finite difference methods for hamilton-jacobi equations ii · pdf filefinite difference...

Finite Difference Methods for Hamilton-JacobiEquations II

Chi-Wang Shu

Division of Applied Mathematics

Brown University

FINITE DIFFERENCE METHODS FOR HAMILTON-JACOBI EQUATIONS II

Outline

• ENO and WENO interpolations to compute left and right derivatives

for Cartesian meshes

• ENO and WENO interpolations to compute gradients for triangular

meshes

• Fast sweeping method for WENO schemes on Cartesian meshes

• Treatment of boundary conditions

Division of Applied Mathematics, Brown University


ENO and WENO interpolations to compute left and right derivatives

From the last lecture, in order to design a high order finite difference

method for solving the 2D HJ equation

ϕt + H(ϕx, ϕy) = 0, ϕ(x, y, 0) = ϕ0(x, y) (1)

on rectangular meshes, we can proceed as follows:

• Find a suitable monotone numerical Hamiltonian,

H(u−, u+; v−, v+), such that

H ( ↑, ↓; ↑, ↓ ) .



• Compute high order approximations to the left and right x-derivatives

and y-derivatives of ϕ at the grid point (x, y) = (xi, yj), in a

dimension by dimension fashion, and denote them as u−

i,j , u+i,j , v−

i,j

and v+i,j , respectively.

• Form the semi-discrete scheme

d

dtϕi,j = −H

(

u−

i,j, u+i,j ; v

−

i,j , v+i,j

)

and discretize it in time by high order TVD time discretization (or

another time discretization method).



Therefore, in order to obtain high order finite difference schemes for HJ

equations on Cartesian meshes, we only need to discuss one dimensional

high order accurate approximations to the left and right derivatives, given

the point values of the function on grid points.

In the first lecture, we indicated that we could use the usual polynomial

interpolation procedure, namely

• Choose a stencil such as S = xi−2, xi−1, xi, xi+1 containing four

grid points, including xi and biased to the left (Note: linear stability for

time dependent calculations restricts how much one can bias the

stencil to one side. The most commonly used “upwind stencil” is one

point biased to the upwind side)



• Find the interpolating polynomial p(x) which interpolates the function

ϕ at each grid point in the stencil S. That is, p(xi−2) = ϕi−2,

p(xi−1) = ϕi−1, p(xi) = ϕi, p(xi+1) = ϕi+1. Because the stencil

S has 4 points, the interpolation polynomial p(x) is of degree 3 (a

cubic polynomial)

• Take u−

i = p′(xi) as the approximation to the left derivative of ϕ at

the location x = xi. One should work out the algebra first to obtain

the explicit formula

u−

i =1

∆x

(

1

6ϕi−2 − ϕi−1 +

1

2ϕi +

1

3ϕi+1

)

(for non-uniform mesh the coefficients will depend on the local mesh

sizes but not on ϕ).



This would lead to linear schemes, namely the schemes are linear when

applied to linear PDEs. These schemes are excellent for smooth solutions

but may generate spurious oscillations when applied to non-differentiable

viscosity solutions.

We would like to have approximations which are

• high order accurate in approximating left and right derivatives in

smooth regions, and

• (essentially) non-oscillatory near kinks, i.e. discontinuities in the

derivatives of the function



The spurious oscillations appear because the interpolation stencil S may

cross a discontinuity of the derivative. In such cases, not only will the

accuracy of the interpolation be completely lost (remember accuracy is

proved based on the assumption of smoothness of the underlying function

over the interpolation stencil), but also the interpolation tends to generate

overshoots or undershoots, especially for approximating derivatives.

For fixed stencils (for example, always use two points to the left, the point

itself, and one point in the right), the stencil will definitely cross the

discontinuity of the derivative somewhere.

One way to overcome this difficulty is to “limit” the approximation, through,

e.g. a minmod limiter. This leads to second order TVD type schemes.



A better way is the ENO (essentially non-oscillatory) or WENO (weighted

ENO) procedure. Let us first describe the ENO interpolation procedure

(Osher and Shu, SINUM 1991).

Suppose we would like to find a polynomial p(x) of degree k ≥ 2

interpolating the function ϕ at the points in the stencil S, which contains

the two points xi−1, xi. Note: it is important to notice that, unlike the

case in conservation laws, here we would start with a stencil containing

two points xi−1, xi, not just one point xi. This is because the first

order scheme already uses a linear interpolation with the stencil

xi−1, xi. The correct ENO stencil should contain the stencil for the first

order base scheme, which provides correct upwinding and stability.



