A New Fifth Order Finite Difference WENO Scheme for Hamilton-Jacobi Equations

Transcription

1 A New Fift Order Finite Difference WENO Sceme for Hamilton-Jacobi Equations Jun Zu, 1 Jianxian Qiu 1 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 10016, People s Republic of Cina Scool of Matematical Sciences and Fujian Provincial Key Laboratory of Matematical Modeling and Hig-Performance Scientific Computing, Xiamen University, Xiamen, Fujian , People s Republic of Cina Received June 016; revised 7 November 016; accepted 15 December 016 Publised online 3 February 017 in Wiley Online Library (wileyonlinelibrary.com). DOI /num.133 In tis continuing paper of (Zu and Qiu, J Comput Pys 318 (016), ), a new fift order finite difference weigted essentially non-oscillatory (WENO) sceme is designed to approximate te viscosity numerical solution of te Hamilton-Jacobi equations. Tis new WENO sceme uses te same numbers of spatial nodes as te classical fift order WENO sceme wic is proposed by Jiang and Peng (SIAM J Sci Comput 1 (000), ), and could get less absolute truncation errors and obtain te same order of accuracy in smoot region simultaneously avoiding spurious oscillations nearby discontinuities. Suc new WENO sceme is a convex combination of a fourt degree accurate polynomial and two linear polynomials in a WENO type fasion in te spatial reconstruction procedures. Te linear weigts of tree polynomials are artificially set to be any random positive constants wit a minor restriction and te new nonlinear weigts are proposed for te sake of keeping te accuracy of te sceme in smoot region, avoiding spurious oscillations and keeping sarp discontinuous transitions in nonsmoot region simultaneously. Te main advantages of suc new WENO sceme comparing wit te classical WENO sceme proposed by Jiang and Peng (SIAM J Sci Comput 1 (000), ) are its efficiency, robustness and easy implementation to iger dimensions. Extensive numerical tests are performed to illustrate te capability of te new fift WENO sceme. 017 Wiley Periodicals, Inc. Numer Metods Partial Differential Eq 33: , 017 Keywords: finite difference framework; Hamilton-Jacobi equation; weigted essentially non-oscillatory sceme Correspondence to: Jianxian Qiu, Scool of Matematical Sciences and Fujian Provincial Key Laboratory of Matematical Modeling and Hig-Performance Scientific Computing, Xiamen University, Xiamen, Fujian , People s Republic of Cina ( jxqiu@xmu.edu.cn) Contract grant sponsor: NSFC (J. Zu); contract grant number: Contract grant sponsor: State Scolarsip Fund of Cina for studying abroad (J. Zu) Contract grant sponsor: NSFC (J. Qiu); contract grant number: Contract grant sponsor: NSAF (J. Qiu); contract grant number: U Wiley Periodicals, Inc.

