Semi-discrete central-upwind schemes with reduced dissipation for Hamilton Jacobi equations

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1 IMA Journal of Numerical Analysis 005 5, doi: /imanum/drh015 Semi-discrete central-upwind schemes with reduced dissipation for Hamilton Jacobi equations STEVE BRYSON Programme in Scientific Computing/Computational Mathematics, Stanford University and the NASA Advanced Supercomputing Division, NASA Ames Research Center, Moffett Field, CA , USA ALEXANDER KURGANOV Department of Mathematics, Tulane University, 683 St. Charles Avenue, New Orleans, LA 70115, USA DORON LEVY Department of Mathematics, Stanford University, Stanford, CA , USA AND GUERGANA PETROVA Department of Mathematics, Texas A&M University, College Station, TX , USA Received on 18 February 004; revised on 10 May 004] We introduce a new family of Godunov-type semi-discrete central schemes for multidimensional Hamilton Jacobi equations. These schemes are a less dissipative generalization of the central-upwind schemes that have been recently proposed in Kurganov, Noelle and Petrova 001, SIAM J. Sci. Comput., 3, pp We provide the details of the new family of methods in one, two, and three space dimensions, and then verify their expected low-dissipative property in a variety of examples. Keywords: Hamilton Jacobi equations; central-upwind schemes; semi-discrete methods. 1. Introduction We consider the multidimensional Hamilton Jacobi equation, ϕ t H x ϕ = 0, x R d, 1.1 with Hamiltonian H. First-order numerical schemes that converge to the viscosity solution of 1.1 were first introduced by Crandall & Lions 1984 and by Souganidis Recent attempts to obtain higher-order approximate solutions of 1.1 include upwind methods Jiang & Peng, 000; Osher & Sethian, 1988; Osher & Shu, 1991, discontinuous Galerkin methods Hu & Shu, 1999, and others. Here, we study a class of proection evolution methods, called Godunov-type schemes. The main bryson@nas.nasa.gov kurganov@math.tulane.edu dlevy@math.stanford.edu gpetrova@math.tamu.edu IMA Journal of Numerical Analysis Vol. 5 No. 1 c Institute of Mathematics and its Applications 005; all rights reserved.

2 114 S. BRYSON ET AL. structure of these schemes is as follows: one starts with the point values of the solution, constructs an essentially non-oscillatory continuous piecewise polynomial interpolant, and then evolves it to the next time level while proecting the solution back onto the computational grid. The key idea in Godunov-type central schemes is to avoid solving generalized Riemann problems, by evolving locally smooth parts of the solution. Second-order staggered Godunov-type central schemes were introduced by Lin & Tadmor 001, 000. L 1 -convergence results for these schemes were obtained in Lin & Tadmor 001. More efficient non-staggered central schemes as well as genuinely multidimensional generalizations of the schemes in Lin & Tadmor 000 were presented in Bryson & Levy 003a, with high-order extensions up to fifth-order proposed in Bryson & Levy 003b,c. Second-order semi-discrete Godunov-type central schemes were introduced in Kurganov & Tadmor 000, where local speeds of propagation were employed to reduce the numerical dissipation. The numerical viscosity was further reduced in the central-upwind schemes Kurganov et al., 001 by utilizing one-sided estimates of the local speeds of propagation. Higher-order extensions of these schemes were introduced in Bryson & Levy 003d, where weighted essentially non-oscillatory WENO interpolants were used to increase accuracy. WENO interpolants were originally developed for numerical methods for hyperbolic conservation laws Liu et al., 1994; Jiang & Shu, 1996, and were first implemented in the context of upwind schemes for Hamilton Jacobi equations in Jiang & Peng 000. Godunov-type central-upwind schemes are constructed in two steps. First, the solution is evolved to the next time level on a non-uniform grid the location of the grid points depends on the local speeds, and thus can vary at every time step. The solution is then proected back onto the original grid. The proection step requires an additional piecewise polynomial reconstruction over the non-uniform grid. In this paper we show that in the semi-discrete setting different choices of such a reconstruction lead to different numerical Hamiltonians, and thus to different schemes. In particular, we can recover the scheme from Kurganov et al A more careful selection of the reconstruction results in a new centralupwind scheme with smaller numerical dissipation. This approach was originally proposed in Kurganov & Petrova 000, where it was applied to one-dimensional 1D systems of hyperbolic conservation laws. It has been recently generalized and implemented for multidimensional systems of hyperbolic conservation laws in Kurganov & Lin in preparation. The paper is organized as follows. In Section, we develop new semi-discrete central-upwind schemes for 1D Hamilton Jacobi equations. We also review the interpolants that are required to complete the construction of the second- and fifth-order schemes. Generalizations to more than one space dimension with special emphasis on the two-dimensional setup are then presented in Section 3, where the corresponding multidimensional interpolants are also discussed. In Section 4, we evaluate the performance of the new schemes with a series of numerical tests. Finally, in Appendix A, we prove the monotonicity of the new numerical Hamiltonian.. One-dimensional schemes.1 Semi-discrete central-upwind schemes for Hamilton Jacobi equations In this section, we describe the derivation of a new family of semi-discrete central-upwind schemes for the 1D Hamilton Jacobi equation, ϕ t H ϕ x = 0, x R,.1