First, recall the definition of Newton divided differences:

ϕ[i] = ϕi, ϕ[i, i + 1] =ϕ[i + 1] − ϕ[i]

xi+1 − xi

and in general

ϕ[i, · · · , i + m] =ϕ[i + 1, · · · , i + m] − ϕ[i, · · · , i + m − 1]

xi+m − xi

Let us assume that all these Newton divided differences have been

computed and stored. Note: for uniform meshes, we only need to compute

undivided differences, hence saving cost.



• Start with the first order stencil S1 = xi−1, xi.

• Decide to add either the left neighbor, xi−2, or the right neighbor,

xi+1, into the stencil. If

|ϕ[i − 2, i − 1, i]| < |ϕ[i − 1, i, i + 1]|

then we believe the stencil xi−2, xi−1, xi is better than

xi−1, xi, xi+1. This is because the interpolation polynomial pl(x)

on the stencil xi−2, xi−1, xi is given by

pl(x) = p1(x) + ϕ[i − 2, i − 1, i] (x − xi−1) (x − xi)

and the interpolation polynomial pr(x) on the stencil xi−2, xi−1, xiis given by

pr(x) = p1(x) + ϕ[i − 1, i, i + 1] (x − xi−1) (x − xi)



where p1(x) is the first degree interpolation polynomial over

S1 = xi−1, xi.

Notice that pl(x) − p1(x) and pr(x) − p1(x) are both a scalar

multiple of the same function (x − xi−1) (x − xi), and the deviation

pl(x) − p1(x) is smaller than the deviation pr(x) − p1(x). If we

believe the first degree interpolation polynomial p1(x) is good

(yielding a good first order monotone scheme), we would like to

deviate from it as little as possible (while still keeping accuracy). This

leads to the ENO choice of pl(x). Hence in this case, the stencil for

the quadratic interpolation polynomial is S2 = xi−2, xi−1, xi.



Otherwise, if

|ϕ[i − 2, i − 1, i]| ≥ |ϕ[i − 1, i, i + 1]|

we choose S2 = xi−1, xi, xi+1.

• We now repeat this process. Suppose S2 = xi−1, xi, xi+1. We

would decide to add either the left neighbor, xi−2, or the right

neighbor, xi+2, into the stencil. If

|ϕ[i − 2, i − 1, i, i + 1]| < |ϕ[i − 1, i, i + 1, i + 2]|

then we choose the stencil for the cubic interpolation polynomial as

S3 = xi−2, xi−1, xi, xi+1. Otherwise, we choose

S3 = xi−1, xi, xi+1, xi+2.



• We continue this process until the target stencil Sk is chosen. The

interpolating polynomial p(x) on this stencil is what we want. We

would then take u−

i = p′(xi). In practice, we do not need to explicitly

compute p(x) and then take its derivative. We simply precompute the

coefficients for each possible stencil. For example, if the final ENO

stencil is S3 = xi−2, xi−1, xi, xi+1, then we have

u−

i =1

∆x

(

1

6ϕi−2 − ϕi−1 +

1

2ϕi +

1

3ϕi+1

)

where the coefficients 16, −1, 1

2and 1

3are precomputed and stored.

For non-uniform meshes, these coefficients depend on the local mesh

sizes but not on ϕ, hence they can still be precomputed and stored.



• For computing u+, the only change to the procedure above is the

starting first order stencil, it is changed to S1 = xi, xi+1.

The ENO procedure is simple to code and to compute, even for

non-uniform meshes. We refer to (Osher and Shu, SINUM 1991) for

computational results.



The WENO interpolation procedure (Jiang and Peng, SISC 2000) is based

on the ENO procedure. If we still use the previous example of trying to

obtain piecewise cubic (third order accurate for derivatives) polynomial

interpolations, and if we insist that the stencil should contain xi−1, xiwhich is the stencil for the first order monotone scheme, then we have the

following three possible stencils which might be chosen by the ENO

procedure:

S0 = xi−3, xi−2, xi−1, xi,

S1 = xi−2, xi−1, xi, xi+1,

S2 = xi−1, xi, xi+1, xi+2



They correspond to three third order approximations to the left derivative

u−

i ≈ ϕx(x−

i ):

u−,0i =

1

∆x

(

1

3∆+ϕi−3 −

7

6∆+ϕi−2 +

11

6∆+ϕi−1

)

,

u−,1i =

1

∆x

(

−1

6∆+ϕi−2 +

5

6∆+ϕi−1 +

1

3∆+ϕi

)

,

u−,2i =

1

∆x

(

1

3∆+ϕi−1 +

5

6∆+ϕi −

1

6∆+ϕi+1

)

,

The ENO procedure would choose one of these and ignore the remaining

two. This is the correct procedure to follow near kinks (discontinuities of

the first derivative), as one or two out of the three candidate stencils S0,

S1 and S2 might contain a kink.