2 1096 ZHU AND QIU I. INTRODUCTION In tis paper, we present a new fift-order accurate finite difference weigted essentially non-oscillatory (WENO) reconstruction metodology for solving te Hamilton-Jacobi (H-J) equations: { φ t + H(x 1,, x n, t, φ, φ) = 0, (x 1,..., x n, t) [0, ), (1.1) φ(x 1,..., x n,0) = φ 0 (x 1,..., x n ), (x 1,..., x n ), were φ = (φ x1,..., φ x n) T. Te Hamilton-Jacobi equations are often used in many applications suc as geometric optics, computer vision, material science, image processing and variational calculus [1 3] and so on. It is well known tat te solutions to (1.1) are continuous but teir associated derivatives migt be discontinuous. And suc solutions may not be unique unless using te pysical implications and ten getting te viscosity solutions [4]. As everyone knows, te H-J equations are closely related to conservation laws, ence we can get te exact solutions of H-J equations from tose of conservation laws and successful numerical metods for conservation laws can be adapted for solving te H-J equations. In tis literature, we refer te crucial works, Oser and Setian [5] presented a second order essentially non-oscillatory (ENO) sceme and Oser and Su [6] put forward some ig order ENO scemes for solving te Hamilton-Jacobi equations. In 000, a finite difference fift-order accurate weigted ENO (WENO) sceme was proposed by Jiang and Peng [7]. Recently, Qiu [8, 9] and togeter wit Su [10] also gained Hermite WENO scemes for solving te Hamilton-Jacobi equations on structured meses. In 1996, Lafon and Oser [11] constructed te ENO scemes for solving te Hamilton-Jacobi equations on unstructured meses. Recently, Zang and Su [1], Li and Can [13] investigated ig order WENO scemes for solving two-dimensional Hamilton-Jacobi equations by te usage of te nodal based WENO polynomial reconstructions on triangular meses. And some finite element metods for arbitrary triangular meses were proposed in [14 17]. In 1999, Hu and Su [16] applied a discontinuous Galerkin (DG) framework on te conservation law system satisfied by te derivatives of te solution to solve H-J equations. Based on suc DG metod, in 007, Ceng and Su presented direct DG metods to solve for te H-J equations (1.1) for φ in [18] and new flux was presented to keep stability of te metod. In [19], Yan and Oser gave a new DG metod to directly solve H-J Eq. (1.1), in wic a local DG metod was applied to approximate derivatives of φ precisely. Following te original idea of WENO metodologies [0, 1], tis paper is mainly based on [7] and is a new generation of te approac of doing WENO in [] from conservation laws to Hamilton-Jacobi equations in te literature. Comparing wit te classical WENO scemes of Jiang and Peng [7], te major advantage of tis new fift order WENO sceme is its easy implementation in te computation. Tis new WENO sceme as a convex combination of a modified fourt degree polynomial wic sould subtract oter two linear polynomials by multiplying proper constant parameters first, ten compound tese tree unequal degree polynomials in a WENO type metodology including artificially setting linear weigts for te robustness, computing smootness indicators and proposing a new type of nonlinear weigts wic are little different from te formula specified in [ 4], for solving te Hamilton-Jacobi equations in one and two dimensions. Te essential merits of suc metodology are its robustness in spatial field by te definition of positive linear weigts, and only one six-point stencil and two tree-point stencils are used to reconstruct tree different polynomials. Te tougt to modify te fourt degree polynomial is very crucial for obtaining te sceme s ig-order accuracy, oterwise, te traditional WENO metodology will inevitably degrade its numerical accuracy. Terefore, we try to use central sceme in smoot regions for simplicity and easy implementation for te

3 WENO SCHEME FOR H-J EQUATIONS 1097 sake of obtaining ig-order accuracy, and switc it to eiter of two second order scemes on compact spatial stencils wen te computing field is adjacent discontinuities for te purpose of avoiding spurious oscillations. Tereafter, new nonlinear weigt formulas are presented for tese polynomials of different degrees based on te information defined on te unequal size stencils [5, 6]. Generally speaking, te primary innovations of tis paper lie in tree aspects: te new way of reconstructing te modified fourt degree polynomial by subtracts two linear polynomials wit different parameters, a robust WENO type communications among tree unequal (modified) polynomials for ig accurate approximations to te derivative quantities in different directions and te new manner of obtaining associated nonlinear weigts. Te organization of tis paper is as follows: In Section II, we review and construct te new fift-order accurate finite difference WENO sceme in 1D and D in detail for solving te Hamilton-Jacobi equations and present extensive numerical results in Section III to verify te accuracy and easy implementation of tese approaces. Concluding remarks are given in Section IV. II. NEW WENO SCHEME FOR HAMILTON-JACOBI EQUATION In tis section, we give te framework for solving te Hamilton-Jacobi equations briefly and ten develop te procedures of te new WENO sceme for bot one and two-dimensional Hamilton-Jacobi equations in detail. A. Te framework for one-dimensional case We take te control Eq. (1.1) in one dimension. For simplicity, te computational field as been divided as te uniform mes wit intervals I i =[x i 1/, x i+1/ ], i = 1,..., N. We denote x i = 1 (x i 1/ + x i+1/ ) to be te interval center and I i =to be te lengt of I i, respectively. (1.1) can be rewritten in one dimension as: { φ t + H(x, t, φ, φ x ) = 0, (.1) φ(x,0) = φ 0 (x). We define φ i φ i (t) = φ(x i, t) and + φ i = φ i+1 φ i. And we would like to omit variable t in te following if not cause confusion. Te semidiscrete form is dφ i (t) dt = L(φ i ) = ˆ H(x i, t, φ i, φ + x,i, φ x,i ). (.) Were Hˆ is a Lipscitz continuous monotone flux consistent wit H, in te sense tat H(x, ˆ t, φ, φ x, φ x ) = H(x, t, φ, φ x ). Te Lax-Friedrics flux is applied ere: ˆ H(x, t, φ, u +, u ) = H ) (x, t, φ, u+ + u α u+ u, (.3) were α = max u H 1 (u). And H 1 stands for te partial derivative of H wit respect to φ x. Now, we would like to reconstruct te approximations of φ x,i from te left and rigt sides of te point x i, and tese procedures confirm te new sceme s ig order of accuracy in smoot regions. Te simple flowcart is elaborated in detail as follows:

4 1098 ZHU AND QIU Step 1. For te purpose of approximating φ x (x i ) on a left-biased six-point stencil {x i 3,..., x i+ } and on a rigt-biased six-point stencil {x i,..., x i+3 }, we do some definitions of te polynomials as: p 1 (x) is a fourt degree polynomial defined on {x i 3, x i, x i 1, x i, x i+1, x i+ } and satisfies 1 xj+1 x j p 1 (x)dx = + φ j, j = i 3, i, i 1, i, i + 1. (.4) p + 1 (x) is a fourt degree polynomial defined on {x i, x i 1, x i, x i+1, x i+, x i+3 } and satisfies 1 xj+1 x j p + 1 (x)dx = + φ j, j = i, i 1, i, i + 1, i +. (.5) p (x) is a linear polynomial defined on {x i, x i 1, x i } and satisfies 1 xj+1 x j p (x)dx = + φ j, j = i, i 1. (.6) p + (x) is a linear polynomial defined on {x i 1, x i, x i+1 } and satisfies 1 xj+1 x j p + (x)dx = + φ j, j = i 1, i. (.7) p 3 (x) is a linear polynomial defined on {x i 1, x i, x i+1 } and satisfies 1 xj+1 x j p 3 (x)dx = + φ j, j = i 1, i. (.8) p + 3 (x) is a linear polynomial defined on {x i, x i+1, x i+ } and satisfies 1 xj+1 x j p + 3 (x)dx = + φ j, j = i, i + 1. (.9) Te fift order approximations of φ x at x i from left and rigt sides are given, respectively, by φ,1 x,i = p 1 (x i) = φ i 3 30 φ +,1 x,i = p + 1 (x i) = 13 + φ i φ i φ i φ i φ i φ i φ i φ i+1, (.10) + φ i+, (.11) te te tird order approximations of φ x at x i from left and rigt sides are given, respectively, by φ, x,i = p (x i) = 1 + φ i φ i 1, (.1)

5 WENO SCHEME FOR H-J EQUATIONS 1099 φ +, x,i = p + (x i) = 1 + φ i φ i, (.13) φ,3 x,i = p 3 (x i) = 1 + φ i φ i, (.14) φ +,3 x,i = p + 3 (x i) = 3 + φ i 1 + φ i+1, (.15) Step. Set any positive linear weigts γ 1, γ, γ 3, suc tat γ 1 + γ + γ 3 = 1, in tis paper, we take γ 1 = and γ = γ 3 = Step 3. Compute te smootness indicators β n ±, wic measure ow smoot te functions p n ± (x) are in te target cells [x i 1, x i ] and [x i, x i+1 ], respectively. Te smaller tese smootness indicators, te smooter te functions are in different target cells. We use te similar recipe for te smootness indicators as in [7, 8]. r xi ( d α β n = pn (x) ) dx, n = 1,, 3, (.16) dx α α=1 x i 1 α 1 and β + n = r xi+1 α 1 α=1 x i ( d α p n + (x) ) dx, n = 1,, 3. (.17) dx α For n =1, r equals four and if n =, 3, r equals one. Te associated smootness indicators are explicitly written and te associated expansions of tem in Taylor series about φ i are obtained as β 1 = 1 ( + φ i φ i ( 11 + φ i 3 ( + φ i 3 ( + φ i φ i φ i + + φ i φ i + φ i ) + φ i φ i φ i 1 ) + + φ i φ i φ i ) + + φ i+1 ) 11 + φ i+1 = (φ i ) 3 (φ i )(φ i ) + O( 4 ), (.18) β = ( + φ i ) + φ i 1 = (φ i ) 3 (φ i )(φ i ) + O( 4 ), (.19) β 3 ( + = φ i 1 β + 1 = 1 ( + φ i ) + φ i = (φ i ) (φ i )(φ (4) i ) + O( 6 ), (.0) 8 + φ i φ i+1 ) + φ i+ ( 11 + φ i φ i φ i φ i φ i+ )

