A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS HASEENA AHMED AND HAILIANG LIU Abstract. High resolution finite difference methods have been successfully employed in solving Hamilton Jacobi equations, and numerical Hamiltonians are as important as numerical flues for solving hyperbolic conservation laws using the finite volume methodology. Though different numerical Hamiltonians have been suggested in the literature, only some simple ones such as the La-Friedrichs Hamiltonian are widely used. In this paper, we identify si Hamiltonians and investigate the performance of Runge-Kutta ENO methods introduced by Shu and Osher in [SIAM J. Numer. Anal. (8)(4):97-9, 99] based on different numerical Hamiltonians, with the objective of obtaining better performance by selecting suitable numerical Hamiltonians. Etensive one dimensional numerical tests indicate that Hamiltonians with the upwinding property perform the best when factors such as accuracy and resolution of kinks are addressed. However, when the order of accuracy of the scheme and the number of grid points are increased, the errors for all Hamiltonians become nearly equal. Numerical tests are also performed for two dimensional problems. Contents. Introduction. Effective Numerical Hamiltonians 4 3. High-resolution discretization 8 4. Numerical Test 9 5. Conclusion 3 Acknowledgments 4 s 4. Introduction In this paper, we investigate the performance of the Runge-Kutta finite difference (RKFD) method with essentially non-oscillatory (ENO) approimation [, 4, 3,, 7, 8, 5, 6] based on different numerical Hamiltonians for solving the Hamilton Jacobi (HJ) equation (.) { t φ + H(Dφ) = in R n (, ), φ = φ in R n {t = }, with the objective of obtaining better performance by choosing suitable numerical Hamiltonians. The Hamilton-Jacobi equation arises in many applications such as geometrical optics, crystal growth, computer vision, and geometric motion. From method of characteristics

HASEENA AHMED AND HAILIANG LIU [7], we know that in general there can be no smooth solution of (.) lasting for all times t. These problems have been approached by Crandall and Lions in 983 [3] by introducing the notion of viscosity. Evans [7] gives the motivation to the formulation of viscosity solutions from maimum principle; eistence and uniqueness is proved in the paper [3]. The viscosity solutions of HJ equations are Lipschitz continuous but with discontinuous derivatives even if the Hamiltonian H and the initial condition φ () is C. We mention in passing that in some other applications such as high frequency wave propagation problems viscosity solutions are not adequate in describing the physical behavior beyond the singularity, and multi-valued solutions have to be considered, see e.g. [6, ] for recent accounts of numerical methods developed in computation of multi-valued solutions. We now illustrate roles of numerical Hamiltonians for D case. The numerical method for multi-d HJ equation can be constructed in a dimension by dimension fashion. A forward Euler in time discretization for D HJ equation gives φ n+ φ n + t Ĥn (φ, φ + ) =, where Ĥ(φ, φ + ) is a numerical approimation of H, and it is required to be consistent in the sense that Ĥ(φ, φ ) = H(φ ). The spatial derivatives such as φ and φ + are discretized simply with first order accurate one-sided differencing. The choice of monotone Hamiltonians leads to an important class of numerical methods for solving (.) which is the class of monotone schemes discussed by Crandall and Lions [4]. Here the monotonicity of the scheme means that φ n+ as defined by the scheme is a nondecreasing function of all the φ n. Crandall and Lions proved that monotone schemes are convergent to the viscosity solution, although they are only first order accurate. Following this framework, we shall construct numerical methods by discretizing the approimate equation of the form (.) t φ + Ĥ(φ, φ + ) =, where Ĥ is an approimation of H. Here φ = φ (, t) and φ + = φ ( +, t) denote the left and right limits of φ crossing the discontinuity. The approimate equation (.) is equivalent to the original equation for smooth solutions, and for solutions with discontinuous derivatives the approimate Hamiltonian Ĥ needs to be selected so that the right solution viscosity solution can be singled out. All terms in equation (.) are now well defined for functions of Lipschitz continuity in spatial variables, then the algorithm for solving it with the same initial data φ () becomes standard. We define point values φ(t) = {φ j (t) = φ( j, t)}, t >, and then construct a continuous piecewise polynomial of the form Evaluation of (.) at = j gives (.3) φ(, t n ) = P φ(t) (), [ j+ j, j+ ]. d dt φ j(t) = Ĥ((φ ) j, (φ + ) j ),

where PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 3 (φ + ) j = P φ(t) ( j+ j ), (φ ) j = P φ(t) ( j j ). We then solve the semi-discrete system (.3) by a high-order discretization in time. In this paper we follow [4] by using the ENO approimation for spatial reconstruction, and TVD Runge-Kutta [8, 9] to achieve higher order time discretization. Though other high resolution numerical methods can be employed for discretizing the approimate equation (.) as well. An important component for making the above algorithm to work is the numerical Hamiltonian, which corresponds to the numerical flu for solving hyperbolic conservation laws using the finite volume methodology. In past decades, various high resolution numerical methods have been proposed for solving Hamilton Jacobi equations, and essentially they differ in either the way of spatial reconstruction or the choice of numerical Hamiltonians. Some Hamiltonians are motivated from the numerical flu for conservation laws [, 5, 9], and others are based on derivation using staggered central approimation [, 7,,, 8]. We note that many eisting reconstruction procedures for high order finite difference schemes must involve wide stencils, thus the numerical result with refined meshes may not seem sensitive to the choice of Hamiltonians. However, for higher-order approimations constructed within each computational cell such as in the discontinuous Galerkin methods [4, 9], the numerical Hamiltonian becomes more important in its role communicating with immediate cell neighbors. The goal of this paper is to identity si popular Hamiltonians and compare the numerical performance of well-developed Runge-Kutta ENO methods based on different numerical Hamiltonians, with the objective of obtaining better performance by choosing a suitable numerical Hamiltonian. All the numerical Hamiltonians Ĥ presented are locally Lipschitz continuous, consistent with H and monotone. Another favorable feature is upwinding. The term upwinding refers to the fact that spatial differencing is performed using mesh points on the side from which information flows. The mathematical formulation for these terminologies is given in Section. Among si Hamiltonians presented within this work, the La-Friedrich s and Central Hamiltonian are not upwinding, but the Godunov, Engquist-Osher, Harten-La-van Leer and the Roe Hamiltonian are all upwinding. For high order spatial discretization, essentially non-oscillatory(eno) schemes (Harten and Osher [3], Harten, Engquist, Osher and Chakravarthy [], Shu and Osher [7, 8]) have been very successful in solving hyperbolic conservation laws. The key idea of ENO approimation is the adaptive stencil procedure which automatically obtains information from the locally smoothest region, and hence yields a uniformly high-order essentially non-oscillatory approimation for piecewise smooth functions. Weighted ENO (WENO) schemes were developed for conservation laws using conve combination of all candidate stencils instead of just one as in the original ENO [, 6]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed in [4, 5]. The plan of this paper is as follows: In Section we introduce a notion of upwind-type Hamiltonian, and give the description of si numerical Hamiltonians under consideration. In Section 3 we review the ENO reconstruction procedure and the Runge-Kutta time discretization, as adapted from the work by Shu and Osher [4]. In Section 4, we present etensive numerical eperiments in one-dimension case and some tests in two-dimensional case to compare the performance. Concluding remarks are given in Section 5.