However, in the smooth region, all three stencils are equally good. It is a

waste to look at all of them and eventually discard two of them. A greedy

algorithm would try to use all of them

u−

i = w0u−,0 + w1u

−,1 + w2u−,2 (2)

Now the question is how to choose the weights w0, w1 and w2.

For consistency we need

w0 + w1 + w2 = 1

and for stability we prefer wk ≥ 0.



In fact, if w0 + w1 + w2 = 1 is satisfied, then any choice of w0, w1 and

w2 would lead to at least third order accuracy, since each u−,k is already

third order accurate. There are WENO schemes in the literature which are

of this type (the weighted average is not higher order accurate than the

approximation in each substencil).



A simple algebra reveals that, if we choose the weights w0, w1 and w2 as

the so-called optimal linear weights

w0 = γ0 = 0.1, w1 = γ1 = 0.6, w2 = γ2 = 0.3,

then the approximation

u−

i = γ0u−,0 + γ1u

−,1 + γ2u−,2

would be fifth order accurate. In fact, this would be the same

approximation obtained from the interpolation polynomial based on the

larger stencil

S = S1 ∪ S2 ∪ S3 = xi−3, xi−2, xi−1, xi, xi+1, xi+2



Of course, this approximation will be oscillatory near kinks. Now, the trick

is to require that the choice of the weights w0, w1 and w2 to be such that

• in smooth regions, the weights w0, w1 and w2 are very close to the

optimal linear weights γ0, γ1, and γ2:

w0 = 0.1+O(∆x2), w1 = 0.6+O(∆x2), w2 = 0.3+O(∆x2),

• near kinks, if the stencil Sk contains a kink while at least one of the

other two stencils does not, then the corresponding weight wk is very

small. In fact, we require wk = O(∆x4) in this case.



The key ingredient in designing a nonlinear weight to satisfying the two

properties listed above is a smoothness indicator, which is a measurement

of how smooth the function being interpolated is inside the interpolation

stencil. The recipe used in (Jiang and Peng, SISC 2000) is similar to that

in (Jiang and Shu, JCP 1996) for conservation laws, namely the

smoothness indicator is a scaled sum of the squares of the L2 norms of

the second and higher derivatives of the interpolation polynomial on the

target interval. We remark that there is a difference to the case for

conservation laws: here the smoothness indicator sums the squares of the

L2 norms of the second and higher derivatives, not the first and higher

derivatives as in the conservation law case.



These smoothness indicators work out to be

IS0 = 13(a − b)2 + 3(a − 3b)2,

IS1 = 13(b − c)2 + 3(b + c)2,

IS2 = 13(c − d)2 + 3(3c − d)2,

where

a =∆2ϕi−2

∆x, b =

∆2ϕi−1

∆x, c =

∆2ϕi

∆x, d =

∆2ϕi+1

∆x(3)

are the second order differences, defined by

∆2ϕi = ϕi+1 − 2ϕi + ϕi−1



With these smoothness indicators, the nonlinear weights are then defined

by

w0 =w0

w0 + w1 + w2

, w1 =w1

w0 + w1 + w2

, w2 =w2

w0 + w1 + w2

,

with

w0 =1

(ε + IS0)2, w1 =

6

(ε + IS1)2, w2 =

3

(ε + IS2)2,

where ε is a small number to prevent the denominator to become zero and

is typically chosen as ε = 10−6.



Finally, after some algebraic manipulations, we obtain the fifth order

WENO approximation to u−

i as

u−

i =1

12∆x(−∆+ϕi−2 + 7∆+ϕi−1 + 7∆+ϕi − ∆+ϕi+1)

−ΦWENO(a, b, c, d)

where

ΦWENO(a, b, c, d) =1

3w0 (a− 2b + c) +

1

6

(

w2 −1

2

)

(b− 2c + d)

with a, b, c, d defined by (3).



By symmetry, the approximation to the right derivative u+i is given by

u+i =

1

12∆x(−∆+ϕi−2 + 7∆+ϕi−1 + 7∆+ϕi − ∆+ϕi+1)

+ΦWENO(e, d, c, b)

with b, c, d defined by (3) and e defined by

e =∆2ϕi+2

∆x.