3 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 115 subect to the initial data ϕx, t = 0 = ϕ 0 x. Wefollow the approach in Kurganov et al. 001 see also Kurganov & Tadmor, 000. For simplicity we assume a uniform grid in space and time with grid spacing x and t, respectively. The grid points are denoted by x := x, t n := n t, and the approximate value of ϕ x, t n is denoted by ϕ n. Assume that the approximate solution at time t n, ϕ n, is given, and that a continuous piecewisepolynomial interpolant ϕx, t n is reconstructed from ϕ n.atevery grid point, the maximal right and left speeds of propagation, a and a, are then estimated by a = max H u, 0 }, a minϕx,ϕ x } u maxϕx,ϕ x = min H u, 0 },. } minϕx,ϕ x } u maxϕx,ϕ x } where ϕ x ± are the one-sided derivatives at x = x, that is ϕ x ± := ϕ xx ± 0, t n. Obviously, the quantities a ± also depend on time, and ϕ ± x depend on both time and location. These dependences are omitted to simplify the notation. If the Hamiltonian is convex,. reduces to a = max H ϕ x, H ϕ x, 0}, a = min H ϕ x, H ϕ x, 0},.3 while in the non-convex case, one has to use directly the expressions in.. We then proceed by evolving the reconstruction ϕ at the evolution points x n ± := x ± a ± t, tothe next time level according to.1. The time step t is chosen so that x n < xn 1 for all. Therefore the solution remains smooth at x n ± for t tn, t n1 ] see Fig. 1 and we can compute the values of the evolved solution ϕ n1 ± } by the Taylor expansion in time: ϕ n1 ± = ϕxn ±, tn th ϕ x x n ±, tn O t..4 Using the values ϕ n1 ± } on the non-uniform grid xn ± },weconstruct a new quadratic interpolant ψ. On the interval x n, xn ], the interpolant takes the form ψ x, t n1 := ϕ n1 ϕn1 ϕn1 x n x x n xn 1 ϕ xx n1 x x n x xn,.5 where ϕ xx n1 is yet to be determined and is an approximation to ϕ xx x n, tn1, x n = x n xn /. The proection back onto the original grid is then carried out by evaluating ψx, t n1 at x, ϕ n1 := ψx, t n1 = a a a ϕ n1 a a a ϕ n1 1 ϕ xx n1 a a t..6 Note that if the Riemann fan is symmetric, that is, if a = a, then xn = x. Substituting.4 in.6 yields ϕ n1 = a a a a a a ϕx n, tn th ϕ x x n, tn ϕx n, tn th ϕ x x n, tn 1 ϕ xx n1 a a t O t..7

4 116 S. BRYSON ET AL. ϕ n1 ϕ n1 ϕ n1 1 ϕ n1 ϕ n1 1 ϕ n1 1 ϕ n ϕ n 1 x n x n x x x x n 1 x n 1/ 1 FIG.1. Central-upwind differencing: 1D. 1 Using the Taylor expansion in space, we arrive at ϕx n ±, tn = ϕ n ± ta± ϕ± x O t,.8 ϕ n1 = ϕ n t a a a a ϕ x ϕx t a a ] a H ϕ x x n, tn a H ϕ x x n, tn 1 ϕ xx n1 a a t O t..9 We then let t 0, and end up with the family of semi-discrete central-upwind schemes: d dt ϕ t = a Hϕ x a Hϕx a a a a ϕ x ϕ x a a 1 ] lim t ϕ xx n1..10 t 0 Here, the one-sided speeds of propagation, a ±, are given by., and ϕ± x are the left and right derivatives at the point x = x of the reconstruction ϕ, t at time t. Finally, in order to complete the construction of the scheme, we must determine ϕ xx n1.for example, selecting ϕ xx n1 to be independent of t gives ] lim t ϕ xx n1 = 0, t 0 and then.10 recovers the central-upwind scheme in Kurganov et al However, since the interpolant ψ, t n1 is defined on the intervals x n, xn ], whose size is proportional to t,itisnatural to choose ϕ xx n1 to be proportional to 1/ t. Inthis case, the approximation of the second derivative in.10 will add a non-zero contribution to the limit as t 0. At the same time, to guarantee a

5 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 117 non-oscillatory reconstruction, we should use a nonlinear limiter. For example, one can use the minmod limiter: ϕx t ϕ xx n1 x n = minmod, tn1 ψ x x n, tn1 ψ x x n a a,, tn1 ϕ x x n, tn1 a a,.11 where ψ x is the derivative of.5, ϕ x x n ±, tn1 are the values of the derivative of the evolved reconstruction ϕ, t n at t = t n1, and the multivariate minmod function is defined by minx }, if x > 0, minmodx 1, x,...:= maxx }, if x < 0,.1 0, otherwise. A different choice of limiter in.11 will result in a different scheme from the same family of centralupwind schemes. All that remains is to determine the quantities used in.11. Since all data are smooth along the line segments x n ±, t, tn t < t n1,wecan use a Taylor expansion in time to obtain ϕ x x n ±, tn1 = ϕ x x n ±, tn O t..13 According to.5, the derivative ψ x x n, tn1 of the new reconstruction ψ at time level t n1 is ψ x x n, tn1 = ϕn1 ϕn1 a a,.14 t and after substituting.4 and.8 into.14, we obtain ψ x x n, tn1 = a ϕ x a ϕ x a a H ϕ xxn, tn H ϕ x xn, tn a a O t..15 Passing to the semi-discrete limit t 0 in.11,.13, and.15 gives lim t 0 t ϕ xx n1 ] = a a minmod ϕ x ψint x,ψint x ϕ x,.16 where ψ int x := lim ψ x x n, t 0 tn1 ]= a ϕ x a ϕ x a a Hϕ x Hϕ x a a..17 Finally, substituting.16 into.10, we obtain the 1D low-dissipative semi-discrete central-upwind scheme: d dt ϕ t = a Hϕ x a Hϕx a a a ϕ a x ϕ x ϕ a a minmod x ψx int a a, ψint x ϕ ] x a a,.18 where ψ int x is given by.17. For future reference, we denote the RHS of.18 by H BKLP.