4 HASEENA AHMED AND HAILIANG LIU. Effective Numerical Hamiltonians In order for the approimation equation (.) to be faithful to the original Hamilton Jacobi equation, the numerical Hamiltonian has to be carefully chosen. An ideal Hamiltonian should satisfy the following requirements. Definition: A Hamiltonian Ĥ is said to be an upwind-type Hamiltonian if (i)(consistency) Ĥ(u, u) = H(u) (ii)(monotonicity) Ĥ(u, u + ) is nondecreasing in u and non-increasing in u +. (ii)(upwind) Ĥ(u, u + ) = H(u + ) when H u < for all u between u + and u ; Ĥ(u, u + ) = H(u ) when all H u >. The monotonicity is required to ensure the entropy stability, but may lead to ecessive dissipation. The characteristics imposed by the upwind requirement is epected to help sharpen things up again. In this section we present different numerical Hamiltonians used in the comparison study for -D and -D case for the RKFD method. Etensive numerical eperiments to compare their performance for this method is given in Section 4. First, we review si numerical Hamiltonians under consideration for the -D case. Notation I(a, b) is used to denote the interval [min(u +, u ), ma(u +, u )].. La-Friedrich s () Hamiltonian and the local () Hamiltonian: The Hamiltonian is one of the simplest and most widely used numerical Hamiltonian for the ENO method. However, the numerical viscosity of the Hamiltonian is also the largest among monotone Hamiltonians for scalar problems. It is defined as ( ) ( ) u Ĥ (u +, u + + u u ) = H σ(u +, u + u ), where for the Hamiltonian, we obtain a bound for the maimal speed of propagation as σ = ma H (u), u [A,B] where [A, B] is the range of u. For a Hamiltonian we have from [4] that σ i = ma u I(u + i,u i ) H (u). For conve Hamiltonian, σ i is given as ma { H (u + i ), H (u i ) }. Clearly the Hamiltonian is not upwinding.. Central Hamiltonian: Recently, high resolution central type schemes having non-oscillatory property have been introduced for Hamilton-Jacobi equations by Lin and Tadmor [], followed by [7,,, 8]. The idea presented in [8] involves a two step process - evolution and projection. Given solution values φ i at the grid points i, we evolve in time the values of φ at the point n i± := i + σi n t, where σi n is the maimum speed of propagation. The evolution is given by a Taylor series epansion as follows: φ n+ i± = φ( i±, t n ) th( φ ( n i±, t n )),

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 5 (.) where φ is the reconstructed piecewise polynomial interpolant, and φ its derivative. The solution is then projected back to the original grid points i, i =,,,..., by using the average φ n+ i = (φn+ i+ + φn+ i ). The resulting scheme has a semi-discrete form d dt φ(t) = ( H(φ + (t)) + H(φ (t)) ) ( ) φ + + σ i (t) (t) φ (t). A k th order scheme is obtained by using a k th order polynomial interpolant for φ and k th order Taylor epansion in time. Using our notation for high order forward and backward difference approimations for the derivative φ as u + and u, we can write the Central Hamiltonian from equation (.) as Ĥ Central (u +, u ) = ( H(u + ) + H(u ) ) ( ) u σ(u +, u + u ). The maimum speed of propagation σ is calculated as in the case of the Hamiltonian. 3. Godunov s Hamiltonian: The Godunov Hamiltonian is obtained by attempting to solve the Riemann problem of the equation (.) eactly with piecewise linear initial condition determined by u ±. It can be given in its closed form as [4] Ĥ God (u +, u ) = min u u u + if u u +, ma u + u u if u > u +. The Godunov Hamiltonian is purely upwind and is the least dissipative among all monotone Hamiltonians [3], but could be very costly to evaluate in the -D case, as it often lacks eplicit formulas and relies on iterative procedures for its evaluation. 4. Engquist-Osher Hamiltonian: The Engquist-Osher flu was originally defined for hyperbolic conservation laws by using the flu splitting technique in [5], details of which could be found in [9]. We propose a numerical Hamiltonian of a similar nature as follows ( ) Ĥ EO (u +, u ) = H(u + ) + H(u ) u + u H (u) du The integral in the EO Hamiltonian can be calculated eactly most times for scalar Hamiltonians. 5. Harten-La-van Leer(HLL) Hamiltonian: The HLL flu was first introduced for hyperbolic conservation laws in [] by Harten, La and van Leer by using approimate Riemann solvers. In the contet of Hamilton Jacobi equations, the corresponding Hamiltonian can be written as Ĥ HLL (u +, u ) = a j H(u+ ) + a + j H(u ) a + j + a j a + j a+ j a j + a j. ( u + u ).

