Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations

Transcription

1 Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations Steve Bryson, Aleander Kurganov, Doron Levy and Guergana Petrova Abstract 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, SIAM J. Sci. Comput., 3 001, pp ]. We provide the details of the new family of methods in one, two, and three space dimensions, and then verify their epected low-dissipative property in a variety of eamples. AMS subect classification: Primary 65M06; Secondary 35L65 Key Words: Hamilton-Jacobi equations, central-upwind schemes, semi-discrete methods. 1 Introduction We consider the multidimensional Hamilton-Jacobi equation, ϕ t H ϕ = 0, 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 and Lions in 6] and by Souganidis in 1]. Recent attempts to obtain higher-order approimate solutions of 1.1 include upwind methods 9, 19, 0], discontinuous Galerkin methods 8], and others. Here, we study a class of proection-evolution methods, called Godunov-type schemes. The main 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 net 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 and Tadmor in 16, 17]. L 1 -convergence results for these schemes were obtained in 16]. More efficient non-staggered central schemes as well as genuinely multidimensional generalizations of the schemes in 17] were presented in ], with high-order etensions up to fifth-order proposed in 3, 4]. Program in Scientific Computing/Computational Mathematics, Stanford University and the NASA Advanced Supercomputing Division, NASA Ames Research Center, Moffett Field, CA ; bryson@nas.nasa.gov Department of Mathematics, Tulane University, 683 St. Charles Avenue, New Orleans, LA 70115; kurganov@math.tulane.edu, tel: , fa: Department of Mathematics, Stanford University, Stanford, CA ; dlevy@math.stanford.edu, tel: , fa: Department of Mathematics, Teas A&M University, College Station, TX ; gpetrova@math.tamu.edu, tel: , fa:

2 S. Bryson, A. Kurganov, D. Levy & G. Petrova Second-order semi-discrete Godunov-type central schemes were introduced in 14], where local speeds of propagation were employed to reduce the numerical dissipation. The numerical viscosity was further reduced in the central-upwind schemes 1] by utilizing one-sided estimates of the local speeds of propagation. Higher-order etensions of these schemes were introduced in 5], where weighted essentially non-oscillatory WENO interpolants were used to increase accuracy. WENO interpolants were originally developed for numerical methods for hyperbolic conservation laws 18, 10], and were first implemented in the contet of upwind schemes for Hamilton-Jacobi equations in 9]. Godunov-type central-upwind schemes are constructed in two steps. First, the solution is evolved to the net time level on a nonuniform 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 nonuniform 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 1]. A more careful selection of the reconstruction results in a new central-upwind scheme with smaller numerical dissipation. This approach was originally proposed in 13], where it was applied to one-dimensional 1-D systems of hyperbolic conservation laws. It has been recently generalized and implemented for multidimensional systems of hyperbolic conservation laws in 11]. The paper is organized as follows. In, we develop new semi-discrete central-upwind schemes for 1-D 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 dimensions with special emphasis on the two-dimensional setup are then presented in 3, where the corresponding multidimensional interpolants are also discussed. In 4, we evaluate the performance of the new schemes with a series of numerical tests. Finally, in the Appendi, 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 1-D Hamilton-Jacobi equation, ϕ t H ϕ = 0, R,.1 subect to the initial data ϕ, t = 0 = ϕ 0. We follow the approach in 1] see also 14]. For simplicity we assume a uniform grid in space and time with grid spacing and t, respectively. The grid points are denoted by :=, t n := n t, and the approimate value of ϕ, t n is denoted by ϕ n Ȧssume that the approimate solution at time t n, ϕ n, is given, and that a continuous piecewisepolynomial interpolant ϕ, t n is reconstructed from ϕ n. At every grid point, the maimal right and left speeds of propagation, a and a, are then estimated by a = ma H u, 0, a minϕ,ϕ u maϕ,ϕ = min H u, 0,. minϕ,ϕ u maϕ,ϕ where ϕ ± are the one-sided derivatives at =, that is ϕ ± := ϕ ± 0, t n.