6 1100 ZHU AND QIU ( + φ i 880 ( + φ i φ i φ i 1 + φ i φ i ) + + φ i+ 4 + φ i+1 ) + + φ i+ = (φ i ) + 3 (φ i )(φ i ) + O( 4 ), (.1) β + = ( + φ i 1 + φ i ) = (φ i ) (φ i )(φ (4) i ) + O( 6 ), (.) β + 3 = ( + φ i + φ i+1 ) = (φ i ) + 3 (φ i )(φ (3) i ) + O( 4 ), (.3) Step 4. We compute te nonlinear weigts based on te linear weigts and te smootness indicators. For instance, as sown in [3, 4], we use new τ ± wic are simply defined as te absolute difference between β ± 1, β±, and β± 3, and are little different from tat specified in [3, 4]. So te associated difference expansions in Taylor series about φ i are ( β ± τ ± = 1 β± + β± 1 β± 3 ω ± n = ω ± ( n 3, ω n = γ n 1 + τ ± l=1 ω± ε + β ± l n ) = O( 6 ). (.4) ), n = 1,, 3. (.5) Here ε is a small positive number to avoid te denominator to become zero. It is directly to verify from (.18) to (.3) and te parameters τ ±, at te smoot regions of te numerical solution satisfying τ ± ε + β ± n = O( 4 ), n = 1,, 3, (.6) on condition tat ε β n ±. Terefore, te nonlinear weigts ω± n satisfy te order accuracy condition ω n ± = γ n + O( 4 ) [3, 4], providing te formal fift-order accuracy to te WENO sceme in [7], 8, [9]. We take ε = 10 6 in our computation. Step 5. Te final nonlinear WENO reconstructions φ ± x,i are defined by a convex combination of te tree (modified) reconstructed polynomial approximations. ( 1 φ ± x,i = ω± 1 φ ±,1 x,i γ φ ±, x,i γ ) 3 φ ±,3 x,i + ω ± γ 1 γ 1 γ φ±, x,i + ω ± 3 φ±,3 x,i. (.7) 1 Te terms on te rigt-and side of (.7) are very sopisticated tan te usual definition ω ± 1 φ±,1 x,i + ω ± φ±, x,i + ω ± 3 φ±,3 x,i wic would degrade te optimal fift-order accuracy because of φ ±, x,i and φ ±,3 x,i are active and te convex combination togeter wit φ ±,1 x,i would not offer ig order approximation at point x i to numerical flux in smoot region. Tus we sould abolis teir contribution in smoot region. By doing suc procedure, if a big spatial stencil permits optimal stability and accuracy to be reaced, (.7) could obtain tat stability and accuracy obviously. Neverteless, wen te solution on te big spatial stencil is roug, it is beneficial to searc for two smaller tree-point stencils. Suc two small spatial stencils tat are used ensure tat (wen te large six-point spatial stencil may be inoperative). In tis way, it is not very essential for us