6 HASEENA AHMED AND HAILIANG LIU For every grid point, the maimal right and left speeds of propagation a + j are estimated by a + j = ma {H (u), }, u I(u +,u ) a j = min {H (u), }. u I(u +,u ) For conve Hamiltonians, this reduces to a + j = ma { H (u + ), H (u ), }, a j = min { H (u + ), H (u ), }. and a j Note that the above Hamiltonian was recently derived for Hamilton Jacobi equations by Bryson, Kurganov, Levy and Petrova in []. 6. Roe with Local La Friedrich fi Hamiltonian: Here upwinding idea is used in the construction of the Hamiltonian. The Roe Hamiltonian is defined in [4] as follows: H(u ) Ĥ Roe (u +, u ) = Ĥ if H (u) does not change sign in the region u I(u +, u ), otherwise, where u is defined from upwinding as, { u u + if H = (u), u if H (u). Ĥ Roe has almost as small dissipation as ĤGod. For the -D case, u + and u are used to denote the high order approimations to the derivative along the -direction, that is φ + and φ and v + and v are used to denote that along the y-direction φ + y and φ y. The numerical Hamiltonians for -D systems are each functions of u +, u, v + and v and can be given as follows:. La-Friedrich s () Hamiltonian and the local () Hamiltonian: It is defined by [4] ( ) ( ) u Ĥ + + u = H, v+ + v u σ (u +, u + u ) ( ) v σ y (v +, v + v ), where for the Hamiltonian, we have the maimum speeds of propagation in the and y direction given by σ = ma u [A,B] v [C,D] H u (u, v), σ y = ma H v (u, v), u [A,B] v [C,D]

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 7 where [A, B] is the range of u and [C, D] is the range of v. For a Hamiltonian we have that σ = ma u I(u,u + ) v [C,D] H u (u, v), σ y = ma v I(v,v + ) u [A,B] H v (u, v).. Central Hamiltonian: High resolution central schemes having non-oscillatory property are derived for -D systems in the same way as the -D system [8]. We can write the Central Hamiltonian as [ H(u +, v + ) + H(u +, v ) + H(u, v + ) + H(u, v ) ] Ĥ Central = 4 ( ) u σ(u +, u, v +, v + u + v + v ), where the maimal local { speed of propagation σ j,k is given } by the maimal value over the square C j,k := (, y) [ j, j+ ] [y j, y j+ ] σj,k n = ma { H u (u, v), H v (u, v) } C j,k 3. Harten-La-van Leer(HLL) Hamiltonian: The HLL Hamiltonian for -D Hamilton Jacobi equations was derived by Bryson, Kurganov, Levy and Petrova in [] and is given by Ĥ HLL = a b H(u +, v + ) + a b + H(u +, v ) + a + b H(u, v + ) + a + b + H(u, v ) (a + + a )(b + + b ) a+ a ( u + u ) b+ b ( v + v ). a + + a b + + b For every grid point ( j, y k ), the maimal right and left speeds of propagation in the direction, a + jk and a jk and in the y direction, b+ jk and b jk are estimated by a + jk = ma C j,k {H u (u, v), }, a jk = min C j,k {H u (u, v), }, b + jk = ma C j,k {H v (u, v), }, b jk = min C j,k {H v (u, v), }, where the square C j,k is given as in the -D Central Hamiltonian. 4. Roe with Local La Friedrich fi Hamiltonian: It is defined in [4] as follows: Ĥ Roe = H(u, v ), ( if H u and) H v do( not change ) sign in u I(u, u + ), v I(v, v + ), u H + +u, v u σ + u, ( otherwise) if H v ( does not ) change sign in u [A, B], v I(v, v + ), H u, v+ +v v σ + v y, otherwise if H u does not change sign in u I(u, u + ), v [C, D], Ĥ (u +, u, v +, v ), otherwise,

8 HASEENA AHMED AND HAILIANG LIU where σ and σ y are given as in Hamiltonian, u and v are defined purely by upwinding as { u u + if H = u (u, v), u if H u (u, v), { v v + if H = v (u, v), v if H v (u, v). 3. High-resolution discretization In this section, we give a brief description of the RKFD method with ENO approimation for the one dimensional case. We use notations in [5]. We wish to solve the one-dimensional Hamilton Jacobi equation (3.) { φ t + H(φ ) = in [a, b] (, ), φ = φ in [a, b] {t = }. The computational domain [a, b] is divided into n cells a = < <... < n < n = b. We assume a uniform grid in space and time with grid spacing and t, respectively. The grid points are denoted by i := i, t n := n t, and the approimate value of φ( i, t n ) is denoted by φ n i. The numerical approimation to the viscosity ( i, t n ) of (3.) at the mesh point ( i, t n ) is denoted by φ n i. We use finite difference methods to calculate φ n i for which the derivatives occurring in (3.) are replaced by appropriate finite difference approimations. Since the solutions to Hamilton Jacobi equations have discontinuous derivatives φ, we consider that the left and right hand limits of the discontinuous derivative of the and φ + eist at the cell interface i and define a numerical Hamiltonian Ĥ as a function of these. If we use the notations φ + = φ i+ φ i and φ i = φ i φ i to denote the first order forward i and backward difference approimations to the right and left derivatives of φ() at the location i. A first order monotone finite difference scheme for (3.) can be given as (3.) φ n+ i = φ n i tĥ(φn, φ n ). + i i The ENO interpolation idea is used to obtain higher order approimations to the derivative φ ±. The key feature of the ENO algorithm is an adaptive stencil high order interpolation which tries to avoid shocks or high gradient regions whenever possible and hence works well with Hamilton-Jacobi equations. If u + i and u i denote high order approimations to the derivative φ + and φ, the finite difference approimation (3.3) φ n+ i = φ n i tĥ(u+ (t), u (t)), gives a higher order finite difference scheme in space. The key feature of ENO to avoid numerical oscillations in the solution of Hamilton-Jacobi equations is through the interpolation procedure to obtain u ± i. Given point values f( i ), i =, ±, ±,... of a (usually piecewise smooth) function f() at discrete nodes i, we associate an r th degree polynomial P f,r i+ interval [ i, i+ ], with the left-most point in the stencil as (r) k min by the algorithm from [4]:. P f, () = f[ i+ i ] + f[ i, i+ ]( i ), k () min = i; () with each, constructed inductively