3 Schemes with reduced dissipation for HJ equations 3 ϕ n ϕ n ϕ n 1 n n n n 1/ Figure.1: Central-upwind differencing: 1-D Obviously, the quantities a ± also depend on time, and ϕ ± depend on both time and location. These dependences are omitted to simplify the notation. If the Hamiltonian is conve,. reduces to a = ma H ϕ, H ϕ, 0, a = min H ϕ, H ϕ, 0,.3 while in the non-conve case, one has to use directly the epressions in.. We then proceed by evolving the reconstruction ϕ at the evolution points n ± := ± a ± t, to the net time level according to.1. The time step t is chosen so that n < n 1 for all. Therefore the solution remains smooth at n ± for t tn, t n1 ] see Figure.1 and we can compute the values of the evolved solution ± by the Taylor epansion in time: ± = ϕn ±, t n th ϕ n ±, t n O t..4 Using the values ± on the nonuniform grid n ±, we construct a new quadratic interpolant ψ. On the interval n, n ], the interpolant takes the form ψ, t n1 := ϕn1 ϕn1 n 1 n n ϕ n1 n n,.5 where ϕ n1 is yet to be determined and is an approimation to ϕ n, tn1, n = n n /. The proection back onto the original grid is then carried out by evaluating ψ, t n1 at, := ψ, t n1 = a a a ϕ n1 a a a 1 ϕ n1 a a t..6 Note that if the Riemann fan is symmetric, that is, if a = a, then n =. Substituting.4 in.6 yields = a a a a a a Using the Taylor epansion in space, ϕ n, t n th ϕ n, t n ϕ n, tn th ϕ n, tn 1 ϕ n1 a a t O t..7 ϕ n ±, t n = ϕ n ± ta ± ϕ± O t,.8

4 4 S. Bryson, A. Kurganov, D. Levy & G. Petrova we arrive at = ϕ n t a a a a ϕ ϕ t a a ] a H ϕ n, tn a H ϕ n, tn 1 ϕ 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 a Hϕ ϕ a ϕ a 1 ] lim t ϕ n1..10 a a a Here, the one-sided speeds of propagation, a ±, are given by., and ϕ± are the left and right derivatives at the point = of the reconstruction ϕ, t at time t. Finally, in order to complete the construction of the scheme, we must determine ϕ n1. For eample, selecting ϕ n1 to be independent of t gives ] t ϕ n1 = 0, lim and then.10 recovers the central-upwind scheme in 1]. However, since the interpolant ψ, t n1 is defined on the intervals n, n ], whose size is proportional to t, it is natural to choose ϕ n1 to be proportional to 1/ t. In this case, the approimation 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 non-oscillatory reconstruction, we should use a nonlinear limiter. For eample, one can use the minmod limiter: t ϕ n1 ϕ n = minmod, tn1 ψ n, tn1 ψ n,, tn1 ϕ n, tn1,.11 a a where ψ is the derivative of.5, ϕ 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 min, if > 0, minmod 1,,... := ma, if < 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 n ±, t, tn t < t n1, we can use a Taylor epansion in time to obtain a ϕ n ±, t n1 = ϕ n ±, t n O t..13 According to.5, the derivative ψ n, tn1 of the new reconstruction ψ at time level t n1 is ψ n, tn1 = ϕn1 ϕn1 a.14 a t, and after substituting.4 and.8 into.14, we obtain ψ n, t n1 = a ϕ a ϕ a a H ϕ n, tn H ϕ n, tn a a O t..15 a a

5 Schemes with reduced dissipation for HJ equations 5 Passing to the semi-discrete limit t 0 in.11,.13, and.15 gives where ] lim t ϕ n1 = ψ int a a minmod ϕ ψ int, ψ int ϕ,.16 := lim ψ n, t n1 ] = a ϕ a ϕ a a Hϕ Hϕ a a..17 Finally, substituting.16 into.10, we obtain the 1-D low-dissipative semi-discrete central-upwind scheme: d dt ϕ t = a Hϕ a Hϕ a a ϕ a a ϕ ϕ a minmod ψ int a a, ψint ϕ ] a a,.18 a where ψ int is given by.17. For future reference, we denote the RHS of.18 by H BKLP. 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 1] increases the accuracy of the resulting fully-discrete scheme: O t r versus O t 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 1] is O r, and 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 ψ int is always in the interval minϕ, ϕ, maϕ, ϕ ], and therefore the absolute value of the term ϕ ϕ in the numerical dissipation in the scheme from 1] is always greater than the absolute value of the new term, that is ϕ ϕ ϕ ϕ minmod ϕ ψint, ψint ϕ. Indeed, we have ψ int = a ϕ a ϕ a a Hϕ Hϕ a a = ϕ ] a H ξ ϕ a a H ξ a a a ],.19 where ξ minϕ, ϕ, maϕ, ϕ. It follows from the definition of the local speeds. that a H ξ 0, H ξ a 0. Thus,.19 is a conve combination of ϕ and ϕ, and therefore ψint minϕ, ϕ, maϕ, ϕ ] Remarks. i We would like to emphasize that the reduced dissipation in the scheme.18 when compared with the scheme of 1] is due to the minmod term on the RHS of.18. This ] additional term arises when we define ˆϕ n1 by.11 such that the lim t ϕ n1 in.10 does not vanish.