6 118 S. BRYSON ET AL. Notice that in the fully-discrete setting the use of the intermediate quadratic reconstruction ψ, t n1 at level t n1 as opposed to the intermediate piecewise linear reconstruction in Kurganov et al., 001 increases the accuracy of the resulting fully-discrete scheme: O t x r versus O t x r, where r is the formal order of accuracy of the continuous piecewise polynomial reconstruction ϕ, t n. When we pass to the semi-discrete limit t 0, both quadratic and linear interpolation errors go to 0, and therefore the formal order of accuracy of both and the semi-discrete scheme in Kurganov et al. 001 is O x r. The temporal error is determined solely by the formal order of accuracy of the ODE solver used to integrate.18. However, the minmod limiter introduces a new term that leads to a reduction of the numerical dissipation without affecting the accuracy of the scheme. To demonstrate this, we show that ψx int is always in the interval minϕ x,ϕ x }, maxϕ x,ϕ x }], and therefore the absolute value of the term ϕ x ϕ x in the numerical dissipation in the scheme from Kurganov et al. 001 is always greater than the absolute value of the new term, that is ϕ x ϕ x ϕ x ϕ x minmod ϕ x ψint x,ψint x ϕx. Indeed, we have ψ int x = a ϕ x a ϕ x a a Hϕ x Hϕ x a a = ϕ x a H ] ξ a a ϕx H ξ a a a ],.19 where ξ minϕ x,ϕ x }, maxϕ x,ϕ x }. Itfollows from the definition of the local speeds. that Thus,.19 is a convex combination of ϕ x and ϕ x Remarks. a H ξ 0, H ξ a 0., and therefore ψint x minϕ x,ϕ x }, maxϕ x,ϕ x }]. 1. We would like to emphasize that the reduced dissipation in the scheme.18 when compared with the scheme of Kurganov et al. 001 is due to the minmod term in the RHS of.18. This ] additional term arises when we define ˆϕ xx n1 by.11 such that the lim t 0 t ϕ xx n1 in.10 does not vanish.. While the new nonlinear limiters in the scheme.18 require additional computational work, the quantities that participate in the limiter do not require any new flux evaluations and hence the increase in the computational complexity is minimal. Such additional work when compared with the original scheme of Kurganov et al., 001 can be worthwhile in cases where the user is interested in increasing the resolution of the solution without increasing the order of accuracy of the method. It was shown in Bryson & Levy 003d that the numerical Hamiltonian H KNP from Kurganov et al. 001 is monotone, provided that the Hamiltonian H is convex. Here, we state a theorem about the monotonicity of H BKLP the new, less dissipative Hamiltonian in.18. The proof is left to the Appendix. We will consider only Hamiltonians for which H changes sign, because otherwise either a 0ora 0 and the Hamiltonian in.18 reduces to the upwind one for which such a theorem is known. THEOREM.1 Let the Hamiltonian H C be convex and satisfy the following two assumptions:

7 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 119 A1 The function Gu,v:= H u u vh v Hu Hv ] H u H v ] 0.0 for all u and v in the set S u,v S u,v,where S Hu Hv u,v:= u,v: u v S u,v:= u,v: Hu Hv u v H u H } v, u u v, H u H } v, u u v,.1 and u is the only point such that H u = 0; A For any v and for an arbitrary interval a, b], the sets S u,v a, b] and S u,v a, b] are either the empty set or finite unions of closed intervals and/or points. Then the numerical Hamiltonian in.18: H BKLP u, u := a Hu a Hu a a a a u u u a a minmod u int a a, uint u ] a a,. where u int := a u a u a a Hu Hu a a, Remarks. and where a := a u, u = maxh u, H u, 0}, a := a u, u = minh u, H u, 0} is monotone, that is H BKLP is a non-increasing function of u and a non-decreasing function of u. 1. Examples of Hamiltonian that satisfy conditions.0.1 are any convex quadratic Hamiltonian Hu = au bu c. Straightforward computation gives that the function Gu,v 0, and the sets in A are either empty or one closed interval, or one point, and therefore the theorem holds. Another example, for which Theorem.1 is valid, is Hu = u 4.In this case, the sets.1 are S u,v= u,v: u v 0 v}, S u,v= u,v: u v 0 v}, and, as one can easily verify, the function Gu,v=8u v 3 u 3 3u v 6uv v 3 0, in S u,v S u,v.asfor the sets in assumption A, they are either empty or one closed interval, or one point.. Notice that assumption A in Theorem.1 is needed only for technical purposes and in fact it is satisfied by almost every Hamiltonian H that arises in applications.

8 10 S. BRYSON ET AL. 3. The monotonicity of the numerical Hamiltonian is an essential ingredient in the theory of Barles & Souganidis The main theorem in Barles & Souganidis 1991 implies that a consistent, stable and monotone approximation of a Hamilton Jacobi equation that satisfies an underlying comparison principle converges to the unique viscosity solution of that equation. In our context, such an approximation can be obtained if we assume a piecewise-linear reconstruction and replace the time derivative by a forward Euler approximation.. A second-order scheme A non-oscillatory second-order scheme can be obtained if one uses a non-oscillatory continuous piecewise quadratic interpolant ϕ. The values of the one-sided derivatives of ϕ at x, t n in.17 and.18 are given by ϕ ± x = ϕ n ± 1 x x ϕ xx n 1, ϕ n 1 := ϕ n 1 ϕn,.3 where the second derivative is computed with a nonlinear limiter. For example, ϕ xx n 1 ϕ n = minmod θ 3 ϕ n 1 x, ϕ n 3 ϕ n 1 x,θ ϕ n ϕ n 1 1 x..4 Here, θ 1, ] and the minmod function is given by.1. As is well-known, larger values of θ correspond to less dissipative limiters see Sweby, The scheme requires an ODE solver that is at least second-order accurate..3 Higher-order schemes In this section, we briefly describe the third- and fifth-order WENO reconstructions. They were derived in Bryson & Levy 003d in the context of central-upwind schemes, and are similar to those used in high-order upwind schemes Jiang & Peng, 000. In smooth regions, the WENO reconstructions use a convex combination of multiple overlapping reconstructions to attain high-order accuracy. In non-smooth regions, a smoothness measure is employed to increase the weight of the least oscillatory reconstruction. Here, we reconstruct the one-sided derivatives ϕ x ± k, at x = x for k = 1,...,d stencils, and write the convex combination d ϕ x ± = w k, ± ϕ± x k,, k=1 d w k, ± = 1, w± k, k=1 0,.5 where the values ϕ ± x are to be used in the scheme The weights w± k, are defined as w ± k, = α± k, d α l, ± l=1, α k, ± = c k ± p..6 ɛ S k, ±

9 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 11 The constants c k ± are set so that the convex combination in.5 is of the maximal possible order of accuracy in smooth regions. We take p = and choose ɛ = 10 6 to prevent the denominator in.6 from vanishing. A third-order WENO reconstruction is obtained in the case d = with ϕ x 1, = ϕ 1 ϕ 1 x ϕ x, = 3ϕ 4ϕ 1 ϕ x, ϕx 1, = ϕ 4ϕ 1 3ϕ, x, ϕx, = ϕ 1 ϕ 1. x k c 1 = c = 3, c = c 1 = 1 3, and the smoothness measures are Here, S r, s] := x S 1, = S 1, 0], S, = S 0, 1], S 1, = S, 1], S, = S 1, 0]. s ϕ i x x i=r s i=r1 ϕ i, ± ϕ := ±ϕ ±1 ϕ..7 x A fifth-order WENO reconstruction is obtained when d = 3. In this case, ϕ x 1, = ϕ 6ϕ 1 3ϕ ϕ 1 6 x ϕ x, = ϕ 1 3ϕ 6ϕ 1 ϕ 6 x ϕ x 3, = 11ϕ 18ϕ 1 9ϕ ϕ 3 6 x The constants c k ± are given by and the smoothness measures are, ϕx 1, = ϕ 3 9ϕ 18ϕ 1 11ϕ, 6 x, ϕx, = ϕ 6ϕ 1 3ϕ ϕ 1, 6 x, ϕx 3, = ϕ 1 3ϕ 6ϕ 1 ϕ. 6 x c 1 = c 3 = 3 10, c 3 = c 1 = 1 10, c± = 3 5, S 1, = S, 0], S, = S 1, 1], S 3, = S 0, ], S 1, = S 3, 1], S, = S, 0], S 3, = S 1, 1]. The time evolution of.18 should be performed with an ODE solver whose order of accuracy is compatible with the spatial order of the scheme. In our numerical examples, we use the strong stability preserving SSP Runge Kutta methods from Gottlieb et al Multidimensional schemes In this section, we derive the two-dimensional D generalization of the semi-discrete centralupwind scheme and then extend it to three space dimensions. We also comment on the multidimensional interpolants that these extensions require.