ENO and WENO interpolations to compute gradients for triangular meshes

While in principle the ENO and WENO interpolation procedures can be

designed in a similar fashion for triangular meshes (Abgrall, JCP 1994;

Zhang and Shu, SISC 2003), the procedure is quite involved. We therefore

skip the details and refer to (Abgrall, JCP 1994; Zhang and Shu, SISC

2003). We will describe an alternative approach, namely the discontinuous

Galerkin method, for solving HJ equations on triangular meshes in next

lecture.



Fast sweeping method for WENO schemes on Cartesian meshes

The fast sweeping method (Boue and Dupuis, SINUM 1999; Zhao, Math

Comp 2005) is an efficient procedure to obtain the solutions of a numerical

scheme for solving steady state HJ equations. As we mentioned before, if

we use a time accurate method, such as the Runge-Kutta method to

march in time until steady state is reached, the computational cost is quite

high.



As of now the fast sweeping method is mainly designed for solving convex

HJ equations. In particular, it is mostly used for solving the Eikonal

equation

H(φx, φy) = f(x, y)

where

H(u, v) =√

u2 + v2.



Let us now describe the fast sweeping procedure:

• Write out the scheme

H

(

∆x−ϕi,j

∆x,

∆x+ϕi,j

∆x;

∆y−ϕi,j

∆y,

∆y+ϕi,j

∆y

)

= fi,j

for the first order scheme and

H(

u−

i,j, u+i,j ; v

−

i,j , v+i,j

)

= fi,j

where u−

i,j , u+i,j , v−

i,j , and v+i,j are nonlinear functions of ϕk,` for (k, `)

near (i, j), for high order WENO schemes.

Notice that both will be a nonlinear system of algebraic equations for

the unknowns ϕi,j for all (i, j) in the computational domain.



• The nonlinear equation at the location (i, j) is local, involving only

finitely many (independent of the mesh sizes) neighboring points. For

example, for a first order scheme, the equation at the location (i, j)

involves only the five unknowns ϕi,j , ϕi−1,j , ϕi+1,j , ϕi,j−1, ϕi,j+1.

• The local solver: If we assume all the neighboring values are known,

e.g. in the first order case we assume ϕi−1,j , ϕi+1,j , ϕi,j−1 and

ϕi,j+1 are known, then we attempt to explicit solve ϕi,j (that is, write

ϕi,j explicitly as a function of ϕi−1,j , ϕi+1,j , ϕi,j−1 and ϕi,j+1).

Note: since the equation is nonlinear, this step is by no means trivial.

Often this cannot be done, or the solution is not unique. However, for

the Eikonal equation and first order upwind scheme, this can be done.



• We use a Gauss-Seidel iteration procedure with four alternating

direction sweepings:

(1) i = 0 : I, j = 0 : J ; (2) i = I : 0, j = 0 : J ;

(3) i = I : 0, j = J : 0; (4) i = 0 : I, j = J : 0.

For Gauss-Seidel, solutions are updated during the iteration instantly.

Thus, when we reach the location i from left to right, the value at i− 1

is already the updated one although the value at i + 1 is still the old

one. When we loop to a point (i, j), we use the local solver to obtain

a new value for ϕi,j .



There is a “causality” check to determine which solution to choose if

the local solver has more than one solution, and if we should use this

new value or keep the old one: for the Eikonal equation, we adopt the

new value if it is smaller than the old value, and we discard it if it is

bigger than the old value.

• The iteration stops when the residue is smaller than a pre-set

threshold (usually taken as slightly above machine zero).

It can be shown (Zhao, Math Comp 2005) that the fast sweeping method

for a first order upwind scheme solving the Eikonal equation converges to

the truncation error of the scheme in O(N) cost, where N is the number

of grid points. This is to say that the number of Gauss-Seidel iterations

needed is independent of the mesh size, or the total number of grid points

N .



For the high order WENO scheme, the local solver is very difficult, if not

impossible, to obtain, because of the complicated nonlinearity caused by

the nonlinear weights in the WENO procedure. Zhang, Zhao and Qian

(JSC 2006) used a very clever trick to avoid using the local solver

associated with the WENO scheme. Rather, they use the local solver

associated with the first order scheme, in such a way that when steady

state is reached, the solution obtained is the solution to the high order

WENO scheme, not the first order monotone scheme. The procedure is

summarized as follows for the third order case:



• The solution from the first-order fast sweeping method is used as the

initial guess for all the grid points

• The following Gauss-Seidel iterations with four alternating direction

sweepings are then performed:

(1) i = 0 : I, j = 0 : J ; (2) i = I : 0, j = 0 : J ;

(3) i = I : 0, j = J : 0; (4) i = 0 : I, j = J : 0.