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 9. If k (l) min f,l and P () are both defined, then let i+ a (l) = f[ (l) k,..., (l) min k +l], min b (l) = f[ (l) k,..., min k (l) min +l], and i. if a (l) b (l), then c (l) = b (l) and k (l) min = k(l) min ; otherwise c(l) = a (l) and k (l) min = k(l) min ; ii. P f,l i+ () = P f,l () + c (l) i+ k (l) min +l i=k (l) min ( i ). The approimations to the left and right derivatives of φ are then taken as u ± i = P φ,r ( i+ i ). Here f[...] denote Newton divided differences. For high order accuracy in time, a high order TVD Runge-Kutta time discretization [8, 9] is used. Time step restriction is taken as t ma u u H(u).3, where the maimum is taken over the relevant ranges of u. Here.3 is just a convenient number used. Also, since the higher order interpolation for derivatives can be etended dimension by dimension, it is easy to implement ENO approimation scheme for higher dimensions as well. 4. Numerical Test In this section, we perform etensive numerical eperiments to compare the performance of the ENO approimation method based on the si different Hamiltonians outlined in the previous section. The detailed numerical study is performed mainly for the one dimensional case addressing issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for -dimensional case. In all the figures, we plot the point values of the numerical solution at the grid points. For CPU time comparison, all the computations are performed on a Dell Digital personal workstation, Intel Pentium 4 CPU, GHz, GB RAM. In our numerical eperiments, the CFL number is taken as.3 for the first, second and third order accuracy. In all the three one-dimensional eamples, the reference eact solution is computed using a fifth order WENO [6, 6] and fourth order TVD Runge-Kutta [9, 8] using 4 cells. We denote the finite difference method with ENO interpolation and Hamiltonian X as ENO-X, such as ENO- for the RKFD scheme with Hamiltonian. In comparing the L and L errors we use notation ENO-X ENO-Y (or ENO-X ENO-Y ) to indicate that error from using Hamiltonian X is greater than (or equal to the order of 5 ) that while using Hamiltonian Y.

HASEENA AHMED AND HAILIANG LIU Eample 4.. We work with an eample in [4] which is a standard test problem for Hamilton-Jacobi equations. Consider, a -dimensional equation given by φ t + (φ + ) =,, φ(, ) = cos π. Here, the Hamiltonian is a conve function and we can use the simplified epressions given in the previous section for the computation of the maimal speed of propagation. We use periodic boundary conditions and compute the solution and test accuracy when it is smooth, that is, up to time T =.5. Since the Hamiltonian is widely used, π we use it as a reference to analyze the performance of other Hamiltonians. In Table, we provide a CPU time comparison for the RKFD scheme with different Hamiltonians. Note that this CPU time comparison depends on our specific implementation of these Hamiltonians and also on the specific test case. The numerical errors and the orders of accuracy for the numerical n i, and ratios of the numerical errors for comparison with the ENO- scheme are shown in Tables -4. All the schemes give the desired accuracy in space. The L error depends on the order of ENO approimation and the number of cells used. For first order scheme, ENO-Central gives the largest and ENO-God gives the smallest L and L errors among all the schemes. From Table, we can see that for small number of cells (N =, 4), the L error by ENO-God, ENO-EO, ENO-HLL and ENO-Roe are about 8 9% of the L error by ENO- and as the number of cells increase (N 8), the error by ENO-God, ENO- EO, ENO-HLL and ENO-Roe are about 94 98% of the L error by ENO-. For the second and third order schemes, we can see from Tables 3 and 4 that for small number of cells ENO-Central gives the largest and ENO-God gives the smallest L error, but as the number of cells is increased, the errors become very close to each other. The main difference comes into play in the computation time. The total CPU time for N =, 4, 8, 6, 3, 64 and 8 cells is recorded in Table. The ENO-Central and the ENO-EO Hamiltonians take about nearly the same computation time as of the ENO- scheme. ENO-God takes about 7 times the time taken by ENO- for the first order scheme, about 55 times the time taken by ENO- for the second order scheme and about 45 times the time taken by ENO- for the third order scheme. ENO-HLL takes about % more time than the ENO-, and ENO-Roe takes about 3 5 times the time taken by ENO- for st, nd and 3rd order schemes. Table : Total CPU time in seconds for the RKFD method with ENO approimation with different Hamiltonians. The sum of the CPU times for N =, 4, 8, 6, 3, 64 and 8 cells is recorded for Eample 4. for orders,, and 3. Order Central God EO HLL Roe.6.6 73..5. 4.7.64.6 4.9.9 3..8 3 4.6 4.34 4.83 4.8 5.59 5.94

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS.5 Central.5.5.9.8.7.6.5 God.5 EO.5.9.8.7.6.5.5 HLL.5.9.8.7.6.5.5 Roe.5.9.8.7.6.5.5.9.8.7.6.5 Figure : Resolution of discontinuities for eample 4., st order, at T =.5 π, 4 cells, zoomed at the region.5 which contains the discontinuous derivative..5 Central.5.5.9.8.7.6.5 God.5 EO.5.9.8.7.6.5.5 HLL.5.9.8.7.6.5.5 Roe.5.9.8.7.6.5.5.9.8.7.6.5 Figure : Resolution of discontinuities for eample 4., nd order, at T =.5 π, 4 cells, zoomed at the region.5 which contains the discontinuous derivative.

HASEENA AHMED AND HAILIANG LIU.5 Central.5.5.9.8.7.6.5 God.5 EO.5.9.8.7.6.5.5 HLL.5.9.8.7.6.5.5 Roe.5.9.8.7.6.5.5.9.8.7.6.5 Figure 3: Resolution of discontinuities for eample 4., 3rd order, at T =.5 π, 4 cells, zoomed at the region.5 which contains the discontinuous derivative..5 Central...3.4.5.5 God.5 EO...3.4.5...3.4.5.5 HLL.5 Roe...3.4.5...3.4.5 Figure 4: Resolution of discontinuities for st order, eample 4. for T =.5 π, 4 cells, zoomed at the region.5 which contains the discontinuous derivative in the solution.

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 3.5 Central...3.4.5.5 God.5 EO...3.4.5...3.4.5.5 HLL.5 Roe...3.4.5...3.4.5 Figure 5: Resolution of discontinuities for nd order, eample 4. for T =.5 π, 4 cells, zoomed at the region.5 which contains the discontinuous derivative in the solution..5 Central.5..5..5.3.35.4.45.5.5 God.5 EO...3.4.5...3.4.5.5 HLL.5 Roe...3.4.5...3.4.5 Figure 6: Resolution of discontinuities for 3rd order, eample 4. for T =.5 π, 4 cells, zoomed at the region.5 which contains the discontinuous derivative in the solution.