10 1 S. BRYSON ET AL., k1, k NW NE, k 1, k 1, k, k -, k SW SE, k, k 1 FIG.. Central-upwind differencing: D. 3.1 A two-dimensional scheme We consider the D Hamilton Jacobi equation, ϕ t Hϕ x,ϕ y = 0, 3.1 and proceed as in Kurganov et al We assume that at time t = t n the approximate point values ϕ n k ϕx, y k, t n are given, and construct a D continuous piecewise-quadratic interpolant, ϕx, y, t n xx, defined on the cells S k := x, y : x yy k y 1}. Oneachcell S k there will be four such interpolants labelled NW, NE, SE, and SW, one for each triangle that constitutes S k see Fig.. Specific examples of ϕx, y, t n are discussed in Section 3.3. Similarly to the 1D case, we use the maximal values of the one-sided local speeds of propagation in the x- and y-directions to estimate the widths of the local Riemann fans. These values at any grid point x, y k can be computed as } a k := max H u ϕ x x, y, t, ϕ y x, y, t, C k a k := } min H u ϕ x x, y, t, ϕ y x, y, t C k, b k } := max H v ϕ x x, y, t, ϕ y x, y, t, C k b k := min C k } H v ϕ x x, y, t, ϕ y x, y, t, 3. where C k := x 1, x 1 ] y k 1, y k 1 ], := max, 0, := min, 0, and H u, H v T is the gradient of H. Note that in order to obtain a monotone scheme in two dimensions, one may need to use global a priori bounds on some of the derivatives in 3. see Osher & Shu, 1991 for details. The reconstruction ϕx, y, t n is then evolved according to the Hamilton Jacobi equation 3.1. Due to the finite speed of propagation, for sufficiently small t, the solution of 3.1 with initial data ϕ is smooth around x n ±, yn k± where xn ± := x ± a ± k t, yn k± := y k ± b ± k t, see Fig.. We denote ϕ n ±,k± := ϕxn ±, yn k±, tn, and use the Taylor expansion to calculate the intermediate values at the next

11 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 13 time level t = t n1 : ϕ n1 ±,k± = ϕn ±,k± t H ϕ xx n ±, yn k±, tn, ϕ y x n ±, yn k±, tn O t. 3.3 We now proect the intermediate values ϕ n1 ±,k± onto the original grid points x, y k. First, similarly to.5, we use new 1D quadratic interpolants in the y-direction, ψx n ±,, tn1,toobtain ψx n ±, y k, t n1 = ϕ n1 ±,k ϕn1 ±,k ϕn1 ±,k b k = b k b k b k b k ϕ n1 ±,k b k b k b k b k 1 ϕ yy n1 ±,k b k b k t ϕ n1 ±,k 1 ϕ yy n1 ±,k b k b k t, 3.4 where ϕ yy n1 ±,k ϕ yy x n ±, ŷn k, tn1 and ŷk n := yk n yn k /. Next, we use the values ψx n ±, y k, t n1 to construct another 1D quadratic interpolant ψ, y k, t n1, this time in the x- direction, whose values at the original grid points are ϕ n1 k := ψx, y k, t n1 = a k a k ψx n a k a, y k, t n1 k a k a k ψx n, y k, t n1 1 ϕ xx n1,k a k a k t. 3.5 Here, ϕ xx n1,k ϕ xx x n, y k, t n1 and x n := x n xn /. We choose ϕ xx n1,k to be the weighted average ϕ xx n1,k = b k b k b k ϕ xx n1,k b k b k b k ϕ xx n1,k, ϕ xx n1,k± ϕ xx x n, yn k±, tn Notice that both ϕ yy n1 ±,k in 3.4 and ϕ xx n1,k± in 3.6 are yet to be determined. We then substitute 3.4 and 3.6 into 3.5, and obtain ϕ n1 k = a k b k ϕn1,k a k b k ϕn1,k a k b k ϕn1,k a k b k ϕn1,k a k a k b k b k a k a k b k b k b k b k a k a k ] t b k ϕ xx n1,k b k ϕ xx n1,k ] t a k ϕ yy n1,k a k ϕ yy n1,k O t. 3.7

12 14 S. BRYSON ET AL. Substituting 3.3 into 3.7 yields ϕ n1 a k k = b k a k a k b k b k ϕ n,k t H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k b k a k a k b k b k ϕ n,k t H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k b k a k a k b k b k ϕ n,k t H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k b k a k a k b k b k ϕ n,k t H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k a ] k t b k b b k ϕ xx n1,k b k ϕ xx n1,k k b k b ] k t a k a a k ϕ yy n1,k a k ϕ yy n1,k O t. 3.8 k The values ϕ n ±,k± are computed by the Taylor expansions: ϕ n ±,k± = ϕn,k ± ta± k ϕ± x ± tb± k ϕ± y O t, 3.9 where ϕ ± x := ϕ x x ± 0, y k, t n and ϕ ± y := ϕ y x, y k ± 0, t n are the corresponding right and left derivatives of the continuous piecewise quadratic reconstruction at x, y k. Next, substituting 3.9 into 3.8 gives ϕ n1 k = ϕ n k t a k a k a k a k ϕ x ϕ x t b k b k b k ϕ b y ϕ y t k a k a k b k b k a k b k H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k b k H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn ] a k b k H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k b k H ϕ xx n, yn k, tn, ϕ y x n, yn k, tn a k a ] k t b k b b k ϕ xx n1,k b k ϕ xx n1,k k b k b ] k t a k a a k ϕ yy n1,k a k ϕ yy n1,k O t k