6 6 S. Bryson, A. Kurganov, D. Levy & G. Petrova ii 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 flu evaluations and hence the increase in the computational compleity is minimal. Such additional work when compared with the original scheme 1] 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 5] that the numerical Hamiltonian H KNP from 1] is monotone, provided that the Hamiltonian H is conve. Here, we state a theorem about the monotonicity of H BKLP the new, less dissipative Hamiltonian in.18. The proof is left to the Appendi. We will consider only Hamiltonians for which H changes sign, because otherwise either a 0 or a 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 conve and satisfy the following two assumptions: 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 and u is the only point such that H u = 0; H u H v H u H v, u u v, u u v,,.1 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: where H BKLP u, u := a Hu a Hu a a u int := a u a u a a a a u u a a minmod Hu Hu a a, u u int a a, uint u a a ],. and a := a u, u = mah 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. Remarks. i Eamples of Hamiltonians that satisfy conditions.0.1 are any conve quadratic Hamiltonians Hu = au bu c. Straightforward computation gives that the function Gu, v 0, the sets in A are either empty or one closed interval, or one point, and therefore the theorem holds. Another eample, 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,

7 Schemes with reduced dissipation for HJ equations 7 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. As for the sets in assumption A, they are either empty or one closed interval, or one point. ii 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. iii We would like to remind the reader that the monotonicity of the numerical Hamiltonian is an essential ingredient in the theory of Barles and Souganidis 1]. The main theorem in 1] implies that a consistent, stable and monotone approimation of a Hamilton-Jacobi equation that satisfies an underlying comparison principle, converges to the unique viscosity solution of that equation. In our contet, such an approimation can be obtained if we assume a piecewise-linear reconstruction and replace the time derivative by a forward Euler approimation.. 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, t n in.17 and.18, are given by ϕ n ϕ ± ± = 1 ϕ n 1, ϕ n := ϕ n 1 1 ϕ n,.3 where the second derivative is computed with a nonlinear limiter. For eample, ϕ n ϕ n ϕ n ϕ n ϕ n ϕ n ϕ n = minmod θ 3 1 1, 3 1, θ Here, θ 1, ] and the minmod function is given by.1. As it is well-known, larger values of θ correspond to less dissipative limiters see ]. 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 weighted essentially non-oscillatory WENO reconstructions. They were derived in 5] in the contet of central-upwind schemes, and are similar to those used in high-order upwind schemes 9]. In smooth regions, the WENO reconstructions use a conve combination of multiple overlapping reconstructions to attain high order accuracy. In nonsmooth regions, a smoothness measure is employed to increase the weight of the least oscillatory reconstruction. Here, we reconstruct the one-sided derivatives ϕ ± k, at = for k = 1,..., d stencils, and write the conve combination: ϕ ± = d w ± k, ϕ± k,, k=1 where the values ϕ ± d w ± k, = 1, w± k, 0,.5 k=1 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,

8 8 S. Bryson, A. Kurganov, D. Levy & G. Petrova The constants c ± k are set so that the conve combination in.5 is of the maimal 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 1, = ϕ 1 ϕ 1 ϕ, = 3ϕ 4ϕ 1 ϕ, ϕ 1, = ϕ 4ϕ 1 3ϕ,, ϕ, = ϕ 1 ϕ 1. The constants c ± k are given by c 1 = c = 3, c = c 1 = 1 3, and the smoothness measures are Here, S r, s] := S 1, = S 1, 0], S, = S 0, 1], S 1, = S, 1], S, = S 1, 0]. s i=r ϕ i s i=r1 ϕ i, ± ϕ := ±ϕ ±1 ϕ..7 A fifth-order WENO reconstruction is obtained when d = 3. In this case, ϕ 1, = ϕ 6ϕ 1 3ϕ ϕ 1 6 ϕ, = ϕ 1 3ϕ 6ϕ 1 ϕ 6 ϕ 3, = 11ϕ 18ϕ 1 9ϕ ϕ 3 6, ϕ 1, = ϕ 3 9ϕ 18ϕ 1 11ϕ, 6, ϕ, = ϕ 6ϕ 1 3ϕ ϕ 1, 6, ϕ 3, = ϕ 1 3ϕ 6ϕ 1 ϕ. 6 The constants c ± k are given by and the smoothness measures are 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 eamples, we use the strong stability preserving SSP Runge-Kutta methods from 7]. 3 Multidimensional Schemes In this section, we derive the two-dimensional -D generalization of the semi-discrete central-upwind scheme.17.18, and then etend it to three space dimensions. We also comment on the multidimensional interpolants that these etensions require.