4 HASEENA AHMED AND HAILIANG LIU Table : The L and L errors for eample 4. using N equally spaced cells with different Hamiltonians at T =.5 for first order ENO approimation. π N Hamiltonian L error L order error ratio L error L order error ratio 5.849E-. 5.6645E-.835. Central 5.8766E-.4 6.45679E-.6.366 God 4.654E-.799 4.96337E-.937.86678 EO 4.66587E-.897 4.96337E-.937.86678 HLL 4.797784E-.8567 4.96337E-.937.86678 Roe 4.87469E-.8784 4.96337E-.937.86678 4.5736E-.866..83454E-.3373. Central.5858E-.76.3935 3.376E-.7.635 God.636E-.6.87987.6757E-.965.943776 EO.76E-.754.88347.6757E-.965.943776 HLL.9488E-.5.89535.6757E-.965.943776 Roe.9647E-.74.89986.6757E-.965.943776 8.9586E-.48..45688E-.9776. Central.9793E-.88.44.57655E-.7.3493 God.85E-.354.9353.45883E-.9449.96543 EO.99E-.4.935963.45883E-.9449.96543 HLL.537E-.487.93789.45883E-.9449.96543 Roe.36E-.488.9383.45883E-.9449.96543 6 5.74498E-3.674. 7.35336E-3.9953. Central 5.74695E-3.687.348 7.4744E-3.5.644 God 5.5556E-3.97.96654 7.345E-3.9676.983574 EO 5.5587E-3.8.96636 7.345E-3.9676.983574 HLL 5.5533E-3.3.96677 7.345E-3.9676.983574 Roe 5.555E-3.34.96799 7.345E-3.9676.983574 3.8335E-3.346. 3.6998E-3.998. Central.839E-3.35.9 3.7383E-3.96.8353 God.764664E-3..9885 3.6634E-3.986.99647 EO.764664E-3..9885 3.6634E-3.986.99647 HLL.76467E-3.8.9887 3.6634E-3.986.99647 Roe.7653E-3.9.9833 3.6634E-3.986.99647 For L error, when N 3: ENO-Central ENO- ENO-Roe ENO-HLL ENO-EO ENO-God. For L error, when N 3: ENO-Central ENO- ENO-Roe ENO-HLL ENO-EO ENO-God. We analyze the resolution of discontinuities for different Hamiltonians at time T =.5 π when the solution has a discontinuous derivative. In Figures -3, we show the resolution of discontinuities when the number of cells is 4 for the st, nd and 3rd order schemes respectively, for the different Hamiltonians in comparison with the reference eact solution, and against the ENO- scheme on the same mesh, zoomed at the region.5 which contains the discontinuous derivative in the solution. Solid lines

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 5 Table 3: The L and L errors for eample 4. using N equally spaced cells with different Hamiltonians at T =.5 for second order ENO approimation. π N Hamiltonian L error L order error ratio L error L order error ratio.9435e-..395e-.584. Central.388E-.9644.34684E-.6985. God.5567E-.97655.96735E-.3797.988734 EO.5593E-.97348.96735E-.3797.988734 HLL.6554E-.976844.96735E-.3797.988734 Roe.5794E-.97487.96735E-.3797.988734 4 3.47373E-3.66. 6.8783E-3.845. Central 3.5375E-3.7364.95 6.88844E-3.83748.47 God 3.487E-3.87.99354 6.85355E-3.875.99734 EO 3.59E-3.3.9936 6.85355E-3.875.99734 HLL 3.556E-3.378.99369 6.85355E-3.875.99734 Roe 3.55E-3.3469.99364 6.85355E-3.875.99734 8 8.34437E-4.54..7977E-3.97456. Central 8.3859E-4.74.45.79958E-3.97683.66 God 8.87E-4.74.998345.79587E-3.9736.999339 EO 8.87E-4.749.998345.79587E-3.9736.999339 HLL 8.87E-4.779.998345.79587E-3.9736.999339 Roe 8.87E-4.77.998345.79587E-3.9736.999339 6.7357E-4.7846. 4.4789E-4.899. Central.7388E-4.834. 4.47953E-4.8958.4 God.75E-4.69.999599 4.47847E-4.7449.999854 EO.75E-4.69.999599 4.47847E-4.7449.999854 HLL.75E-4.69.999599 4.47847E-4.7449.999854 Roe.75E-4.69.999599 4.47847E-4.7449.999854 3 5.666E-5.33..4883E-4.8353. Central 5.675E-5.45.8.495E-4.855.9 God 5.695E-5.89.9999.4845E-4.89.999965 EO 5.695E-5.89.9999.4845E-4.89.999965 HLL 5.695E-5.89.9999.4845E-4.89.999965 Roe 5.695E-5.89.9999.4845E-4.89.999965 For L error, for N 4: ENO-Central ENO- ENO-HLL ENO-Roe ENO-EO ENO-God, For L error, for 8 N 3: ENO-Central ENO- ENO-HLL ENO-Roe ENO-EO ENO-God. For L error, for N 3: ENO-Central ENO- ENO-HLL ENO-Roe ENO-EO ENO-God. indicate the reference eact solution, hollow circles indicate scheme with ENO-, and the plus symbols represent the scheme with respective schemes given in the graphs. For small number of cells N, and first, second and third order of accuracy, the resolution of the ENO-Central scheme is the worst among all the schemes. The resolution of the