(4)

When we loop to a point (i, j), the solution is updated as follows,

φnewi,j =

min(φxmini,j , φymin

i,j ) + fi,j h, if |φxmini,j − φymin

i,j | ≤ fi,j h,

φxmini,j

+φymin

i,j+(2f2

i,jh2

−(φxmini,j

−φymin

i,j)2)

1

2

2, otherwise

(5)



where fi,j = f(xi, yj), and

φxmini,j = min

(

φi,j − h (φx)−

i,j, φi,j + h (φx)+i,j

)

φymini,j = min

(

φi,j − h (φy)−

i,j, φi,j + h (φy)+i,j

)

, (6)

with

(φx)−

i,j = (1−w−)

(

φi+1,j − φi−1,j

2h

)

+w−

(

3φi,j − 4φi−1,j + φi−2,j

2h

)

,

(7)

(φx)+i,j = (1−w+)

(

φi+1,j − φi−1,j

2h

)

+w+

(−3φi,j + 4φi+1,j − φi+2,j

2h

)

,

(8)

w− =1

1 + 2r2−

, r− =ε + (φi,j − 2φi−1,j + φi−2,j)

2

ε + (φi+1,j − 2φi,j + φi−1,j)2, (9)



w+ =1

1 + 2r2+

, r+ =ε + (φi,j − 2φi+1,j + φi+2,j)

2

ε + (φi+1,j − 2φi,j + φi−1,j)2. (10)

Here ε is a small number in the WENO nonlinear weights. The

definitions for (φy)−

i,j and (φy)+i,j are of course analogous.

• Convergence is declared if

∥

∥φnew − φold∥

∥ ≤ δ, (11)

where δ is a given convergence threshold value.

We refer to (Zhang, Zhao and Qian, JSC 2006) for more details and

numerical results.



Treatment of boundary conditions

We now discuss the treatment of (non-periodic) boundary conditions. Here

we mainly discuss the result contained in (Huang, Shu and Zhang, JCM

2008) for steady state Eikonal equations using fast sweeping method, but

the methodology applies to more general cases.

For the convenience of the algorithm description below, we divide the set

of mesh points (xi, yj) into the following four categories:

• Category I contains the points at the inflow part of the domain

boundary. The numerical solution φi,j in Category I is fixed at the

prescribed physical boundary condition and does not change during

the fast sweeping iteration.



• Category II contains the points at the outflow part of the domain

boundary, where no physical boundary condition is given, and the

ghost points outside the computational domain near the outflow

boundary which are necessary for the wide stencil WENO

interpolation. The numerical solution φi,j in Category II is obtained by

extrapolation of suitable accuracy, based on the numerical solution

inside the computational domain.



• Category III contains the few points inside the computational domain

and near the inflow boundary. These points cannot be updated by the

WENO scheme because of its wide stencil. For the third order WENO

scheme under consideration, any point which has a horizontal or

vertical distance less than 3h from the inflow boundary belongs to this

category. The standard strategy in the literature of fast sweeping

method to treat points in Category III is to fix the numerical solution

φi,j as the exact solution of the PDE and it does not change during

the fast sweeping iteration. This of course is not always feasible.

Another strategy used in the literature is to fix the numerical solution

φi,j as that of a first order fast sweeping solution and it does not

change during the fast sweeping iteration. This could of course lead to

a loss of local and hence global accuracy, since information will flow

from this part of the boundary into the computational domain.



• Category IV contains all the remaining points, which are updated

during the fast sweeping iterations until convergence.

Boundary treatment strategy I: Richardson extrapolation

The first strategy that we propose to treat points in Category III is to obtain

several first order accurate solutions with different mesh sizes, then use

Richardson extrapolation to obtain accurate point values for those points in

Category III. This is feasible without excessive computational cost

because points in Category III are close to the inflow boundary, hence the

first order fast sweeping iterations can be performed locally, greatly

reducing the computational cost.



Richardson extrapolation is a well-known idea so we will only describe our

application of this idea briefly. Assume Ih is the numerical solution of the

first order fast sweeping scheme with mesh size h at the location (x∗, y∗),

which is a grid point in Category III. If we further assume

Ih − I = αh + βh2 + O(h3)

with constants α and β, where I is the exact solution at the location

(x∗, y∗), which is reasonable when the exact solution is smooth, then

clearly

Ih =1

3Ih − 2Ih/2 +

8

3Ih/4 (12)

would be a third order approximation to I :

Ih − I = O(h3).



This boundary treatment strategy is suitable for most types of inflow

boundaries, including the source boundary consisting of a single point.