9 Schemes with reduced dissipation for HJ equations 9, k1 -, k NW NE, k -1, k 1, k, k -, k- SW SE, k-, k A Two-Dimensional Scheme We consider the -D Hamilton-Jacobi equation, ϕ t Hϕ, ϕ y = 0, Figure 3.1: Central-upwind differencing: -D and proceed as in 1]. We assume that at time t = t n the approimate point values ϕ n k ϕ, y k, t n are given, and construct a -D continuous piecewise-quadratic interpolant, ϕ, y, t n, defined on the cells S k :=, y : yy k y 1. On each cell S k there will be four such interpolants labeled NW, NE, SE, and SW, one for each triangle that constitutes S k see Figure 3.1. Specific eamples of ϕ, y, t n are discussed in 3.3. Similarly to the 1-D case, we use the maimal values of the one-sided local speeds of propagation in the - and y-directions to estimate the widths of the local Riemann fans. These values at any grid point, y k can be computed as a k := ma H u ϕ, y, t, ϕ y, y, t, C k a k := min H u ϕ, y, t, ϕ y, y, t C k, b k where C k := 1 := ma H v ϕ, y, t, ϕ y, y, t, C k b k :=, 1 ] y k 1 min C k H v ϕ, y, t, ϕ y, y, t, , y k 1 ], := ma, 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 0] for details. The reconstruction ϕ, 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 n ±, yn k± where n ± := ± a ± k t, yn k± := y k ± b ± k t, see Figure 3.1. We denote by ϕ n ±,k± := ϕn ±, yn k±, tn, and use the Taylor epansion to calculate the intermediate values at the net time level t = t n1 ±,k± = ϕn ±,k± t H ϕ n ±, y n k±, tn, ϕ y n ±, y n k±, tn O t. 3.3 We now proect the intermediate values ±,k± onto the original grid points, y k. First, similarly to.5, we use new 1-D quadratic interpolants in the y-direction, ψ n ±,, t n1, to obtain ψ n ±, y k, t n1 = ±,k ϕn1 ±,k ϕn1 ±,k b k b k b k 1 ϕ yy n1 ±,k b k b k t

10 10 S. Bryson, A. Kurganov, D. Levy & G. Petrova = b k b k b k ±,k b k b k b k ±,k 1 ϕ yy n1 ±,k b k b k t, 3.4 where ϕ yy n1 ±,k ϕ yy n ±, ŷn k, tn1 and ŷ n k := yn k yn k /. Net, we use the values ψ n ±, y k, t n1 to construct another 1-D quadratic interpolant ψ, y k, t n1, this time in the -direction, whose values at the original grid points are k := ψ, y k, t n1 3.5 = a k a k ψ n a k, y k, t n1 a k a k a k ψ n, y k, t n1 1 ϕ n1,k a k a k t. Here, ϕ n1,k ϕ n, y k, t n1 and n := n n /. We choose ϕ n1,k average to be the weighted ϕ n1,k = b k b k b k ϕ n1,k b k b k b k ϕ n1,k, ϕ n1,k± ϕ n, yn k±, tn Notice that both ϕ yy n1 ±,k in 3.4 and ϕ n1,k± in 3.6 are yet to be determined. We then substitute 3.4 and 3.6 into 3.5, and obtain 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 t b k b b k ϕ n1,k b k ϕ 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 3.7 Substituting 3.3 into 3.7 yields k = a k b k a k a k b k b k ϕ n,k t H ϕ n, yn k, tn, ϕ y n, yn k, tn a k b k a k a k b k b k ϕ n,k t H ϕ n, yn k, tn, ϕ y n, yn k, tn a k b k a k a k b k b k ϕ n,k t H ϕ n, yn k, tn, ϕ y n, yn k, tn a k b k a k a k b k b k ϕ n,k t H ϕ n, yn k, tn, ϕ y n, yn k, tn a k a k b k b k b k b k a k a k ] t b k ϕ n1,k b k ϕ n1,k ] t a k ϕ yy n1,k a k ϕ yy n1,k O t. 3.8 The values ϕ n ±,k± are computed by the Taylor epansions: ϕ n ±,k± = ϕn,k ± ta± k ϕ± ± tb ± k ϕ± y O t, 3.9

