NUMERICAL STUDIES ON HAMILTON-JACOBI EQUATION IN NON-ORTHOGONAL COORDINATES

J. Appl. Math. & Computing Vol. 21(2006), No. 1-2, pp. 603-622 NUMERICAL STUDIES ON HAMILTON-JACOBI EQUATION IN NON-ORTHOGONAL COORDINATES MINYOUNG YUN Abstract. The level-set method is quite efficient and robust to track the motion of interface for interested object. It is necessary to deal with the Hamilton-Jacobi equation through reinitialization process so that the added dimension has a signed distance. However, during the computation, the level set function becomes irregular. In order to prevent the irregularities of the calculated level set function, the reinitialization of the level set function is needed during the entire computation. Since the zero level set represents a moving boundary interface, it must not move during reinitialization. However, this zero level set moves in actual computation and numerical error deviated from the zero level set is introduced. In fact, the volume with the zero level set shrinks and the volume is not conserved during the reinitialization process. In the present study, the proper treatment is used to conserve the volume during the reinitialization process. Special emphasis is given to the effects of numerical schemes on the reconstruction of the signed distance function in the context of level set methods. So far, most of the level set method is based on the orthogonal grid systems. In order to handle the complex geometry encountered in the real engineering problems, the extension to the formulations and discretized equations based on the non-orthogonal grids has been made. Numerical results indicate that the present procedure successfully demonstrated the capability to reconstruct the signed distance function. AMS Mathematics Subject Classification : 35L55. Key words and phrases : Level set method, Hamilton-Jacobi equation, reinitialization, non-orthogonal coordinates. 1. Introduction The main characteristic of front propagating problems is that the location of the moving boundary inherently influences the evolution process. Thus, it Received June 13, 2005. c 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 603

604 Minyoung Yun is crucial to determine the position of the boundary. There are several frontcapturing methods including the Lagrangian tracking method, the volume of fluid method, and the level set method. Among these front capturing methods, the level set approach is the most promising procedure which can circumvent serious drawbacks of the other two methodologies. The advantages of this level set technique include the no requirement of explicit interface information, flexibility for treating complex interface geometries, use of the fixed-grid discretization, numerical stability, no complicated bookkeeping procedure, and the easy extension to three-dimensional problems. Level set methods has been widely used to solve the front propagating problems crystal growth[1], multi-phase flow[2,3], and image processing[4-6]. Especially in the techniques of image science which demands multi-disciplinary knowledge and flexible but still robust methods, some relevant level set techniques are potentially useful. The level set method for capturing moving fronts was introduced by Osher and Sethian [7]. The image science using level set techniques has been applied successfully to the image/video processing, computer vision, and graphics. These include special research efforts for medical imaging and Hollywood-type special image effects. However, during the computation, the level set function becomes irregular. In order to prevent the irregularities of the calculated level set function, the reinitialization of the level set function is needed after a small number of evolution time steps. Thus, the reinitialization is continuously applied during the entire computation. Since the zero level set represents a moving boundary which may be an interface of two differently characterizing fields or materials, it must not move during reinitialization. However, this zero level set moves in actual computation and numerical error deviated from the zero level set is introduced. Moreover, the error is not qualitatively arbitrary but tends always to one direction. In fact, the volume with the zero level set shrinks and the volume is not conserved in the reinitialization process. These numerical errors are progressively accumulated during the entire calculation. These deviations eventually results in the substantial degree of the volume shrinks and grid anisotropy during the re-initialization process. In order to overcome the undesirable defects including the volume shrinks and grid anisotropy, the present study uses the proper treatment[8] to conserve the volume during the reinitialization process and the high-order upwind scheme[9,10] to minimize numerical diffusion. A high order WENO (Weighted Essentially Non-Oscillatory) scheme[9] has been employed to approximate the Hamilton-Jacobi equation. Roughly speaking, WENO schemes are central schemes in regions where the solution is smooth but emulate ENO (Essentially Non- Oscillatory) schemes near the singularities of the solution. This is achieved by weighting the substencils of the base ENO scheme with the weights adapted to the relative smoothness of the solution on these substencils. It is recognized