13 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 15 Finally, the limit t 0 generates a family of D semi-discrete central-upwind schemes: d dt ϕ kt = a k b k Hϕ x,ϕ y a k b k Hϕ x,ϕ y a k b k Hϕ x,ϕ y a k b k Hϕ x,ϕ y a k a k b k b k a k a k a k ϕ a x ϕ x b k b k k b k b k b k a k a k b k b k b k b k a k a k a k ϕ y ϕ y } lim t ϕ xx n1,k b k t 0 } lim t ϕ yy n1,k a k t 0 } ] lim t ϕ xx n1,k t 0,k}] lim t ϕ yy n t 0 We still need to specify ϕ xx n1,k± and ϕ yy n1 ±,k.ifthey are proportional to ϕ x n1 / x and to ϕ y n1 / y respectively, then lim t 0 t ϕ xx n1,k± } } = 0, lim t ϕ yy n1 ±,k = 0, t 0 and we obtain the original D central-upwind scheme from Kurganov et al However, similarly to the 1D case, we can choose ϕ xx n1,k± and ϕ yy n1 ±,k to be proportional to 1/ t, sothat the above limit will not vanish. For example, one can use the minmod limiter: ϕx t ϕ xx n1,k± = minmod n1,k± φ x x n, yn k±, tn1 φ x x n a k,, yn k±, tn1 ϕ x n1,k± a k a k, 3.1 a k ϕy t ϕ yy n1 ±,k = minmod n1 ±,k ψ y x ±, ŷk n, tn1 ψ y x ±, ŷk n b k,, tn1 ϕ y n1 ±,k b k b k, 3.13 b k where ϕ x n1 ±,k± := ϕ xx n ±, yn k±, tn1 and ϕ y n1 ±,k± derivative φ x in 3.1 are given by := ϕ yx n ±, yn k±, tn1. The values of the φ x x n, yn k±, tn1 = ϕn1,k± ϕn1,k± a k a k t, 3.14 and after using 3.3 and 3.9, we obtain φ x x n, yn k±, tn1 = H ϕ xx n, yn k±, tn, ϕ y x n, yn k±, tn H ϕ x x n, yn k±, tn, ϕ y x n, yn k±, tn a k a k a k ϕ x a k ϕ x a k a k O t Since the data are smooth along the line segments x n ±, yn k±, t, tn t < t n1,itisclear that lim ϕ x n1,k± = t 0 ϕ x, lim ϕ x n1,k± = t 0 ϕ x, lim ϕ y n1 ±,k = t 0 ϕ y, lim ϕ y n1 ±,k = t 0 ϕ y. 3.16

14 16 S. BRYSON ET AL. Therefore using 3.1, 3.16, and 3.15, we obtain } lim t ϕ xx n1,k± = t 0 a k a k minmod ϕ x ϕint± x,ϕx int± ϕx where ϕ int± x, 3.17 := a k ϕ x a k ϕ x a k a k Hϕ x,ϕ± y Hϕ x,ϕ± y a k a k Likewise, using 3.13, we obtain } lim t ϕ yy n1 ±,k = t 0 b k b k minmod ϕ y ϕint± y,ϕy int± ϕy where ϕ int± y, 3.19 := b k ϕ y b k ϕ y b k b k Hϕ± x,ϕ y Hϕ± x,ϕ y b k b k. 3.0 Finally, we substitute 3.17 and 3.19 into The resulting D semi-discrete central-upwind scheme is d dt ϕ kt = a k b k Hϕ x,ϕ y a k b k Hϕ x,ϕ y a k b k Hϕ x,ϕ y a k b k Hϕ x,ϕ y a k a k b k b k a k a k b k b k ϕ x ϕ b x k ϕ a k a k b k minmod x ϕx int b k a k a k b k ϕ b k minmod x ϕx int b k a k a k ϕ y ϕy a k b k b k a k a k a k a k a k ϕ y ϕy int minmod b k b k ϕ y ϕy int minmod b k b k, ϕint x ϕ x a k a k, ϕint x, ϕint y ϕ x ] a k a k, ϕint y ϕy b k b k ϕy ]. 3.1 b k b k Here, ϕx int± and ϕy int± are given by 3.18 and 3.0, respectively; the one-sided local speeds, a ± k and b ± k, are given by 3.; and formulae for ϕ± x and ϕ± y are discussed in Section 3.3 below. Remark. In practice, for convex Hamiltonians H the one-sided local speeds are computed as a k = max ± b k = max ± } H x ϕ x ±,ϕ± y, 0 } H y ϕ x ±,ϕ± y, 0, a k = min ±, b k = min ± H x ϕ x ±,,ϕ± y, 0} H y ϕ x ±,,ϕ± y, 0} where the maximum and minimum are taken over all the possible permutations of ±. 3.

15 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES A three-dimensional scheme We consider the three-dimensional 3D Hamilton Jacobi equation, ϕ t Hϕ x,ϕ y,ϕ z = 0. We use the maximal values of the one-sided local speeds of propagation, a ± kl, b± kl, and c± kl,inthe x-, y- and z-directions, respectively. These values at any grid point x, y k, z l are given by the obvious generalizations of 3. and } c kl :=max H w ϕ x x, y, z, t, ϕ y x, y, z, t, ϕ z x, y, z, t, C kl } c kl := min H w ϕ x x, y, z, t, ϕ y x, y, z, t, ϕ z x, y, z, t C kl, where C kl := x 1, x 1 ] y k 1, y k 1 ], z l 1, z l 1 ]. Proceeding as in two dimensions, the 3D semi-discrete central-upwind scheme is suppressing the indices, k, l 1 ] = a ± a a b b c c b ± c ± Hϕx,ϕ y,ϕ z ± a a ϕ a a x ϕx b b b b ϕ y ϕ y c c ϕ c c z ϕz 1 a a b b c c a a ] b ± c ± Dx ϕn1,k,l ± b b ] a ± c ± D y ϕn1 c c,k,l ± ± dϕ dt a ± b ± D z ϕn1,k,l] }, 3.3 where the summations are taken over all possible permutations of and.for example, in the first sum a b c should be multiplied by Hϕx,ϕ y,ϕ z.in3.3, we use the notation where Dx ϕn1,k±,l± :=minmod ϕ x ϕint x,k±,l±,ϕint x,k±,l± ϕ x, D y ϕn1 ±,k,l± :=minmod ϕ y ϕint y ±,k,l±,ϕint y ±,k,l± ϕ y, D z ϕn1 ±,k±,l :=minmod ϕ z ϕx int ϕ,k±,l± :=a x a ϕx a a ϕy int ϕ ±,k,l± :=b y b ϕy b b ϕ int z ±,k±,l := c ϕ z c ϕ z c c ϕ int z ±,k±,l,ϕz int ±,k±,l ϕz, Hϕ x,ϕ± y,ϕ± z Hϕ x,ϕ± y,ϕ± z a a, Hϕ± x,ϕ y,ϕ± z Hϕ± x,ϕ y,ϕ± z b b, Hϕ± x,ϕ± y,ϕ z Hϕ± x,ϕ± y,ϕ z c c.