11 Schemes with reduced dissipation for HJ equations 11 where ϕ ± := ϕ ±0, y k, t n and ϕ ± y := ϕ y, y k ±0, t n are the corresponding right and left derivatives of the continuous piecewise quadratic reconstruction at, y k. Net, substituting 3.9 into 3.8 gives k = ϕ n k t a k a k a k ϕ a ϕ t b k b k k b k ϕ b y ϕ y t k a k a k b k b k a k b k H ϕ n, yk n, tn, ϕ y n, yk n, tn a k b k H ϕ n, yk n, tn, ϕ y n, yk n, tn ] a k b k H ϕ n, yk n, tn, ϕ y n, yk n, tn a k b k H ϕ n, yk n, tn, ϕ y n, yk n, tn a k a k b k b k b k b k a k a k ] b k ϕ n1,k b k ϕ n1 t,k ] t a k ϕ yy n1,k a k ϕ yy n1,k O t Finally, the limit t 0 generates a family of -D semi-discrete central-upwind schemes: d dt ϕ kt = a k b k Hϕ, ϕ y a k b k Hϕ, ϕ y a k b k Hϕ, ϕ y a k b k Hϕ, ϕ y a k a k b k b k a k a k a k a k a k a k b k b k b k b k a k a k ϕ ϕ b k b k b k b k b k a k lim lim t ϕ n1,k ϕ y ϕ y t ϕ yy n1,k b k a k lim t ϕ n1,k ],k] lim t ϕ yy n We still need to specify ϕ n1,k± and ϕ yy n1 ±,k. If they are proportional to ϕ n1 / and to ϕ y n1 / y respectively, then lim t ϕ n1,k± = 0, lim t ϕ yy n1 ±,k = 0, and we obtain the original -D central-upwind scheme from 1]. However, similarly to the 1-D case, we can choose ϕ n1,k± and ϕ yy n1 ±,k to be proportional to 1/ t, so that the above limit will not vanish. For eample, one can use the minmod limiter: ϕ t ϕ n1,k± = minmod n1,k± φ n, yn k±, tn1 φ n a k,, yn k±, tn1 ϕ n1,k± a k a k, 3.1 a k ϕy t ϕ yy n1 ±,k = minmod n1 ±,k ψ y ±, ŷ k, t n1 b k, b k ψ y ±, ŷ k, t n1 ϕ y n1 ±,k b k, 3.13 b k where ϕ n1 ±,k± := ϕ n ±, yn k±, tn1 and ϕ y n1 ±,k± := ϕ y n ±, yn k±, tn1. The values of the derivative φ in 3.1 are given by φ n, yk± n, tn1 = ϕn1,k± ϕn1,k± a k, 3.14 a k t