The efficiency of this strategy however depends on how fast we can

compute the first order approximations Ih, Ih/2 and Ih/4 for all grid points

inside Category III. When the characteristics from the inflow boundary do

not intersect with each other, such first order fast sweeping computation

can be performed locally and is very fast. When the characteristics from

the inflow boundary do intersect with each other, the efficiency of this

strategy would decrease. Fortunately, in this case the inflow boundary

would not be a single point, hence the second strategy described in next

subsection would usually be applicable.



Boundary treatment strategy II: a Lax-Wendroff type procedure

The original Lax-Wendroff scheme (Lax and Wendroff, CPAM 1960) uses

an important idea of converting the time derivatives to spatial derivatives,

by repeatedly using the PDE. We propose to use the same idea to obtain

high order approximations to the solution values for the points in Category

III.

To fix the ideas, let us assume that the left boundary

Γ = (x, y) : x = 0, 0 ≤ y ≤ 1 (13)

of the computational domain [0, 1]2 is the inflow boundary, on which the

solution is given as

φ(0, y) = g(y), 0 ≤ y ≤ 1.



We would like to obtain a high order approximation to the solution value

φi,j ≈ φ(xi, yj) for i = 1, 2 and a fixed j, which corresponds to a point

(xi, yj) in Category III. A simple Taylor expansion gives, for i = 1, 2,

φ(xi, yj) = φ(0, yj) + ih φx(0, yj) +(ih)2

2φxx(0, yj) + O(h3)

hence our desired approximation for the third order WENO scheme is

φi,j = φ(0, yj) + ih φx(0, yj) +(ih)2

2φxx(0, yj).

We already have φ(0, yj) = g(yj). The Eikonal PDE, evaluated at the

point (0, yj), becomes

H(φx(0, yj), g′(yj)) = f(0, yj) (14)

in which the only unknown quantity is φx(0, yj). Solving this (usually



nonlinear) equation should give us φx(0, yj). There might be more than

one root, in which case we should choose the root so that

∂uH(φx(0, yj), g′(yj)) > 0 (15)

where ∂u refers to the partial derivative with respect to the first argument

in H(u, v). The condition (15) guarantees that the boundary Γ in (13) is

an inflow boundary. If the condition (15) still cannot pin down a root, then

we would choose the root which is closest to the value from the first order

fast sweeping solution at the same grid point. To obtain φxx(0, yj), we

first take the derivative with respect to y on the original PDE, and then

evaluate it at the the point (0, yj), which yields

∂uH(φx(0, yj), g′(yj))φxy(0, yj)+∂vH(φx(0, yj), g

′(yj))g′′(yj) = fy(0, yj),

(16)



where ∂u and ∂v refer to the partial derivatives with respect to the first and

second arguments in H(u, v), respectively. In this equation the only

unknown quantity is φxy(0, yj), hence we obtain easily its value, thanks

to (15). We then take the derivative with respect to x on the original PDE,

and evaluate it at the the point (0, yj) to obtain

∂uH(φx(0, yj), g′(yj))φxx(0, yj)+∂vH(φx(0, yj), g

′(yj))φxy(0, yj) = fx(0, yj),

This time, the only unknown quantity is φxx(0, yj), which we can obtain

readily from this equality.



It is clear that this procedure can be carried out to any desired order of

accuracy. Also, the inflow boundary Γ in (13) can be any piece of a

smooth curve: we only need to change the x and y partial derivatives to

normal and tangential derivatives with respect to Γ. However, for this

approach to work, Γ can not consist of a single point.



Numerical examples

Example 1. We solve the Eikonal equation with

f(x, y) =π

2

√

sin2(π

2x)

+ sin2(π

2y)

.

The inflow boundary Γ is the single point (0,0). The computational domain

is [−1, 1]2. The exact solution for this problem is

φ(x, y) = − cos(π

2x)

− cos(π

2y)

.

Since the inflow boundary Γ consists of a single point, the second strategy

described above does not apply. We use the first strategy described above

to handle the inflow boundary condition. Namely, in the small box

[−2h, 2h]2, we apply the first order fast sweeping method with three

different mesh sizes h, h/2 and h/4, and then use the Richardson



extrapolation (12) to obtain a third order approximation to the grid values

φi,j in this small box, which are then fixed as the initial values during the

third order fast sweeping WENO process. Notice that this process has

very little computational cost since the box [−2h, 2h]2 is very small. For

the outflow boundary, which is the boundary of the box [−1, 1]2, we take a

simple third order extrapolation to provide solution values in the ghost

points outside the computational domain. The The results are given in

Table 1. We can see clearly that the scheme with the numerical boundary

treatment gives the correct order of accuracy.