16 18 S. BRYSON ET AL. 3.3 Multidimensional interpolants The schemes developed in Sections 3.1 and 3. require a multidimensional non-oscillatory reconstruction. The simplest option is to use straightforward multidimensional extensions of the 1D interpolants from Sections. and.3, obtained via a dimension-by-dimension approach. Forexample, a D non-oscillatory second-order central-upwind scheme is given by 3.1 with ϕ n ϕ x ± = ± 1,k x ϕn x ϕ xx n 1,k, ϕ± y = y ϕ xx n 1 = minmod θ,k,k± 1 y ϕ n 3,k ϕn ϕ n 1,k x, 3,k ϕn 1,k x, n ϕyy,k 1, ϕ n θ 1,k ϕn 1,k x, ϕ yy n,k 1 = minmod θ ϕ n,k 3 ϕ n,k 1 x, ϕ n,k 3 ϕ n,k 1 x, ϕ n ϕ n,k θ 1,k 1 x, where θ 1, ], and the minmod function is given by.1. Similarly, the corresponding dimensionby-dimension D extensions of the WENO interpolants from Section.3 can be used to reconstruct the derivatives in 3.18, 3.0, and 3.1. For more details see Bryson & Levy 003d. 4. Numerical examples In this section, we test the performance of the new semi-discrete central-upwind schemes on a variety of numerical examples. We compare the methods developed in this paper, labelled BKLP, with the secondorder scheme from Kurganov et al. 001 and the fifth-order scheme from Bryson & Levy 003d, both of which are referred to as KNP. Our results demonstrate that the BKLP schemes achieve a better resolution of singularities in comparison with the corresponding KNP schemes. Note that in regions where the solution is sufficiently smooth, a a and b b are either equal to zero or very small for smooth Hamiltonians and sufficiently small x and y. Hence, the BKLP and KNP schemes of the same order will be almost identical in these areas, and thus there will be practically no difference in the resolution of smooth solutions. We therefore only examine results after the formation of singularities, for which a a and/or b b may be large. The ODE solver that was used in all our simulations is the fourth-order strong stability preserving Runge Kutta method SSP-RK of Gottlieb et al Assuming an ODE of the form d dt ϕ =

17 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 19 H ϕ and initial data ϕ n, the fourth-order SSP-RK method is ϕ 1 = ϕ n 1 th ϕ n, ϕ = ϕn th ϕ n ϕ th ϕ 1, ϕ 3 = ϕn th ϕ n ϕ th ϕ ϕ th ϕ, ϕ n1 = 1 5 ϕn 1 10 th ϕ n ϕ1 1 6 th ϕ ϕ 1 3 ϕ3 1 6 th ϕ 3. The intermediate values of the gradient that are required at every stage of the RK method 4.1 are computed using WENO reconstructions. 4.1 One-dimensional problems A convex Hamiltonian. We first test the performance of our schemes for the Hamilton Jacobi equation with a convex Hamiltonian: ϕ t 1 ϕ x 1 = 0, 4. subect to the periodic initial data ϕx, 0 =cosπ x. The change of variables u x, t = ϕ x x, t 1 transforms the equation into the Burgers equation u t 1 u = 0, which can be easily solved via the x method of characteristics. The solution develops a singularity in the form of a discontinuous derivative at time t = 1/π. The computed solutions at T = 5/π after the singularity formation are shown in Figure 3, where the second- and fifth-order BKLP and KNP schemes are compared. There is a significant improvement in the resolution of the singularity for the BKLP schemes compared with the KNP schemes. The secondorder BKLP scheme has a smaller error at the singularity than the fifth-order KNP scheme, while the fifth-order BKLP scheme has the smallest error. In Table 1 we show the relative L 1 - and L -errors. A non-convex Hamiltonian. In this example, we compute the solution of the 1D Hamilton Jacobi equation with a non-convex Hamiltonian: ϕ t cos ϕ x 1 = 0, 4.3 subect to the periodic initial data ϕ x, 0 = cos π x. This initial-value problem has a smooth solution for t 1 049/π, after which a singularity forms. A second singularity forms at t 1 9/π. The solutions at time T = /π, computed with N = 100, are shown in Fig. 4, with a close-up of the singularities in Fig. 5. The convergence results after the singularity formation are given in Table. In this example, the local speeds of propagation were estimated by.. The results are similar to the convex case, though the improvement here is somewhat less dramatic.

18 130 S. BRYSON ET AL. 0 4 convex example, N=100 at t= 5/π closeup: convex example, N=100 at t= 5/π φ 0 4 φ x x FIG. 3. Problem 4.. Left: the solution. Right: a close-up of the solution near the singularity. : second-order KNP; o: secondorder BKLP, : fifth-order KNP, : fifth-order BKLP. The exact solution is the dashed line. TABLE 1 Problem 4.. Relative L 1 - and L -errors for the KNP and BKLP schemes Convex example ϕ t 1 ϕ x 1 = 0 second-order after singularity T = 5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP fifth-order after singularity T = 5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP Next, we examine the convergence of the numerical solutions of 4. and 4.3, computed by the fifth-order BKLP and KNP schemes. These results, together with the fifth-order methods from Jiang & Peng 000 and Bryson & Levy 003c, are shown in Fig. 6. The reader may note that the convergence rates in these examples are erratic. However, this investigation of the relative L 1 -errors for many different grid spacings shows that the behaviour is due to super-convergence at some grid spacings. Notice that for all grid spacings the L 1 -error of the BKLP method is less than the others.