7 WENO SCHEME FOR H-J EQUATIONS 1101 to deliberately coose linear weigts for te purpose of obtaining ig-order accuracy in smoot region and could keep ig resolutions for te singularities of te derivatives in nonsmoot region. Step 6. Te ODE (.) is rewritten as te formula dφ i (t) dt = L(φ i ). (.8) Ten we use tird order version TVD Runge-Kutta time discretization metod [30]: φ (1) = φ n + tl(φ n ), φ () = 3 4 φn φ(1) tl(φ(1) ), φ n+1 = 1 3 φn + 3 φ() + 3 tl(φ() ), (.9) to obtain fully discrete sceme bot in space and time. B. Te Framework for Two-Dimensional Case We take te control Eq. (1.1) in two dimensions. For simplicity, we also assume as been divided as an uniform mes wit intervals I i,j =[x i 1/, x i+1/ ] [y j 1/, y j+1/ ], i = 1,..., N, j = 1,..., M. We denote (x i, y j ) = ( 1 (x i 1/ + x i+1/ ), 1 (y j 1/ + y j+1/ )), I i,j =(x i+1/ x i 1/ ) (y j+1/ y j 1/ ) = to be te interval center and te area of I i,j, respectively. Ten (1.1) can be reformulated as: { φ t + H(x, y, t, φ, φ x, φ y ) = 0, (.30) φ(x, y,0) = φ 0 (x, y). We define φ i,j = φ(x i, y j ), + x φ i,j = φ i+1,j φ i,j and + y φ i,j = φ i,j+1 φ i,j. Te semidiscrete form in two dimensions is dφ i,j (t) dt = L(φ i,j ) = ˆ H(x i, y j, t, φ i,j, φ + x,i,j, φ x,i,j, φ+ y,i,j, φ y,i,j ). (.31) were Hˆ is a Lipscitz continuous monotone flux consistent wit H, in te sense tat H(x, ˆ y, t, φ, φ x, φ x, φ y, φ y ) = H(x, y, t, φ, φ x, φ y ). Te Lax-Friedrics flux is applied ere: ) H(x, ˆ y, t, φ, u +, u, v +, v ) = H (x, y, t, φ, u+ + u, v+ + v α u+ u β v+ v, (.3) were α = max u H 1 (u, v) and β = max v H (u, v). H 1 stands for te partial derivative of H wit respect to φ x and H stands for te partial derivative of H wit respect to φ y. Here φ ± x,i,j are WENO approximations to φ(x i,y j ), and φ ± x y,i,j are WENO approximations to φ(x i,y j ), wic are similar to y one-dimensional approximations (.7). Te approximation procedures of φ ± x,i,j and φ± y,i,j use a dimension-by-dimension fasion wit fixed subscript j and i, respectively.

8 110 ZHU AND QIU TABLE I. φ t + φ x = 0. φ(x,0) = cos(πx). Periodic boundary conditions. t =.. Grid points L 1 error Order L error Order L 1 error Order L error Order WENO-ZQ (1) sceme WENO-JP sceme E 1.95 E.71 E 4.63 E E E E E E E E E E E E E E E E E E E E E WENO-ZQ () sceme WENO-ZQ (3) sceme E E 1.47 E E E E E E E E E E E E E E E E E E E E E E III. NUMERICAL TESTS In tis section, we set CFL number to be 0.6 and present te results of numerical tests of te new finite difference WENO sceme as WENO-ZQ wic is specified in te previous section bot in one and two dimensions, and classical finite difference WENO sceme as WENO-JP wic is specified in [7]. For te purpose of testing weter te random coice of te linear weigts would pollute te optimal fift-order accuracy of WENO-ZQ sceme or not, we set different type of linear weigts in te numerical accuracy cases as: (1) γ 1 =0.998, γ =0.001, and γ 3 =0.001; () γ 1 =1.0/3.0, γ =1.0/3.0, and γ 3 =1.0/3.0; (3) γ 1 =0.01, γ =0.495, and γ 3 = And set γ 1 =0.998, γ =0.001, and γ 3 =0.001 in te oter examples. Example 3.1. We solve te following linear equation: φ t + φ x = 0, 1 <x<1, (3.1) wit te initial condition φ(x,0) = cos(πx) and periodic boundary conditions. Wen t = te solution is still smoot. Te errors and numerical orders of accuracy are sown in Table I. We can see te WENO-ZQ acieves its designed order of accuracy in one dimension and sustains less absolute quantity of L 1 and L errors. We can see tat te WENO-ZQ sceme wit different type of linear weigts acieves close to its designed order of accuracy. And te Fig. 1 sows tat WENO-ZQ sceme needs less CPU time tan WENO-JP sceme does to obtain te same quantities of L 1 and L errors. Example 3.. equation: We solve te following nonlinear scalar one-dimensional Hamilton-Jacobi φ t cos(φ x + 1) = 0, 1 <x<1, (3.) wit te initial condition φ(x, 0) = cos(πx) and periodic boundary conditions. We compute te result up to t = 0.5/π. Te errors and numerical orders of accuracy are sown in Table II. Again, we can see WENO-ZQ acieves its designed order of accuracy in one dimension. For comparison, errors and numerical orders of accuracy by te classical WENO-JP sceme are sown in