Table 1: Example 1. Richardson extrapolation for the inflow boundary. N

is the number of mesh points in each direction.

N L1 error order L∞ order iteration number

40 8.00E-04 6.36E-04 40

80 5.92E-05 3.76 3.33E-05 4.25 30

160 3.64E-06 4.02 1.54E-06 4.44 38

320 4.00E-07 3.19 1.58E-07 3.28 50

640 4.98E-08 3.00 2.02E-08 2.97 81



Example 2 (shape-from-shading). We solve the Eikonal equation with

case (a): f(x, y) =√

(1 − |x|)2 + (1 − |y|)2; (17)

case (b): f(x, y) = 2√

y2(1 − x2)2 + x2(1 − y2)2. (18)

The computational domain Ω = [−1, 1]2. The inflow boundary for this

example is the whole boundary of the box [−1, 1]2, namely

Γ = (x, y) : |x| = 1 or |y| = 1. The boundary condition

φ(x, y) = 0 is prescribed on Γ. The exact solutions for these two cases

are given by

case (a): φ(x, y) = (1 − |x|)(1 − |y|); (19)

case (b): φ(x, y) = (1 − x2)(1 − y2). (20)



For this example, we can apply both of the strategies described above. We

first apply the Richardson extrapolation strategy for the points inside the

computational domain which are of distance at most 2h away from the

inflow boundary Γ, using the results of the first order fast sweeping

method with three different mesh sizes h, h/2 and h/4. The results are

given in Table 2. We can again see clearly that the scheme with this

numerical boundary treatment gives the correct order of accuracy.



Table 2: Example 2. Richardson extrapolation for the inflow boundary. N

is the number of mesh points in each direction.

N L1 error order iter L1 error order iter

case (a) case (b)

80 3.45E-06 21 3.38E-05 35

160 3.04E-07 3.51 28 3.44E-06 3.30 50

320 2.58E-08 3.56 46 2.23E-07 3.95 81

640 2.14E-09 3.59 81 9.25E-09 4.59 149



Next, we apply the Lax-Wendroff type procedure to obtain third order

approximations to the values of the numerical solution corresponding to

the points inside the computational domain which are of distance at most

2h away from the inflow boundary Γ. The results are given in Table 3.

This time, since the solution is a polynomial of degree lower than the order

of the scheme, we are able to obtain the exact solution with only round-off

errors, as the Lax-Wendroff type procedure is able to prescribe the values

to the points inside the computational domain which are of distance at

most 2h away from the inflow boundary Γ exactly by Taylor expansion.

We remark that the first strategy of using the Richardson extrapolation is

not able to provide the solution values to the points inside the

computational domain which are of distance at most 2h away from the

inflow boundary Γ exactly, but only to the designed third order accuracy,

hence the final fast sweeping WENO results in Table 2 are also only high



order accurate but not exact to round-off errors.

Table 3: Example 2. Lax-Wendroff type procedure for the inflow boundary.

N is the number of mesh points in each direction.

N L1 error L∞ iter L1 error L∞ iter

case (a) case (b)

80 2.06E-14 9.26E-13 1 8.64E-07 9.99E-04 35

160 1.58E-14 2.34E-12 1 5.25E-08 2.34E-04 47

320 1.09E-14 5.49E-12 1 4.43E-15 4.26E-12 62

640 1.01E-14 1.08E-11 2 1.41E-15 1.14E-12 90



Example 3. We solve the Eikonal equation with f(x, y) = 1. The

computational domain is [−1, 1]2, and the inflow boundary Γ is the unit

circle of center (0,0) and radius 0.5, that is

Γ =

(x, y) : x2 + y2 =1

4

.

The boundary condition φ(x, y) = 0 is prescribed on Γ. The exact

solution for this problem is the distance function to the circle Γ. This exact

solution has a singularity at the center of the circle to which the

characteristics converge, hence we exclude the box [−0.15, 0.15]2 when

measuring the errors.



For this problem, it is not easy to apply the Richardson extrapolation

strategy, since we use rectangular meshes and the inflow boundary Γ is

not on grid points. However, the Lax-Wendroff type procedure can be

easily used to obtain the values of the numerical solution corresponding to

the points inside the computational domain which have a horizontal or

vertical distance less than 3h from the inflow boundary Γ. For the outflow

boundary, which is the boundary of the box [−1, 1]2, we take a simple

third order extrapolation to provide solution values in the ghost points

outside the computational domain. The results are given in Table 4. We

can see happily again that the scheme with this numerical boundary

treatment gives the correct order of accuracy away from the singularity at

the center.




