth initial data
φ(x, y, 0) =
1
(cos(2πx) − 1) (cos(2πy) − 1) − 1
4
and computational domain [0, 1]2 and periodic boundary conditions. The results at
time t = 0.6 are shown in Figure 10. The AE scheme provides high resolution in the
formation of the singularity.
Example 5.7. We now consider an example with a nonconvex Hamiltonian:
φt − cos(φx + φy + 1) = 0
with initial data
φ(x, y, 0) = − cos
π(x + y)
2
on the computation domain [−2, 2]2 with periodic boundary conditions. At time
t = 0.5
π 2 , the solution is still smooth and we test the order of accuracy for second
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A145
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AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
−1
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−1.8
−2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−2
−1
−0.8
−0.6
(a) AE scheme.
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(b) AE1 scheme.
Reference
AE
AE1
AE2
−1
−1
−1.2
−1.2
−1.4
−1.4
−1.6
−1.6
−1.8
−1.8
−2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−2
−1
−0.8
(c) AE2 scheme.
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(d) Comparison with all schemes.
Fig. 7. Comparison of plots for HJ equation with Riemann initial data, Example 5.2, at
discontinuity on [−1, 1], T = 1, N = 80, Δt = 0.8, third order scheme.
0.6
0.4
Reference
Order 1
Order 2
Order 3
0.55
0.5
0.2
0.45
0
0.4
−0.2
0.35
−0.4
0.3
−0.6
0.25
−0.8
0.2
0.15
−1
0
1
2
3
4
(a) AE scheme.
5
6
7
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
(b) Comparison with all schemes.
Fig. 8. Comparison of plots for Eikonal equation, Example 5.3, at discontinuity on [0, 2π],
T = 1, N = 80, Δt = 0.9, first, second, and third order schemes.
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A146
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
−0.05
−0.1
−1
−0.5
0
0.5
1
−0.1
−1
(a) AE scheme, N=80.
0.25
0.2
0.2
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
−0.05
−0.5
0
0.5
1
−0.1
−1
(c) AE scheme, N=160.
0.5
1
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
−0.05
−0.5
0
0.5
(e) AE scheme, N=320.
−0.5
0
0.5
1
(d) AE1 scheme, N=160.
0.25
−0.1
−1
0
0.25
0.15
−0.1
−1
−0.5
(b) AE1 scheme, N=80.
1
−0.1
−1
−0.5
0
0.5
1
(f) AE1 scheme, N=320.
Fig. 9. Plots of second order global and local AE scheme along the line x = y for Example 5.4
at time t = 2π, where Δt = .95 and θ = 2.
and third order schemes using the dimension-by-dimension approach. The results in
Tables 6 and 7 show that we get the desired order in all norms. At time t = π12 , the
solution develops a discontinuous derivative and results at time t = 1.5
π 2 are shown in
Figure 11.
6. Concluding remarks. In this work, we have developed several AE schemes
based on the alternating evolution approximation to HJ equations, motivated by
our prior work on AE schemes for hyperbolic conservation laws [21, 26]. The AE
system captures the exact solution when initially both components are chosen as
the given initial data for the HJ equation. We numerically approximate the HJ
equation by sampling two components of the AE system over alternating grids. High
order accuracy is achieved by a combination of high order nonoscillatory polynomial
reconstruction from the obtained grid values and an ODE solver in time discretization
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A147
AE SCHEMES FOR HAMILTON–JACOBI EQUATIONS
L1 error
5.45124474
L1 order
L2 error
0.918161204
L∞ error
0.225967148
Scheme
AE
40
AE
2.831282163 0.945130159 0.491727041 0.900889782 0.130491793 0.792153966
80
AE
1.409606128 1.006163433
160
AE
0.666738528 1.080099099 0.118422976 1.068541847 0.033402227 1.035964816
320
AE
0.287707844
1.21251654
0.24836999
L2 order
L∞ order
N
20
0.985366826 0.068490748 0.929978053
0.051304179 1.206800783 0.014630959 1.190920009
Table 5
L1 , L2 , and L∞ comparison for Example 5.5 at time t = 0.1 using second order global AE
scheme.