12 1 S. Bryson, A. Kurganov, D. Levy & G. Petrova and after using 3.3 and 3.9, we obtain φ n, y n k±, tn1 = H ϕ n, yn k±, tn, ϕ y n, yn k±, tn H ϕ n, yn k±, tn, ϕ y n, yn k±, tn a k a k a k ϕ a k ϕ a k a k O t Since the data is smooth along the line segments n ±, yn k±, t, tn t < t n1, it is clear that lim ϕ n1,k± = ϕ, lim ϕ n1,k± = ϕ, Therefore using 3.1, 3.16, and 3.15, we obtain where lim t ϕ n1,k± ϕ int± = a k a k minmod lim ϕ y n1 ±,k = ϕ y, ϕ ϕ int±, ϕ int± ϕ lim ϕ y n1 ±,k = ϕ y. 3.16, 3.17 := a k ϕ a k ϕ a k a k Hϕ, ϕ ± y Hϕ, ϕ ± y a k a k Likewise, using 3.13, we obtain where lim t ϕ yy n1 ±,k ϕ int± y = b k b k minmod ϕ y ϕint± y, ϕ int± y ϕ y, 3.19 := b k ϕ y b k ϕ y b k b k Hϕ±, ϕ y Hϕ±, 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, ϕ y a k b k Hϕ, ϕ y a k b k Hϕ, ϕ y a k b k Hϕ, ϕ y a k a k b k b k a ϕ k a ϕ b k ϕ k a k a k b k minmod ϕ int b k a k, ϕint ϕ a k a k a k b k ϕ b k minmod ϕ int b k a k, ϕint ϕ ] a k a k a k ϕ b y ϕ k b y a k ϕ y ϕ int y k b k b k a k minmod a k b k, ϕint y ϕ y b k b k b k a k ϕ y ϕy int a k minmod a k b k, ϕint y ϕ ] y b k b k. 3.1 b k Here, ϕ 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 formulas for ϕ± and ϕ ± y are discussed in 3.3 below.

13 Schemes with reduced dissipation for HJ equations 13 Remark. In practice, for conve Hamiltonians H the one-sided local speeds are computed as a k = ma H ϕ ± ±, ϕ ± y, 0, a k = min H ϕ ± ±, ϕ ± y, 0, 3. b k = ma Hy ϕ ± ±, ϕ ± y, 0, b k = min Hy ϕ ± ±, ϕ ± y, 0, where the maimum and minimum are taken over all the possible permutations of ±. 3. A Three-Dimensional Scheme We consider the three-dimensional 3-D Hamilton-Jacobi equation, ϕ t Hϕ, ϕ y, ϕ z = 0. We use the maimal values of the one-sided local speeds of propagation, a ± kl, b± kl, and c± kl, in the -, y- and z-directions, respectively. These values at any grid point, y k, z l are given by the obvious generalizations of 3. and c kl := ma H w ϕ, y, z, t, ϕ y, y, z, t, ϕ z, y, z, t, C kl c kl := min H w ϕ, y, z, t, ϕ y, y, z, t, ϕ z, y, z, t C kl, where C kl := 1, 1 ] y k 1, y k 1 ], z l 1, z l 1 ]. Proceeding as in two dimensions, the 3-D semi-discrete central-upwind scheme is suppressing the indices, k, l dϕ dt = 1 a ± a a b b c c b ± c ± Hϕ, ϕ y, ϕ z ] ± a a ϕ a a ϕ b b ϕ b b y ϕ c c y ϕ c c z ϕ z 1 a a b b c c a a ] b ± c ± D ϕn1,k,l ± b b ] a ± c ± Dyϕ n1 c c a ± b ± D,k,l zϕ,k,l] n1, 3.3 ± ± where the summations are taken over all possible permutations of and. For eample, in the first sum a b c should be multiplied by Hϕ, ϕ y, ϕ z. In 3.3, we use the notation D ϕn1,k±,l± := minmod ϕ ϕint,k±,l±, ϕint,k±,l± ϕ, Dy ϕn1 ±,k,l± := minmod ϕ y ϕint y ±,k,l±, ϕ int y ±,k,l± ϕ y, where D zϕ n1 ±,k±,l := minmod ϕ int,k±,l± := a ϕ a ϕ a a ϕ int y ±,k,l± := b ϕ y b ϕ y b b ϕ int z ±,k±,l := c ϕ z c ϕ z c c ϕ z ϕ int z ±,k±,l, ϕ int z ±,k±,l ϕ z, Hϕ, ϕ ± y, ϕ ± z Hϕ, ϕ ± y, ϕ ± z a a, Hϕ±, ϕ y, ϕ± z Hϕ±, ϕ y, ϕ± z b b, Hϕ±, ϕ ± y, ϕ z Hϕ ±, ϕ ± y, ϕ z c c.