Numerical studies on Hamilton-Jacobi equation 605 that WENO scheme are more accurate and robust than the base ENO schemes and still maintain a compact stencil, which could be very desirable for certain boundary treatments. Another crucial issue in the level set approach for the practical problems might be how to deal with the complex geometry encountered in the real engineering problems. So far, most of the level set method is based on the orthogonal grid systems. In the present study, we extend to the formulations of the level set approach based on the non-orthogonal grids. The discretized equations in the body-fitted coordinate are also presented in detail. In the reconstruction of the signed distance function in the context with the level set method, the present study has focused on numerically investigating the effects of numerical schemes together with the volume conservation treatment as well as the level-set formulations based on the non-orthogonal grids. Based on numerical results, the detailed discussions have been made. 2. Reinitialization Level set methods have proven to be useful tools for computing interface evolution. In this approach the interface is represented as the zero level set of a continuous level set function defined in a domain. The level sets always move according to the velocity field means that φ 0 = 0 does not along the trajectory, x(t). φ(x(t),t)=φ 0 The function φ 0 is defined everywhere in the domain. The interface is updated by solving a transport equation obtained by the chain rule, t + φ(x(t),t) x (t) =0 (1) The velocity function, F outwardly normal to the interface can be expressed as, F = x (t) n = x (t) Substituting (2) into (1) yields φ φ F φ = x (t) φ (2) t + F φ =0 : φ 0 = φ(x 0,t= 0) (3) which is the level set equation introduced by Osher and Sethian[13]. For the certain speed function, this is a first-order partial differential equation of Hamilton- Jacobi type. It makes the zero level set to move exactly as the interface moves with the flow. Thus, equation (3) could be an evolution equation for the interface. However, in many applications the level set function obtained by the solution of equation (3) may become distorted, which means that its gradient

606 Minyoung Yun may become very large or very small around the interface. It is therefore useful to replace the level set function with a better behaved function which has the same zero level set. This process is called reinitialization [11]. The simplest and most useful choice is to replace the level set function by the signed distance function. A signed distance function referenced at φ 0 =0is obtained by solving the steady-state solutions of the following equation (4) with a constraint, φ = 1. t = sgn(φ 0)(1 φ ) : φ(x 0, 0) = φ 0 (4) Here the sign function, sgn(x) is defined as 1 if x>0, sgn(x) = 0 if x =0, (5) 1 if x<0. Using equation(4), the idea behind this reinitialization procedure is to impose the steady state solution to be a distance function for the zero level, φ =1 with the same zero level set as the initial function φ 0 = 0. Hence, the distance function may be computed for any given zero level without changing its position. In this reinitialization process, without moving the zero level set itself as well as without knowing explicitly the position of the zero level set, the distance function can be determined from the zero level set. In numerical computations, the smoother sign function could be used. 3. Discretization of Hamilton-Jacobi equation in orthogonal coordinate and numerical results First-order upwind scheme It is well known that the solution of equation (4) typically are continuous but with discontinuous derivatives even when the initial condition φ is smooth, and such solutions are generally not unique. The viscosity solution of Hamilton- Jacobi equation was obtained by Crandal and Lions[3]. They introduced an important class of monotone schemes for the Hamilton-Jacobi equation. Using a first-order upwind scheme[2], the two-dimensional Hamilton-Jacobi equation is discretized by, where φ n+1 i,j = φ n i,j ts(φ 0 i,j)g(φ) i,j (6) max(a 2 +,b 2 )+max(c 2 +,d 2 ) 1 if φ 0 i,j G(φ) i,j = max(a 2,b2 + )+max(c2,d2 + ) 1 if φ0 i,j < 0 (7)

Numerical studies on Hamilton-Jacobi equation 607 For any real number h, h + = max(h, 0), h = min(h, 0). In equation (7), the nearby gradients for grid points are given as a = Dx φ i,j = φ i,j φ i 1,j, b = D x + φ i,j = φ i+1,j φ i,j x x c = Dy φ i,j = φ i,j φ i,j 1, d = D y + y φ i,j = φ i,j+1 φ i,j y The smoothed sign function, S(φ) can be discretized by ( S(φ) =2 H h 1 ) 2 (8) (9) Here, H h (φ) is the Heaviside function described below. 1, if φ i,j > h H h (φ i,j )= 0, ( if φ i,j < h 1 2 1+ φi,j h + 1 πφi,j π sin( h ), ) otherwise(i.e., φ i,j < h) (10) where h = α min( x, y) and α = 1 2. The computational time-step can be obtained to satisfy the CFL condition. t = α 2 min( x, y) with α<1 As a two-dimensional validation case, we consider the zero level set is chosen a circle with radius 4. φ 0 (x, y) = x 2 + y 2 4 (11) Here φ 0 (x, y) is the signed distance function. If the reinitialization algorithm is applied to this distance function, there must be no movement for the initial level set. Figure 1 displays the orthogonal mesh arrangement. In this study, 16 16 grid is all calculations for the orthogonal mesh arrangement. Figure 2 shows the distance function constructed by the first-order scheme with the initial condition given by equation (11). Here we have plotted the zero level set of φ is N = 0, 160, 320, 480, 640, 800. The temporal and spatial intervals are chosen as t =0.5 x, x =10/16. It can be clearly seen that there exist not only the circle shrinks but also the considerable grid anisotropy. Numerical diffusion is partially responsible for these undesirable effects. These deviations could be reduced by utilizing the higher-order upwind schemes. Thus, this algorithm produces an error that is