6 HASEENA AHMED AND HAILIANG LIU Table 4: The L and L errors for eample 4. using N equally spaced cells with different Hamiltonians at T =.5 for third order ENO approimation. π N Hamiltonian L error L order error ratio L error L order error ratio.998844e-3. 3.85539E-3.84. Central.854E-3.4838 3.865339E-3.9774.67 God.98644E-3.993786 3.84454E-3.95.99777 EO.986447E-3.993798 3.84454E-3.95.99777 HLL.986756E-3.993953 3.84454E-3.95.99777 Roe.9888E-3.994589 3.84454E-3.95.99777 4.649E-4 3.4549. 5.865E-4.8569. Central.693E-4 3.4894.549 5.86887E-4.8968.3 God.68595E-4 3.3438.99974 5.858379E-4.8999.9997 EO.68595E-4 3.34335.99974 5.858379E-4.8999.9997 HLL.68595E-4 3.34568.99974 5.858379E-4.8999.9997 Roe.68595E-4 3.3555.99974 5.858379E-4.8999.9997 8 3.8535E-5 3.45347. 7.6553E-5.9935. Central 3.8749E-5 3.4657.65 7.6533E-5.997.36 God 3.847E-5 3.4449.9999 7.649783E-5.989939.999965 EO 3.847E-5 3.4449.9999 7.649783E-5.989939.999965 HLL 3.847E-5 3.4449.9999 7.649783E-5.989939.999965 Roe 3.847E-5 3.4449.9999 7.649783E-5.989939.999965 6 4.67356E-6 3.398..8E-5.945598. Central 4.67388E-6 3.39886.8.85E-5.945646.3 God 4.673E-6 3.3969.999989.78E-5.94555.999997 EO 4.673E-6 3.3969.999989.78E-5.94555.999997 HLL 4.673E-6 3.3969.999989.78E-5.94555.999997 Roe 4.673E-6 3.3969.999989.78E-5.94555.999997 3 5.85965E-7 3.38..35564E-6.955695. Central 5.8597E-7 3.38..35564E-6.955699. God 5.85959E-7 3.34.999999.35563E-6.95569. EO 5.85959E-7 3.34.999999.35563E-6.95569. HLL 5.85959E-7 3.34.999999.35563E-6.95569. Roe 5.85959E-7 3.34.999999.35563E-6.95569. For L error, for N = : ENO-Central ENO- ENO-Roe ENO-HLL ENO-EO ENO-God. For L error, for 4 N 3: ENO-Central ENO- ENO-Roe ENO-HLL ENO-EO ENO-God. For L error, for N 3: ENO-Central ENO- ENO-Roe ENO-HLL ENO-EO ENO-God. ENO-God and ENO-EO scheme are the best, followed by the ENO-Roe scheme, followed closely by that of the ENO- scheme, followed by the ENO-HLL scheme. As the number of cells is increased, for orders of accuracy, and 3, the resolution of the ENO- Central scheme continues to remain the worst among all the schemes and that of the

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 7..4.6.8 Central.5.5..4.6.8 God.5.5..4.6.8 EO.5.5..4.6.8 HLL.5.5..4.6.8 Roe.5.5 Figure 7: Resolution of discontinuities for st order, eample 4.3 for T =, 4 cells, on the whole domain. ENO-God and ENO-EO scheme the best. However, there is a change in the performance of the other schemes. ENO-God and ENO-EO schemes are followed by the ENO-HLL scheme which is followed by ENO-Roe scheme which is followed by ENO- scheme for the resolution of discontinuities. Eample 4.. We repeat the numerical eperiments of the previous eample for another standard test case from [4], a -dimensional HJ equation given by φ t cos(φ + ) =,, φ(, ) = cos π. The Hamiltonian is non-conve. Here again, as in the previous eample we use periodic boundary conditions and compute the solution and test accuracy when it is smooth, that is, at t =.5. π All the schemes give the desired accuracy in space. The numerical results for accuracy and resolution of discontinuities are very similar to that of the conve Hamiltonian (Eample 4.) case, however for the first order case, the efficiency of using particular

8 HASEENA AHMED AND HAILIANG LIU..4.6.8 Central.5.5..4.6.8 God.5.5..4.6.8 EO.5.5..4.6.8 HLL.5.5..4.6.8 Roe.5.5 Figure 8: Resolution of discontinuities for nd order, eample 4.3 for T =, 4 cells, on the whole domain. Hamiltonian is more pronounced than in the case of Eample 4.. For first order scheme, ENO-Central gives the largest and ENO-God gives the smallest L and L errors among all the schemes. For small number of cells (N =, 4), the L error by ENO-God, ENO-EO are about 7 8% of the L error by ENO- and the L error by ENO-HLL and ENO-Roe are about 8 85% of the L error by ENO-. As the number of cells increase (N 8), the error by ENO-God, ENO-EO, ENO-HLL and ENO-Roe are about 9 97% of the L error by ENO-. For the second and third order schemes, we can see from Tables 3 and 4 that for small number of cells ENO-Central gives the largest and ENO-God gives the smallest L error, but as the number of cells is increased, the errors become very close to each other as in the conve Hamiltonian case. We analyze the resolution of discontinuities for different Hamiltonians at time T =.5 π when the solution has two discontinuous derivatives. In Figures 4-6, we show the resolution of discontinuities when the number of cells is 4 for the first, second and third order schemes respectively, for the different Hamiltonians in comparison with the reference eact solution, and against the ENO- scheme on the same mesh, zoomed at the

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 9..4.6.8 Central.5.5..4.6.8 God.5.5..4.6.8 EO.5.5..4.6.8 HLL.5.5..4.6.8 Roe.5.5 Figure 9: Resolution of discontinuities for 3rd order, eample 4.3 for T =, 4 cells, on the whole domain. region.5 which contains one of the discontinuous derivatives. In regard to resolution of discontinuities all the Hamiltonians behave eactly the same as the case of Eample 4.. Eample 4.3. We work with an eample in [8], a -dimensional Riemann problem with non-conve Hamiltonian given by φ t + 4 (φ )(φ 4) =,, φ(, ) =. Since the initial condition is linear, we use linear etrapolation for the boundary points. We analyze the resolution of discontinuities for different Hamiltonians at time T = when the solution has discontinuous derivatives. In Figures 7, 8 and 9 we show the resolution of discontinuities when the number of cells is 4 for the first, second and third order schemes respectively, for the different Hamiltonians in comparison with the reference