N is the number of mesh points in each direction. The errors are measured

in the computational domain but outside the box [−0.15, 0.15]2.


80 0.573E-05 0.129E-03 25

160 0.122E-05 2.23 0.407E-05 4.98 32

320 0.191E-06 2.68 0.122E-05 1.74 46

640 0.246E-07 2.95 0.161E-06 2.92 62



Example 4. We solve the Eikonal equation with f(x, y) = 1. The

computational domain is [−3, 3]2, and the inflow boundary Γ consists of

two circles of equal radius 0.5 with centers located at (−1, 0) and

(√

1.5, 0), respectively, that is

Γ =

(x, y) : (x + 1)2 + y2 =1

4or (x −

√1.5)2 + y2 =

1

4

.

The boundary condition φ(x, y) = 0 is prescribed on Γ. The exact

solution for this problem is the distance function to Γ. The exact solution

for this problem is the distance function to the circle Γ. This exact solution

has singularities at the centers of the circles and on the line that has the

same distance to the two circles, on which the characteristics converge,

hence we exclude the boxes [−1.15, −0.85] × [−0.15, 0.15],

[√

1.5 − 0.15,√

1.5 + 0.15] × [−0.15, 0.15] and



[√

0.375 − 0.65,√

0.375 − 0.35] × [−3, 3] when measuring the

errors. Again, for this problem, it is not easy to apply the Richardson

extrapolation strategy, since we use rectangular meshes and the inflow

boundary Γ is not on grid points. However, the Lax-Wendroff type

procedure can again be easily used to obtain the values of the numerical

solution corresponding to the points inside the computational domain

which have a horizontal or vertical distance less than 3h from the inflow

boundary Γ. For the outflow boundary, which is the boundary of the box

[−3, 3]2, we take a simple third order extrapolation to provide solution

values in the ghost points outside the computational domain. The results

are given in Table 5. We can see happily again that the scheme with this

numerical boundary treatment gives the correct order of accuracy away

from the singularities.




N is the number of mesh points in each direction. The errors are measured

in the computational domain but outside the boxes [−1.15, −0.85] ×[−0.15, 0.15], [

√1.5 − 0.15,

√1.5 + 0.15] × [−0.15, 0.15] and

[√

0.375 − 0.65,√

0.375 − 0.35] × [−3, 3].


80 0.569E-02 0.274E-02 38

160 0.346E-03 4.04 0.766E-03 1.84 47

320 0.240E-04 3.85 0.294E-04 4.71 47

640 0.470E-05 2.35 0.336E-05 3.13 67



References for the second lecture:

[1] R. Abgrall, On essentially non-oscillatory schemes on unstructured

meshes: analysis and implementation, Journal of Computational Physics,

v114 (1994), pp.45-54.

[2] M. Boue and P. Dupuis, Markov chain approximations for deterministic

control problems with affine dynamics and quadratic cost in the control,

SIAM Journal on Numerical Analysis, v36 (1999), pp.667-695.

[3] G. Jiang and D.P. Peng, Weighted ENO schemes for Hamilton-Jacobi

equations, SIAM Journal on Scientific Computing, v21 (2000),

pp.2126-2143.

[4] G. Jiang and C.-W. Shu, Efficient implementation of weighted ENO

schemes, Journal of Computational Physics, v126 (1996), pp.202-228.



[5] L. Huang, C.-W. Shu and M. Zhang, Numerical boundary conditions for

the fast sweeping high order WENO methods for solving the Eikonal

equation, Journal of Computational Mathematics, v26 (2008), pp.336-346.

[6] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes

for Hamilton-Jacobi equations, SIAM Journal on Numerical Analysis, 28

(1991), 907-922.

[7] C.-W. Shu, High order numerical methods for time dependent

Hamilton-Jacobi equations, in Mathematics and Computation in Imaging

Science and Information Processing, S.S. Goh, A. Ron and Z. Shen,

Editors, Lecture Notes Series, Institute for Mathematical Sciences,

National University of Singapore, volume 11, World Scientific Press,

Singapore, 2007, pp.47-91.



[8] Y.-T. Zhang and C.-W. Shu, High-order WENO schemes for

Hamilton-Jacobi equations on triangular meshes, SIAM Journal on

Scientific Computing, v24 (2003), pp.1005-1030.

[9] Y.-T. Zhang, H.-K. Zhao and J. Qian, High order fast sweeping methods

for static Hamilton-Jacobi equations, Journal of Scientific Computing, v29

(2006), pp.25-56.

[10] H.-K. Zhao, A fast sweeping method for Eikonal equations,

Mathematics of Computation, v74 (2005), pp.603-627.



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