608 Minyoung Yun proportional to the number of iterations. In most applications a small number of iterations of reinitialization procedure are applied each time step. Thus the total number iterations will be large; consequently, the error due to the reinitialization algorithm could be rather large. Figure 1. Numerical grid arrangement (16 16 mesh, x = y =10/16). High-order WENO scheme As shown in Figure 2, the first-order result clearly indicates that there are the considerable degree of circle shrinks and anisotropy. These numerical errors are increased with the number of iterations. Figure 2. Construction of the distance function using the 1 st upwind scheme without volume conservation treatment: Dashed(160), Dash Dot(320), Dotted(480), Long Dash(640), Solid(800). In the level set approach, since the total number of iterations could be large,

Numerical studies on Hamilton-Jacobi equation 609 numerical errors are continuously accumulated and eventually becomes quite large. Since numerical diffusion associated with the first order scheme is partially responsible for these deviations, we need to check the third order WENO( Weighted Essentially Non-Oscillatory) scheme. In order to minimize numerical diffusion, a high order accurate weighted ENO (WENO) scheme[9] has been employed to approximate the Hamilton-Jacobi equation. This scheme is constructed by weighting the substencils of the base ENO scheme with the weights adapted to the relative smoothness of the solution on these substencils. The weights can be defined in such a way that in smooth regions it approaches certain optimal weights to achieve a higher order of accuracy while in regions near discontinuities, the stencils which contain the discontinuities are assigned as a nearly zero weight. Thus essentially nonoscillatory property is achieved by emulating ENO schemes around discontinuities and a higher order of accuracy is obtained by emulating upstream central schemes with the optimal weights away form discontinuities. It is recognized that WENO scheme are more accurate and robust than the base ENO schemes and still maintain a compact stencil, which could be very desirable for certain boundary treatments. Another advantage of WENO schemes is that its flux is smoother than that of ENO schemes. The present study employs the high-order WENO scheme suggested by Jiang et al.[9]. The first order gradient can be expressed in the abbreviated form D x ± φ i,j = ±(φ i±1,j φ i,j ) (x axis) x D y ± φ i,j = ±(φ i,j±1 φ i,j ) (y axis) (12) y The gradient for the WENO scheme can be expressed as follows. Here j index is fixed and index is interchangeable for the y-axis. D x ± φ i = 1 ( D x + φ i 2 +7D x + φ i 1 +7D x + φ i D x + φ i+1 ± x 12 ) Φ WENO (Dx D+ x φ i±2,dx D+ x φ i±1,dx D+ x φ i,dx D+ x φ i 1 (13) Φ WENO (a, b, c, d) = 1 3 ω 0(a 2b + c)+ 1 6 The weighting factors ω 0, ω 2 are given as, α 0 = ω 0 = α 0 α 2, ω 2 =, α 0 + α 1 + α 2 α 0 + α 1 + α 2 1 (ε + β 0 ) 2, α 1 = 1 (ε + β 1 ) 2, α 2 = ( ω 2 1 ) (b 2c + d) (14) 2 1 (ε + β 2 ) 2, (15)