HASEENA AHMED AND HAILIANG LIU eact solution, and against the ENO- scheme on the same mesh, over the domain which contains the discontinuous derivatives in the solution. From the figures, we can see that the approimate solution becomes closer to the reference solution when the number of cells is increased and the order of the approimation is increased. In the previous two eamples, ENO-Central was the worst in the resolution of discontinuities. However for the Riemann problem, ENO- scheme serves to be the worst for orders of accuracy, and 3. The resolution of the ENO-God and ENO- EO scheme are the best, followed by the ENO-HLL scheme. There is a change in the performance of the ENO-Roe and ENO-Central as the order is increased and as the number of cells is increased. For st order and small number of cells, the ENO-Roe scheme is superior to the ENO-Central scheme. However as the order is increased along with the number of cells, the ENO-Central scheme takes over and seems to be slightly better than the ENO-Roe at resolving discontinuities. We perform tests on one two-dimensional eample. Study is performed addressing issues of CPU cost and numerical accuracy. The reference eact solution is computed using a fifth order WENO [6, 6] and fourth order TVD Runge-Kutta [9, 8] using 8 8 cells. The CFL number is taken to be. for the first, second and third order accuracy. Eample 4.4. We work with an eample in [4], a -dimensional HJ equation given by φ t + (φ + φ y + ) =,, y, φ(, y, ) = cos π( + y ). For the two-dimensional eample we make comparison only between four Hamiltonians ENO-, ENO-Central, ENO-HLL and ENO-Roe. We use periodic boundary conditions and compute the solution and test accuracy when it is smooth, that is, up to time T =.. The reference eact solution is computed using a fifth order WENO [6, 6] and π fourth order TVD Runge-Kutta [8, 9] using 8 8 cells as in the one-dimensional case. In Table 5, we provide a CPU time comparison for the ENO scheme with different Hamiltonians. The numerical errors and the orders of accuracy for the numerical solution φ n j, and ratios of the numerical errors for cells up to 3 3 in comparison with the scheme are shown in Tables 6-8. The sum of the CPU times for N =,, 4 4, 8 8, 6 6 and 3 3 cells is recorded in Table 5. The ENO- takes the least amount of time followed closely by the ENO-Central scheme. ENO-Roe takes about times the time taken by the Table 5: Total CPU time in seconds for the RKFD method with ENO approimation with different Hamiltonians. The sum of the CPU times for N =,, 4, 8, 6, 3, 64 and 8 cells is recorded for Eample 4.4 for orders,, and 3. Order Central HLL Roe 54.69 6.97 867.6 84.9 38.98 33.6 733.9 569.4 3 67. 646.3 3466.33 37.67

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS Table 6: The L and L errors for eample 4. using N equally spaced cells with different Hamiltonians at T =. for first order ENO approimation. π N N Hamiltonian L error L order error ratio L error L order error ratio 3.44963E-..76378E-. Central.68796E-.78485.6E-.94589 HLL.7968E-.5449.8568E-.67733 Roe.4738E-.5885.8568E-.67733.87766E-.773765..77E-.456957. Central 9.433678E-.6998.8675.374E-.4799.9636 HLL 7.636378E-.366.74 8.389839E-3.7989.77897 Roe 7.673543E-.4986.7544 8.389839E-3.7989.77897 4 4 4.446E-.4837. 4.749E-3.68. Central 4.495E-.67496.999963 4.675746E-3.944.9865 HLL 3.5654E-.385.8847 4.33883E-3.994.99697 Roe 3.57398E-.4346.883955 4.33883E-3.994.99697 8 8.855344E-.4989..46983E-3.96958. Central.85533E-.4945.999993.7747E-3.566.3469 HLL.733488E-.598.9343.86449E-3.99865.9736 Roe.734896E-.64.9358.86449E-3.99865.9736 6 6 8.847578E-3.7796..343E-3.6. Central 8.84757E-3.77953.999999.5E-3.37.68 HLL 8.53954E-3.37.9654.9866E-3.36.986379 Roe 8.5464E-3.334.96553.9866E-3.36.986379 3 3 4.989E-3.4648. 5.5549E-4.84. Central 4.989E-3.4647. 5.534388E-4.9.344 HLL 4.63E-3.7.98955 5.47745E-4.876.9938 Roe 4.5E-3.5.986 5.47745E-4.876.9938 For L error, for N 3: ENO- ENO-Central ENO-Roe ENO-HLL. For L error, for N 4: ENO- ENO-Central ENO-Roe ENO-HLL. For L error, for 8 N 3: ENO-Central ENO- ENO-Roe ENO-HLL. ENO- for all orders and the ENO-HLL takes about 5 6 times the time taken by the ENO- for all orders. Unlike Eamples 4. and 4. of -dimensional case, for first order scheme, ENO- gives the largest and ENO-HLL gives the smallest L error among all the Hamiltonians. For small number of cells (N =, ), the L error by ENO-Central is about 78 87% of the L error by ENO- and as the number of cells increase (N 4 4), ENO-Central and ENO- errors every close to each other. Also for small number of cells (N =, ), the L error by ENO-HLL and ENO-Roe are about 5 7%

HASEENA AHMED AND HAILIANG LIU Table 7: The L and L errors for eample 4. using N equally spaced cells with different Hamiltonians at T =. for second order ENO approimation. π N N Hamiltonian L error L order error ratio L error L order error ratio 9.33585E-..48644E-. Central 9.3564E-.73.58833E-.567 HLL 8.43839E-.93.439668E-.96877 Roe 8.43839E-.93.439668E-.96877.9386E-.7687. 4.757439E-3.7654. Central.958E-.734.53 4.776947E-3.778644.4 HLL.489E-.4756.977343 4.788E-3.7553.9977 Roe.4336E-.46848.977537 4.788E-3.7553.9977 4 4 5.9843E-3.88..6735E-3.5357. Central 5.9888E-3.943.3.783E-3.559.97 HLL 5.6639E-3.93753.99387.447E-3.4748.9984 Roe 5.6639E-3.945.99387.447E-3.4748.9984 8 8.6898E-3.74. 3.8475E-4.94. Central.69E-3.76.4 3.96E-4.9438.3 HLL.6748E-3.646.998477 3.6948E-4.938959.99956 Roe.6748E-3.646.998477 3.6948E-4.938959.99956 6 6 3.5373E-4.894. 8.844E-5.99363. Central 3.5393E-4.89.6 8.84648E-5.9935.5 HLL 3.498E-4.656.9996 8.8357E-5.99747.99988 Roe 3.498E-4.656.9996 8.8357E-5.99747.99988 3 3 7.855E-5.83..88876E-5.979. Central 7.85535E-5.838..889E-5.97956. HLL 7.854484E-5.49.99996.8883E-5.97887.999969 Roe 7.854484E-5.49.99996.8883E-5.97887.999969 For L error, for N = : ENO- ENO-Central ENO-Roe ENO-HLL. For L error, for N 4: ENO-Central ENO- ENO-Roe ENO-HLL. For L error, for 8 N 3: ENO-Central ENO- ENO-Roe ENO-HLL. For L error, for N 3: ENO-Central ENO- ENO-Roe ENO-HLL. of the L error by ENO- and as the number of cells increase (N 4 4), error by ENO-HLL and ENO-Roe are about 9 98% of the L error by ENO-. For the second and third order schemes, we can see from Tables 7 and 8 that for small number of cells ENO-Central gives the largest and ENO-HLL gives the smallest L error among all the Hamiltonians but as the number of cells is increased, the errors become very close to each other.