14 14 S. Bryson, A. Kurganov, D. Levy & G. Petrova 3.3 Multidimensional Interpolants The schemes developed in require a multidimensional non-oscillatory reconstruction. The simplest option is to use straightforward multidimensional etensions of the 1-D interpolants from..3, obtained via a dimension-by-dimension approach. For eample, a -D non-oscillatory second-order central-upwind scheme is given by 3.1 with ϕ n ϕ ± ± = 1,k ϕ n ϕ n 1,k, ϕ±,k± y = 1 y ϕ ϕ θ n n 1,k = minmod 3,k ϕn 1,k, ϕ yy n,k 1 = minmod θ ϕ n,k 3 ϕ n,k 1, y ϕ yy n,k 1, ϕ n 3,k ϕn 1,k ϕ n, θ 1,k ϕn 1,k, ϕ n,k 3 ϕ n,k 1 ϕ n ϕ n,k, θ 1,k 1 where θ 1, ], and the minmod function is given by.1. Similarly, the corresponding dimensionby-dimension -D etensions of the WENO interpolants from.3 can be used to reconstruct the derivatives in 3.18, 3.0, and 3.1. For more details see 5]. 4 Numerical Eamples In this section, we test the performance of the new semi-discrete central-upwind schemes on a variety of numerical eamples. We compare the methods developed in this paper, labeled BKLP, with the second-order scheme from 1] and the fifth-order scheme from 5], 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 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 eamine 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 7]. Assuming an ODE of the form d dtϕ = H ϕ and initial data ϕ n, the fourth-order SSP-RK method is ϕ 1 = ϕ n 1 th ϕn, ϕ = ϕn th ϕn ϕ th ϕ 1, ϕ 3 = ϕn th ϕn ϕ th ϕ ϕ th ϕ, = 1 5 ϕn 1 10 th ϕn 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.,

15 Schemes with reduced dissipation for HJ equations conve eample, N=100 at t=.5/π closeup: conve eample, N=100 at t=.5/π φ 0.4 φ Figure 4.1: Problem 4.. Left: The solution. Right: A close-up of the solution near the singularity. : nd-order KNP, o : nd-order BKLP, : 5th-order KNP, : 5th-order BKLP. The eact solution is the dashed line. 4.1 One-Dimensional Problems A Conve Hamiltonian We first test the performance of our schemes for the Hamilton-Jacobi equation with a conve Hamiltonian: ϕ t 1 ϕ 1 = 0, 4. subect to the periodic initial data ϕ, 0 = cosπ. The change of variables u, t = ϕ, t 1 transforms the equation into the Burgers equation u t 1 u = 0, which can be easily solved via the 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 4.1, where the second- and fifth-order BKLP and KNP schemes are compared. There is 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 4.1 we show the relative L 1 - and L -errors. A Non-Conve Hamiltonian In this eample, we compute the solution of the 1-D Hamilton-Jacobi equation with a non-conve Hamiltonian: ϕ t cos ϕ 1 = 0, 4.3 subect to the periodic initial data ϕ, 0 = cos π. 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 Figure 4., with a close-up of the singularities in Figure 4.3. The convergence results after the singularity formation are given in Table 4.. In this eample, the local speeds of propagation were estimated by.. The results are similar to the conve case, though the improvement here is somewhat less dramatic.

16 φ 16 S. Bryson, A. Kurganov, D. Levy & G. Petrova Conve eample ϕ t 1 ϕ 1 = 0 nd-order after singularity T =.5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP th-order after singularity T =.5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP Table 4.1: Problem 4.. Relative L 1 - and L -errors for the KNP and BKLP schemes. 1.5 non-conve eample, N=100 to t=.0/π Figure 4.: Problem 4.3 the KNP and BKLP numerical solutions.

17 Schemes with reduced dissipation for HJ equations 17 closeup: nonconve eample, N=100 to t=.0/π 1.08 closeup: nonconve eample, N=100 to t=.0/π φ φ Figure 4.3: Problem 4.3. Right: The singularity near = 0.5. Left: The singularity near = : nd-order KNP, o : nd-order BKLP, : 5th-order KNP, : 5th-order BKLP. The eact solution is the dashed line. Non-conve eample ϕ t cos ϕ 1 = 0 nd-order after singularity T =.0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP th-order after singularity T =.0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP Table 4.: Problem 4.3. Relative L 1 - and L -errors for the KNP and BKLP schemes.