9 WENO SCHEME FOR H-J EQUATIONS 1103 FIG. 1. φ t + φ x = 0. φ(x,0) = cos(πx). Computing time and error. Number signs and a solid line denote te results of WENO-ZQ sceme wit different linear weigts (1), (), and (3); squares and a solid line denote te results of WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] TABLE II. φ t cos(φ x + 1) = 0. φ(x,0) = cos(πx). Periodic boundary conditions. t = 0.5/π. Grid points L 1 error Order L error Order L 1 error Order L error Order WENO-ZQ (1) sceme WENO-JP sceme E E E E E E E E E E E E E E E E E E E E E E E E WENO-ZQ () sceme WENO-ZQ (3) sceme E E 6.74 E E 0.94 E E E E E E E E E E E E E E E E E E E E te same table. We can see tat bot WENO-ZQ and WENO-JP scemes acieve teir designed order of accuracy. Figure sows tat WENO-ZQ sceme needs less CPU time tan WENO-JP does to obtain te same quantities of L 1 and L errors, so WENO-ZQ sceme is more efficient tan WENO-JP sceme in tis test case. Example 3.3. We solve te following nonlinear scalar two-dimensional Burgers equation: φ t + (φ x + φ y + 1) = 0, x, y<, (3.3) wit te initial condition φ(x, y, 0) = cos(π(x + y)/) and periodic boundary conditions. We compute te result to t = 0.5/π and te solution is still smoot at tat time. Te errors

10 1104 ZHU AND QIU FIG.. φ t cos(φ x + 1) = 0. φ(x,0) = cos(πx). Computing time and error. Number signs and a solid line denote te results of WENO-ZQ sceme wit different linear weigts (1), (), and (3); squares and a solid line denote te results of WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] TABLE III. φ t + t = 0.5/π. (φx +φy +1) = 0. φ(x, y, 0) = cos(π(x + y)/). Periodic boundary conditions. Grid points L 1 error Order L error Order L 1 error Order L error Order WENO-ZQ (1) sceme WENO-JP sceme E 4 6. E 4.70 E E E E E E E E E E E E E E E E E E WENO-ZQ () sceme WENO-ZQ (3) sceme E E E E E E E E E E E E E E E E E E E E and numerical orders of accuracy by te WENO-ZQ are sown in Table III and te numerical error against CPU time graps are in Fig. 3. We can observe tat te teoretical order is actually acieved and te WENO-ZQ sceme can get better results and is more efficient tan WENO-JP sceme in tis test case. Example 3.4. equation: We solve te following nonlinear scalar two-dimensional Hamilton-Jacobi φ t cos(φ x + φ y + 1) = 0, x, y<, (3.4) wit te initial condition φ(x, y,0) = cos(π(x + y)/) and periodic boundary conditions. We also compute te result until t = 0.5/π. Te errors and numerical orders of accuracy by te WENO-ZQ sceme wit different linear weigts in comparison wit WENO-JP sceme are sown

11 WENO SCHEME FOR H-J EQUATIONS 1105 (φx +φy +1) FIG. 3. φ t + = 0. φ(x, y, 0) = cos(π(x+y)/). Computing time and error. Number signs and a solid line denote te results of WENO-ZQ sceme wit different linear weigts (1), (), and (3); squares and a solid line denote te results of WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] TABLE IV. t = 0.5/π. φ t cos(φ x + φ y + 1) = 0. φ(x, y,0) = cos(π(x + y)/). Periodic boundary conditions. Grid points L 1 error Order L error Order L 1 error Order L error Order WENO-ZQ (1) sceme WENO-JP sceme E E 4.16 E E E E E E E E E E E E E E E E E E WENO-ZQ () sceme WENO-ZQ (3) sceme E E E 4.19 E E E E E E E E E E E E E E E E E in Table IV and te numerical error against CPU time graps are in Fig. 4. WENO-ZQ sceme wit different type of linear weigts is better tan WENO-JP sceme in tis two-dimensional test case. Example 3.5. We solve te linear equation: φ t + φ x = 0, (3.5) wit te initial condition φ(x,0) = φ 0 (x 0.5) togeter wit te periodic boundary conditions, were:

12 1106 ZHU AND QIU FIG. 4. φ t cos(φ x + φ y + 1) = 0. φ(x, y,0) = cos(π(x + y)/). Computing time and error. Number signs and a solid line denote te results of WENO-ZQ sceme wit different linear weigts (1), (), and (3); squares and a solid line denote te results of WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] φ 0 (x) = ( π 3 ( ) 3πx cos 3, 1 x< 1 ), 3 3 (x + 1) cos(πx), 1 x<0, cos(πx), 0 x< 1, 3 8+4π+cos(3πx) + 6πx(x 1), x<1. (3.6) We plot te results wit 100 cells at t = and t = 8 in Fig. 5. We can observe tat te results by te WENO-ZQ ave good resolution for te corner singularity. Example 3.6. We solve te one-dimensional nonlinear Burgers equation: φ t + (φ x + 1) = 0, (3.7) wit te initial condition φ(x,0) = cos(πx) and te periodic boundary conditions. We plot te results at t = 3.5/π wen discontinuous derivative appears. Te solutions of te WENO-ZQ are given in Fig. 6. We can see te sceme gives good results for tis problem. Example 3.7. We solve te nonlinear equation wit a non-convex flux: φ t cos(φ x + 1) = 0, (3.8) wit te initial data φ(x,0) = cos(πx) and te periodic boundary conditions. Ten we plot te results at t = 1.5/π in Fig. 7 wen te discontinuous derivative appears in te solution. We can see tat te WENO-ZQ gives good results for tis problem.

13 WENO SCHEME FOR H-J EQUATIONS 1107 FIG. 5. One dimensional linear equation. 100 grid points. Left: t = ; rigt: t = 8. Solid line: te exact solution; cross symbols: WENO-ZQ sceme; square symbols: WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] FIG. 6. One dimensional Burgers equation. Left: 40 grid points; rigt: 80 grid points. t = 3.5/π. Solid line: te exact solution; cross symbols: WENO-ZQ sceme; square symbols: WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] Example 3.8. We solve te one-dimensional Riemann problem wit a non-convex flux: { φ t (φ x 1)(φ x 4) = 0, 1 <x<1, φ(x,0) = x. (3.9) Tis is a demanding test case, for many scemes ave poor resolutions or could even converge to a non-viscosity solution for tis case. We plot te results at t = 1 by te sceme wit 80 grid points to verify te numerical solution in Fig. 8. We can see tat suc sceme gives good results for tis problem.

14 1108 ZHU AND QIU FIG. 7. Problem wit te non-convex flux H(φ x ) = cos(φ x + 1). Left: 40 cells; rigt: 80 grid points. t = 3.5/π. Solid line: te exact solution; cross symbols: WENO-ZQ sceme; square symbols: WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] FIG. 8. Problem wit te non-convex flux H(φ x ) = 1 4 (φ x 1)(φ x 4). 80 grid points. t = 1. Solid line: te exact solution; cross symbols: WENO-ZQ sceme; square symbols: WENO-JP sceme. [Color figure can be viewed at wileyonlinelibrary.com.] Example 3.9. We solve te same two-dimensional nonlinear Burgers Eq. (3.3) as in Example 3.3 wit te same initial condition φ(x, y,0) = cos(π(x + y)/), except tat we now plot te results at t = 1.5/π in Fig. 9 wen te discontinuous derivative as already appeared in te solution. We observe good resolutions for tis example.

15 WENO SCHEME FOR H-J EQUATIONS 1109 FIG. 9. Two dimensional Burgers equation grid points. t = 1.5/π. WENO-ZQ sceme. Left: contours of te solution; rigt: te surface of te solution. FIG. 10. Two dimensional Riemann problem wit a non-convex flux H(φ x, φ y ) = sin(φ x + φ y ) grid points. t = 1. WENO-ZQ sceme. Left: contours of te solution; rigt: te surface of te solution. Example Te two-dimensional Riemann problem wit a non-convex flux: { φ t + sin(φ x + φ y ) = 0, 1 x, y<1, φ(x, y,0) = π( y x ). (3.10) Te solution of te WENO-ZQ is plotted at t = 1 in Fig. 10. We can also observe good resolutions for tis numerical simulation.

16 1110 ZHU AND QIU FIG. 11. Te optimal control problem grid points. t = 1. WENO-ZQ sceme. Left: te surface of te solution; rigt: te optimal control ω = sign (φ y ). FIG. 1. Eikonal equation wit a non-convex Hamiltonian grid points. t = 0.6. WENO-ZQ sceme. Left: contours of te solution; rigt: te surface of te solution. Example A problem from optimal control: { φ t + sin(y)φ x + (sin(x) + sign(φ y ))φ y 1 sin (y) (1 cos(x)) = 0, φ(x, y,0) = 0, π x, y<π, (3.11) wit periodic conditions, see [6]. Te solutions of WENO-ZQ are plotted at t = 1 and te optimal control ω = sign (φ y ) is sown in Fig. 11. Example 3.1. A two-dimensional Eikonal equation wit a non-convex Hamiltonian, wic arises in geometric optics [31], is given by:

17 WENO SCHEME FOR H-J EQUATIONS 1111 FIG. 13. Propagating surface grid points. Left: ε = 0; rigt: ε = 0.1. WENO-ZQ sceme. { φt + φx + φ y + 1 = 0, 0 x, y<1, φ(x, y,0) = 1 (cos(πx) 1)(cos(πy) 1) 1, (3.1) 4 Te solutions of te WENO-ZQ are plotted at t = 0.6 in Fig. 1. Good resolutions are observed wit te proposed sceme. Example Te problem of a propagating surface [5]: { φt (1 εk) φx + φ y + 1 = 0, 0 x, y<1, φ(x, y,0) = 1 1 (cos(πx) 1)(cos(πy) 1), (3.13) 4 were K is te mean curvature defined by: K = φ xx(1 + φ y ) φ xy φ x φ y + φ yy (1 + φ x ) (1 + φ x + φ y )3/, and ε is a small constant. Periodic boundary conditions are used. Te approximation of K is constructed by te metods similar to te first derivative terms and tree different second order derivatives of associated Hermite reconstruction polynomials are needed. Te results of ε = 0

18 111 ZHU AND QIU (pure convection) and ε = 0.1 by WENO-ZQ are presented in Fig. 13. Te surfaces at t = 0 for ε = 0 and for ε = 0.1, and at t = 0.1 for ε = 0.1, are sifted downward to sow te detail of te solution at later time. IV. CONCLUDING REMARKS In tis paper, we ave constructed a new fift order finite difference WENO sceme wic is a WENO type combination of a modified fift order sceme and two second order scemes for solving te Hamilton-Jacobi equations in one and two dimensions. Te main advantages of suc metodology are its simplicity in spatial field by te definition of positive linear weigts, and only one six-point stencil and two tree-point stencils are needed on constructing different polynomials. Terefore, we try to use central sceme in smoot regions for simplicity and obtain ig order numerical accuracy, and switc to eiter of two lower order scemes wen adjacent discontinuities for te purpose of avoiding spurious oscillations. For te sake of confining te new WENO sceme converge to te optimal fift-order accuracy in smoot regions, we modify te fourt degree polynomial by subtracting two parameterized linear polynomials. Tereafter, new nonlinear weigt s formulas are presented for tese polynomials of different degrees based on te information tat defined on tese unequal spatial stencils. Generally speaking, te constructions of suc new WENO sceme are based on WENO interpolation in spatial field and ten Runge-Kutta discretization is used for solving te ODE. In te new WENO sceme, te nodal point information is used via time approacing and is easier to be implemented tan te classical finite difference WENO sceme. Tis sceme sustains ig-order accuracy in te smoot regions and obtains ig resolutions for te singularities of te derivatives robustly. Extensive numerical experiments are performed to illustrate te effectiveness of te new WENO sceme. References 1. C. K. Can, K. S. Lau, and B. L. Zang, Simulation of a premixed turbulent Name wit te discrete vortex metod, Int J Numer Metods Eng 48 (000), P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman, London, J. A. Setian, Level set metods and fast marcing metods, evolving interface, computational geometry, fluid mecanics, computer vision, and materials science, Cambridge University Press, Cambridge, R. Abgrall and T. Sonar, On te use of Muelbac expansions in te recovery step of ENO metods, Numer Mat, 76 (1997), S. Oser and J. Setian, Fronts propagating wit curvature dependent speed: Algoritms based on Hamilton-Jacobi formulations, J Comput Pys 79 (1988), S. Oser and C.-W. Su, Hig-order essentially nonoscillatory scemes for Hamilton-Jacobi equations, SIAM J Numer Anal 8 (1991), G. S. Jiang and D. Peng, Weigted ENO scemes for Hamilton-Jacobi equations, SIAM J Sci Comput 1 (000), J. Qiu, WENO scemes wit Lax-Wendroff type time discretizations for Hamilton-Jacobi equations, J Comput Appl Mat 00 (007), J. Qiu, Hermite WENO scemes wit Lax-Wendroff type time discretizations for Hamilton-Jacobi equations, J Comput Mat 5 (007), J. Qiu and C.-W. Su, Hermite WENO scemes for Hamilton-Jacobi equations, J Comput Pys 04 (005), 8 99.

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