N
20
Scheme
L1 error
AE
5.487583632
L1 order
L2 error
0.913422858
0.27577215
L∞ error
0.214771031
L2 order
L∞ order
40
AE
1.584797823 1.791872218
80
AE
0.39345259
1.727806122 0.070029702 1.616760554
160
AE
0.095952595 2.035796076
320
AE
0.022627344 2.084254579 0.004184165 2.075319525 0.001182064 2.074586915
2.010037092 0.071300662 1.951489386 0.019686339 1.830772081
0.01763365
2.01558435
0.004979134 1.983228179
Concentration at time 0.60000
Concentration at time 0.60000
1
0.9
−1.3
0.8
−1.35
−1.4
0.7
−1.45
0.6
−1.5
y
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Table 4
L1 , L2 , and L∞ comparison for Example 5.5 at time t = 0.1 using first order global AE scheme.
0.5
−1.55
0.4
−1.6
0.3
−1.65
1
−1.7
0
0.8
0.2
0.2
0.6
0.4
0.4
0.6
0.8
0.1
0.2
1
0
y
0
0
0.1
0.2
0.3
x
(a) AE scheme, Δx = Δy =
1
.
80
0.4
0.5
x
0.6
0.7
(b) AE scheme, Δx = Δy =
0.8
0.9
1
1
.
80
Fig. 10. Surface and contour plots at time t = 0.6 for Example 5.6.
Table 6
L1 , L2 , and L∞ comparison for Example 5.7 at time t =
by-dimension global AE scheme.
N Scheme
L1 error
20
AE 0.3807545007
L1 order
L2 error
0.0961004734
0.5
π2
L2 order
using second order dimension-
L∞ error
0.0347144173
L∞ order
40
AE
0.0926199682 2.0394659222 0.0251604319 1.9333868472 0.0102422250 1.7610057921
80
AE
0.0234684717 1.9805993682 0.0064820603 1.9566323356 0.0027617696 1.8908642702
160
AE
0.0058630945 2.0009896787 0.0016286995 1.9927319161 0.0007481906 1.8841152186
320
AE
0.0014408791 2.0247129301 0.0004013811 2.0206757509 0.0001926201 1.9576475279
640
AE
0.0003416173 2.0764961041 0.0000952364 2.0753878851 0.0000465560 2.0487171898
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A148
HAILIANG LIU, MICHAEL POLLACK, AND HASEENA SARAN
Table 7
L1 , L2 , and L∞ comparison for Example 5.7 at time t =
dimension global AE scheme.
N Scheme
L1 error
20
AE 0.2885167659
L1 order
L2 error
0.0822233536
0.5
π2
using third order dimension-by-
L∞ error
0.0445854569
L2 order
L∞ order
40
AE
0.0608972496 2.2442061801 0.0208873181 1.9769209512 0.0140420171 1.6668230099
80
AE
0.0122271144 2.3162931238 0.0044368628 2.2350153966 0.0034367453 2.0306353513
160
AE
0.0017583509 2.7977889587 0.0006572209 2.7550895638 0.0005444127 2.6582702568
320
AE
0.0002266868 2.9554505249 0.0000826895 2.9906021610 0.0000873968 2.6390477645
640
AE
0.0000273512 3.0510218754 0.0000097165 3.0891842742 0.0000111797 2.9666891406
1.5
1
0.5
0
−0.5
−1
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
−1
0
1
2
y
x
Fig. 11. Surface plot for Example 5.7 at time t =
dimension-by-dimension approach.
1.5
π2
with uniform mesh Δx = Δy =
1
10
using
with matching accuracy. Local AE schemes are made possible by choosing the scale
parameter to reflect the local distribution of nonlinear waves. The AE schemes have
the advantage of easy formulation and implementation and efficient computation of
the solution. For the first order local AE scheme and the second order local AE scheme
with limiters, we have proved the desired maximum-satisfying property. Numerical
tests for both one- and two-dimensional HJ equations presented in this work have
demonstrated the high order accuracy and capacity of the AE schemes.
Acknowledgment. The authors thank Professor Chiwang Shu for helpful discussions on the dimension-by-dimension ENO selection.
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