19 φ SEMI-DISCRETE CENTRAL-UPWIND SCHEMES non-convex example, N=100 to t= 0/π x FIG.4. Problem 4.3: the KNP and BKLP numerical solutions. closeup: non convex example, N=100 to t= 0/π 1 08 closeup: non convex example, N=100 to t= 0/π φ φ x x FIG.5. Problem 4.3. Right: the singularity near x = 0 5. Left: the singularity near x = : second-order KNP, o: second-order BKLP, : fifth-order KNP, : fifth-order BKLP. The exact solution is the dashed line. 4. Two-dimensional problems In this section, we test the D BKLP schemes on Hamilton Jacobi equations with convex and nonconvex Hamiltonians. We start with the convex problem compare with 4. ϕ t 1 ϕx ϕ y 1 = 0, 4.4 which can be reduced to a 1D problem via the coordinate transformation ξ 1/ 1/ x =. η 1/ 1/ y The relative L 1 - and L -errors for the periodic initial data ϕ x, y, 0 = cos πx y/ = cos πξ after the singularity formation at T = 5/π are shown in Table 3. The results show that

20 13 S. BRYSON ET AL. TABLE Problem 4.3. Relative L 1 - and L -errors for the KNP and BKLP schemes Non-convex example ϕ t cos ϕ x 1 = 0 second-order after singularity T = 0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP fifth-order after singularity T = 0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP convex H, T= 5/π non convex H, T= 0/π N N relative L 1 -error relative L 1 -error number of points number of points FIG.6. Convergence of the 1D examples. Left: problem 4., T = 5/π. Right: problem 4.3, T = 0/π.:fifth-order BKLP, : fifth-order KNP, : the fifth-order method from Jiang & Peng 000, The solid lines show example rates of convergence. while the order of accuracy of the new reduced dissipation method does not change, the relative L 1 - and L -errors are smaller with the new method when compared with the results obtained with the method of Kurganov et al., 001. In Table 4, we present similar results for the non-convex problem compare with 4.3 ϕ t cos ϕ x ϕ y 1 = 0, 4.5 with the periodic initial data ϕ x, y, 0 =cos πx y/. Similarly to the convex case, also with

21 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 133 TABLE 3 Problem 4.4. Relative L 1 - and L -errors for the D KNP and BKLP schemes D convex example second-order after singularity T = 5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP fifth-order after singularity T = 5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP TABLE 4 Problem 4.5. Relative L 1 - and L -errors for the D KNP and BKLP schemes. D non-convex example second-order after singularity T = 0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP fifth-order after singularity T = 0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP the non-convex problem we observe relative L 1 - and L -errors that are smaller with the new method than the errors that are obtained with the method of Kurganov et al Acknowledgments The work of AK was supported in part by the NSF Grants DMS and DMS The work of DL was supported in part by the NSF under Career Grant DMS The work of GP was supported in part by the NSF Grant DMS

22 134 S. BRYSON ET AL. REFERENCES BARLES, G.& SOUGANIDIS, P. E.1991 Convergence of approximate schemes for fully nonlinear second order equations. Asymp. Anal., 4, BRYSON, S.&LEVY, D.003a Central schemes for multidimensional Hamilton Jacobi equations. SIAM J. Sci. Comput., 5, BRYSON, S. & LEVY, D. 003b High-order central WENO schemes for 1D Hamilton Jacobi equations. Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 001. F. Brezzi et al., eds. Berlin: Springer, pp BRYSON, S.& LEVY, D.003c High-order central WENO schemes for multi-dimensional Hamilton Jacobi equations. SIAM J. Numer. Anal., 41, BRYSON, S. & LEVY, D. 003d High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton Jacobi equations. J. Comput. Phys., 189, CRANDALL, M.G.&LIONS, P.-L Two approximations of solutions of Hamilton Jacobi equations. Math. Comp., 43, GOTTLIEB, S., SHU, C.-W. & TADMOR, E.001 Strong stability-preserving high order time discretization methods. SIAM Rev., 43, HU, C.& SHU, C.-W A discontinuous Galerkin finite element method for Hamilton Jacobi equations. SIAM J. Sci. Comput., 1, JIANG, G.-S.& PENG, D.000 Weighted ENO schemes for Hamilton Jacobi equations. SIAM J. Sci. Comput., 1, JIANG, G.-S. & SHU, C.-W Efficient implementation of weighted ENO schemes. J. Comput. Phys., 16, 0 8. KURGANOV, A.& LIN, C.-T. On the reduction of numerical dissipation in central-upwind schemes. in preparation. KURGANOV, A.&PETROVA, G.000 Central schemes and contact discontinuities. MAN Math. Model. Numer. Anal., 34, KURGANOV, A. & TADMOR, E. 000 New high-resolution semi-discrete schemes for Hamilton Jacobi equations. J. Comput. Phys., 160, KURGANOV, A., NOELLE, S.& PETROVA, G.001 Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton Jacobi equations. SIAM J. Sci. Comput., 3, VAN LEER, B.1979 Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov s method. J. Comput. Phys., 3, LIN, C.-T. & TADMOR, E.000 High-resolution non-oscillatory central schemes for Hamilton Jacobi Equations. SIAM J. Sci. Comput., 1, LIN, C.-T. & TADMOR, E.001 L 1 -stability and error estimates for approximate Hamilton Jacobi solutions. Numer. Math., 87, LIU, X.-D., OSHER, S.&CHAN, T.1994 Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115, OSHER, S.& SETHIAN, J.1988 Fronts propagating with curvature dependent speed: algorithms based on Hamilton Jacobi formulations. J. Comput. Phys., 79, OSHER, S.& SHU, C.-W High-order essentially nonoscillatory schemes for Hamilton Jacobi equations. SIAM J. Numer. Anal., 8, SOUGANIDIS, P. E.1985 Approximation schemes for viscosity solutions of Hamilton Jacobi equations. J. Differ. Eqns, 59, SWEBY, P. K.1984 High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal., 1,