18 18 S. Bryson, A. Kurganov, D. Levy & G. Petrova conve H, T=.5/π nonconve H, T=.0/π N N relative L 1 error relative L 1 error number of points number of points Figure 4.4: Convergence of the 1-D eamples. Left: Problem 4., T =.5/π. Right: Problem 4.3, T =.0/π. : 5th-order BKLP, : 5th-order KNP, : the 5th-order method from 9], The solid lines show eample rates of convergence. Net, we eamine 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 9] and 4], are shown in Figure 4.4. The reader may note that the convergence rates in these eamples are erratic. However, this investigation of the relative L 1 -errors for many different grid spacings shows that the behavior 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. 4. Two-Dimensional Problems In this section, we test the -D BKLP schemes on Hamilton-Jacobi equations with conve and nonconve Hamiltonians. We start with the conve problem compare with 4.: ϕ t 1 ϕ ϕ y 1 = 0, 4.4 which can be reduced to a 1-D problem via the coordinate transformation ξ 1/ 1/ =. η 1/ 1/ y The relative L 1 - and L -errors for the periodic initial data ϕ, y, 0 = cos π y/ = cos πξ after the singularity formation at T =.5/π are shown in Table 4.3. The results show that 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 1]. In Table 4.4, we present similar results for the non-conve problem compare with 4.3: ϕ t cos ϕ ϕ y 1 = 0, 4.5

19 Schemes with reduced dissipation for HJ equations 19 -D Conve eample nd-order after singularity T =.5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP th-order after singularity T =.5/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP Table 4.3: Problem 4.4. Relative L 1 - and L -errors for the -D KNP and BKLP schemes. with the periodic initial data ϕ, y, 0 = cos π y/. Similarly to the conve case, also with the non-conve 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 1]. -D non-conve eample nd-order after singularity T =.0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP th-order after singularity T =.0/π relative L 1 -error relative L -error N KNP BKLP KNP BKLP Table 4.4: Problem 4.5. Relative L 1 - and L -errors for the -D KNP and BKLP schemes.

20 0 S. Bryson, A. Kurganov, D. Levy & G. Petrova A Appendi: A Proof of Theorem.1 Proof. First, we fi u and show that H BKLP u, u given by. is a non-increasing function of u the proof that H BKLP is non-decreasing function of its second argument is similar. We denote by Hu Hu Qu := u u, u u, A.1 H u, u = u, and Au := 1 a u a u, where a u = mah u, H u, 0 and a u = minh u, H u, 0, and 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 = =1I and V = =1J. 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 BKLP u, u := where and H BKLP 1 u, u, u V 1, H BKLP u, u, u V, A.5 A.6 H1 BKLP u, u := a uhu a uhu a u a a ua u u a u a u u u 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 a ua u u a u a u u u a ua u a u a u u u a u Qu ]. A.8 Notice that for those u for which Qu = Au, we have 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

21 Schemes with reduced dissipation for HJ equations 1 use formulas A.5 and A.6 and continuity arguments to show that H BKLP, u is a non-increasing 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, mau, u, and mau, u, in case u = u, on the first and third interval only and continuous on IR. Case 1. Let u, minu, u. Then d du a u = 0 since a u = mah u, H u, 0 and H is a non-decreasing function of u. In this case we have d H BKLP u, u = a u a ua uu u a du a u a u 3 u Qu ] Note that there eists ξ u, u such that a u a u H u a u a u. 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, d and therefore the derivative du H BKLP u, u 0. Hence H BKLP u, u is non-increasing on, minu, u. Similarly, for u, minu, u we have d H BKLP 1 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 fi 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, mau, u. In this case the derivatives are 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, mau, u. Likewise d H BKLP 1 u, u = a ua ua u H u du a u a u a uh u a u a u. d du a u = 0 and The first term on the RHS is 0 since a u H u. As for 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, mau, u, and in particular on I minu, u, mau, u for every.

22 S. Bryson, A. Kurganov, D. Levy & G. Petrova Case 3a. Let u u u. Then a u 0 and therefore H BKLP H1 BKLP u, u H BKLP u, u Hu. In particular, 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 IR and non-increasing on each of the intervals, minu, u, minu, u, mau, u, and mau, 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. In this 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, we have that d 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 fi 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 on 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 H BKLP 1 u, u = du a uh u 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 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 u, u, i = 1,, give that H BKLP is non-increasing on a, b]. H BKLP i Similarly, one proves that H BKLP u, u when u is fied is a non-decreasing function of the second argument u. Here, in the case corresponding to A.11 in Case 3b above, we have d H BKLP u, u = du a uh u a u a u 3 Gu, u, u < u < u. 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]. Acknowledgment: The work of A. Kurganov was supported in part by the NSF Grants DMS and DMS The work of D. Levy was supported in part by the NSF under Career Grant DMS The work of G. Petrova was supported in part by the NSF Grant DMS

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