PERFORMANCE FOR DIFFERENT NUMERICAL HAMILTONIANS 3 Table 8: The L and L errors for eample 4. using N equally spaced cells with different Hamiltonians at T =. for third order ENO approimation. π N N Hamiltonian L error L order error ratio L error L order error ratio.5659e-..68e-3. Central.564968E-.999974.679E-3.9984 HLL.5397E-.973737.633E-3.98753 Roe.5397E-.973737.633E-3.98753.883497E-3 3.74447..89E-4 3.7756. Central.88373E-3 3.74..37E-4 3.554.999658 HLL.87784E-3 3.3797.996977.8678E-4 3.485.997948 Roe.87784E-3 3.3797.996977.8678E-4 3.485.997948 4 4.4377E-4 3.6697..46495E-5 3.469. Central.44E-4 3.6853.6.464863E-5 3.4566.999964 HLL.359E-4 3.57775.999596.4644E-5 3.4337.999785 Roe.359E-4 3.57775.999596.4644E-5 3.4337.999785 8 8.538874E-5 3.9. 3.89839E-6 3.366. Central.538879E-5 3.3. 3.89876E-6 3.396. HLL.538743E-5 3.949.999948 3.89763E-6 3.885.999976 Roe.538743E-5 3.949.999948 3.89763E-6 3.885.999976 6 6 3.87E-6 3.5563. 4.74357E-7.995599. Central 3.87E-6 3.55633. 4.74363E-7.99564. HLL 3.85E-6 3.55565.999993 4.74346E-7.995569.999997 Roe 3.85E-6 3.55565.999993 4.74346E-7.995569.999997 3 3 3.856748E-7 3.594. 5.435E-8.9993. Central 3.856748E-7 3.594. 5.435E-8.9995. HLL 3.856744E-7 3.5896.999999 5.4358E-8.999. Roe 3.856744E-7 3.5896.999999 5.4358E-8.999. For L error, for N = : ENO- ENO-Central ENO-Roe ENO-HLL. For L error, for N 6: ENO-Central ENO- ENO-Roe ENO-HLL. For L error, for N = 3: ENO-Central ENO- ENO-Roe ENO-HLL. For L error, for N 4: ENO- ENO-Central ENO-Roe ENO-HLL. For L error, for 8 N 3: ENO-Central ENO- ENO-Roe ENO-HLL. 5. Conclusion In this paper, we have studied and compared si different numerical Hamiltonians for the RKFD methods. Etensive -dimensional numerical tests performed on Hamilton- Jacobi equations indicate Hamiltonians with upwinding property perform the best when

4 HASEENA AHMED AND HAILIANG LIU factors such as numerical accuracy and resolution of kinks in the solution are addressed. The RKFD method with ENO approimation with and Central Hamiltonian cost the least CPU time among all, but the numerical errors and resolution of discontinuities are also the worst among all. The RKFD methods with Godunov Hamiltonian seem to cost significantly more CPU time than the RKFD- method. The HLL, Roe, EO Hamiltonians might be good choices as Hamiltonians when all factors such as the cost of CPU time, numerical accuracy and resolution of discontinuities in the solution are considered. However we should keep in mind that as we increase the order of accuracy of the ENO approimation and increase the number of cells, the errors for all Hamiltonians are nearly equal. Acknowledgments This research was supported by the National Science Foundation under Grant DMS5-5975. s [] S. Bryson, A. Kurganov, D. Levy, and G. Petrova. Semi-discrete central-upwind schemes with reduced dissipation for Hamilton-Jacobi equations. IMA J. Numer. Anal., 5():3 38, 5. [] S. Bryson and D. Levy. High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton-Jacobi equations. J. Comput. Phys., 89():63 87, 3. [3] M. G. Crandall and P.-L. Lions. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 77(): 4, 983. [4] M. G. Crandall and P.-L. Lions. Two approimations of solutions of Hamilton-Jacobi equations. Math. Comp., 43(67): 9, 984. [5] B. Engquist and S. Osher. One-sided difference approimations for nonlinear conservation laws. Math. Comp., 36(54):3 35, 98. [6] B. Engquist and O. Runborg. Computational high frequency wave propagation. Acta Numer., :8 66, 3. [7] L. C. Evans. Partial differential equations, volume 9 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 998. [8] S. Gottlieb and C.-W. Shu. Total variation diminishing Runge-Kutta schemes. Math. Comp., 67():73 85, 998. [9] S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43():89 (electronic),. [] A. Harten. Preliminary results on the etension of ENO schemes to two-dimensional problems. In Nonlinear hyperbolic problems (St. Etienne, 986), volume 7 of Lecture Notes in Math., pages 3 4. Springer, Berlin, 987. [] A. Harten, S. Engquist, Björn and, and S. R. Chakravarthy. Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys., 7():3 33, 987. [] A. Harten, P. D. La, and B. van Leer. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 5():35 6, 983. [3] A. Harten and S. Osher. Uniformly high-order accurate nonoscillatory schemes. I. SIAM J. Numer. Anal., 4():79 39, 987. [4] C. Hu and C.-W. Shu. A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J. Sci. Comput., ():666 69 (electronic), 999. [5] G.-S. Jiang and D. Peng. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput., (6):6 43 (electronic),.

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