23 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 135 Appendix A. Proof of Theorem.1 Proof. First, we fix u and show that H BKLP u, u given by. is a non-increasing function of u the proof that H BKLP is a non-decreasing function of its second argument is similar. We denote Hu Hu Qu := u u, u = u, A.1 H u, u = u, and Au := 1 a u a u, where a u = maxh u, H u, 0} and a u = minh u, H u, 0}, and denote by U 1, U, V 1, V, the sets U 1 := U 1 u =u : Qu Au <0}, U := U u =u : Qu Au 0}, A. V 1 := V 1 u =u : Qu Au 0}, V := V u =u : Qu Au >0}. A.3 Both U 1 and V are open sets Q and A are continuous and as such can be represented as a union of at most countably many disoint open intervals I and J, respectively, i.e. U 1 = =1 I and V = =1 J. A.4 In the new notation it is easy to verify that the Hamiltonian H BKLP can be written as H H BKLP u, u 1 BKLP u, u, u U 1, := H BKLP u, u, u U, or as H H BKLP u, u 1 BKLP u, u, u V 1, := H BKLP u, u, u V, where and H1 BKLP u, u := a uhu a uhu a u a u a ua u a u a u u u A.5 A.6 a ua u a u a u u u Qu a u ], A.7 H BKLP u, u := a uhu a uhu a u a u a ua u a u a u u u a ua u a u a u u u a u Qu ]. A.8

24 136 S. BRYSON ET AL. Notice that for those u for which Qu = Au, wehave that H1 BKLP u, u = H BKLP u, u, and therefore H BKLP, u is a well defined function consisting of the pieces Hi BKLP, u, i = 1,. We will use formulae A.5 and A.6 and continuity arguments to show that H BKLP, u is a nonincreasing function on the whole real line. Consider the point u such that H u = 0 see assumption A1. Then Hi BKLP u, u, i = 1,, are continuously differentiable on the intervals, minu, u, minu, u, maxu, u, and maxu, u, in the case u = u,onthe first and third interval only and continuous on R. CASE 1. Let u, minu, u. Then du d a u = 0 since a u = maxh u, H u, 0} and H is a non-decreasing function of u. Inthis case we have d H BKLP u, u = a u a ua uu u a du a u a u 3 u Qu ] a u a u H u a u a u. Note that there exists ξ u, u such that Qu := Hu Hu u u = H ξ H u a u, a is a smooth non-increasing function on, minu, u, a u 0, a u 0, H u a u, and therefore the derivative du d H BKLP u, u 0. Hence H BKLP u, u is non-increasing on, minu, u. Similarly, for u, minu, u we have d H1 BKLP u, u = a u a u u u du a u a u 3 Qu Au] a ua ua u H u a u a u a uh u a u a u. A.9 Now we fix and consider the corresponding open interval I, minu, u see A.4 and representation A.5. As above, a is a smooth non-increasing function on, minu, u, a u 0, a u 0. On each I, Qu <Au, and therefore the first term on the right-hand side RHS of A.9 0. The second term is non-positive since a u H u.the last term is 0 because H u 0 for u, minu, u. This proves that H1 BKLP u, u is a non-increasing function of u on I, minu, u, for every. CASE. Let u minu, u, maxu, u. Inthis case the derivatives are d a u = 0 and d du a u = 0. Therefore d H BKLP u, u = a u a u H u du a u a u 0, since a u H u, which shows that H BKLP, u is non-increasing on minu, u, maxu, u. Likewise d H1 BKLP u, u = a ua ua u H u du a u a u a uh u a u a u. du

25 SEMI-DISCRETE CENTRAL-UPWIND SCHEMES 137 The first term on the RHS is 0 since a u H u.asfor the second term, we have two possibilities. If u < u < u then H u <0, which will make the whole term 0. If u < u < u, then a u 0 and the second term is 0. Therefore, the derivative of H1 BKLP is 0, and thus H1 BKLP, u is nonincreasing on minu, u, maxu, u, and in particular on I minu, u, maxu, u for every. CASE 3a. Let u u u. Then a u 0 and therefore H BKLP H1 BKLP u, u H BKLP u, u Hu.Inparticular, H BKLP is a non-increasing function on u,, and H1 BKLP is a non-increasing function on I u,, for every. Combining the results from Cases 1, and 3a, we obtain that H BKLP is non-increasing on the whole real line since it is continuous on R and non-increasing on each of the intervals, minu, u, minu, u, maxu, u, and maxu, u,, and H1 BKLP is a non-increasing function on every open interval I from U 1 same reasoning. Since H BKLP u, u = H1 BKLP u, u for u I it will follow from A.5 that H BKLP, u is non-increasing on the whole real line. CASE 3b. Let u u u. Inthis case we will utilize representation A.6 for H BKLP and, using the results from Cases 1 and, we will show that H BKLP is a non-increasing function on the interval a, b] for any a and b, and therefore on the whole real line. Notice that in this case V 1 S u, u, and then, by assumption A, V 1 a, b] is either empty or a finite number of points and/or a finite union of closed intervals T k. Note also that we have u, a, b] = =1 J u, a, b] ] m k=1 T k, for some m. A.10 For u u d, we have that du a u = 0, and hence d H BKLP u, u = a u a u u u du a u a u 3 Qu Au] a u a u H u a u a u. As in Case 1, we fix and consider this time the corresponding interval J u,. Since a is a smooth non-decreasing function on u, and Qu >Au on J, the first term in the RHS 0. Also a u H u and hence the second term is also 0. This gives that H BKLP u, u is a non-increasing function of u on J u, for every. When u < u < u, a u = H u, a u =H u, and hence d H1 BKLP u, u a uh u = du a u a u 3 Gu, u, A.11 where Gu, u is given by.0. Since H u 0 for u > u,conditions.0.1 ensure that the RHS in A.11 is 0 for u T k. This shows that H1 BKLP u, u is a non-increasing function on each of the intervals T k constituting V 1 a, b] if V 1 a, b] consists of a finite number of points, then H1 BKLP H BKLP at these points. Since H1 BKLP u, u = H BKLP u, u on J, all the above arguments and A.6 prove that H BKLP is non-increasing on u,. This, together with the conclusion in Cases 1 and and the continuity of Hi BKLP u, u, i = 1,, gives that H BKLP is non-increasing on a, b]. Similarly, one proves that H BKLP u, u when u is fixed is a non-decreasing function of the second argument u. Here, in the case corresponding to A.11 in Case 3b, we have d H BKLP u a uh u, u = du a u a u 3 Gu, u, u < u < u.

26 138 S. BRYSON ET AL. Since H u 0 for u < u, conditions.0.1 guarantee that the derivative is non-negative, and hence H BKLP u, u is a non-decreasing function of u on the finite union of closed intervals S u, u a, b].