610 Minyoung Yun β 0 = 13(a b) 2 +3(a 3b) 2, β 1 = 13(b c) 2 +3(b + c) 2, β 2 = 13(c d) 2 + 3(3c d) 2, Here, ε is the small value, 10 6 to prevent the singularity. This WENO scheme is the fifth-order accurate in smooth regions of the flux function while it is just the third-order accurate at discontinuities. Figure 3 shows the distance function constructed by the third-order WENO scheme with the initial condition given by equation (11). Numerical results clearly indicate that the circle shrinks and the grid anisotropy are nearly invisible. However, there still exist the low level of errors which could create numerical noises in actual applications with a large number of iterations. Figure 3. Construction of the distance function using WENO scheme without volume conservation treatment. In order to evaluate the accuracy of numerical schemes, numerical error is computed by a following expression. nx ny error = i=1 j=1 φ n i,j φexact i,j nx ny Numerical errors of two schemes are compared at Table 1. Numerical errors of the first-order and third-order scheme without a volume conservation treatment are 6.57e-01 and 7.30e-03, respectively. Even if the third-order WENO scheme yields the much lower level of numerical error, numerical errors need to be minimized for the actual applications. In this aspect, to minimize numerical errors in a robust efficient way, this study employs the volume conservation treatment [10] described below.

Numerical studies on Hamilton-Jacobi equation 611 Table 1. Numerical error of two schemes with and without volume conservation treatment (16 16 Mesh) Error Orthogonal Grid Non-orthogonal Grid without volume with volume without volume with volume conservation conservation conservation conservation 1st upwind 6.57e-01 3.47e-02 1.93e+00 6.92e-02 WENO 7.30e-03 1.31e-03 7.28e-02 3.47e-02 Treatment for Volume Conservation The evolution equation for the interface must conserve the volume of the domain bounded by the curve defined implicitly by the equation (4). However, in actual computations this is not true anymore. The total volume surrounded by the zero level set φ 0 can be written as, V = Ω S(φ)dΩ (Ω = {φ Ω:φ(x, y) <φ 0 }) (16) Here, S(φ) is the smoothed sign function. Recently Sussman et al.[8] proposed a constraint for equation (4) to conserve the volume enclosed by the zero level set during the computational process of reinitialization. This can be achieved by requiring that S(φ)dΩ = 0 (17) t Ω where Ω is any fixed domain. By defining L(φ 0,φ) = S(φ 0 )(1 φ ) and inserting additional term to correct numerical errors, the original evolution equation can be modified as t = L(φ 0,φ)+λf(φ) (18) where f(φ) is any function and λ is a function of t only determined by requiring S(φ)dΩ = S (φ) t Ω Ω t dω= S (φ)(l(φ 0,φ)+λf(φ))dΩ = 0 (19) Ω Thus λ can be expressed as

612 Minyoung Yun λ = Ω S (φ)l(φ 0,φ)dΩ Ω S (φ)f(φ)dω We choose f(φ) =S (φ) φ to ensure that the correction is applied only at the interface without disturbing the function property distant from the interface. If equation (4) is solved exactly, λ will be zero. This is because L(φ 0,φ) will be zero in regions where S (φ) is not zero. In numerical computations, this is not true anymore, since the zero level set of φ 0 may differ from that of φ due to numerical error. Since the zero level set should be preserved by the reinitialization step, we need to numerically impose that the mass remain unchanged in any subset of the domain. For numerical purposes, the discretized formulation for the above equation need to preserve the volume in every grid cell, Ω ij. S (φ)(l(φ 0,φ)+λS (φ) φ )dω =0 Ω ij S (φ)l(φ 0,φ)dΩ+λ S (φ) 2 φ dω = 0 (20) Ω ij Ω ij λ ij = Ω ij S (φ)l(φ 0,φ)dΩ Ω ij S (φ) 2 φ dω The numerical integration over the domain { Ω ij = (x, y) x i 1 <x<x 2 i+ 1 2 is computed using a 9-point stencil: 1 g Ωij h2 16g ij + 24 where h 2 = x + y. m= 1;m =0n= 1;n =0 } and y i 1 <y<y 2 i+ 1 2 1 g i+m,j+n (21) (22) Figure 4 shows the construction of the distance function using the first order scheme with volume conservation. Numerical results clearly indicate that the volume conservation treatment greatly improves the solution quality of the first-order scheme and drastically reduces the departure from the initial level-set function. Compared to the first-order scheme without any treatment, the volume conservation procedure reduce numerical error(3.47e-02) with the order-ofmagnitude. However, this error is still higher than the error(7.30e-03) obtained by the third-order WENO scheme without a volume conservation treatment.

Numerical studies on Hamilton-Jacobi equation 613 Figure 4. Construction of the distance function using 1 st upwind scheme with volume conservation treatment Next we test the third-order WENO scheme with the volume conservation procedure. Figure 5 displays the construction of the distance function using the third-order WENO scheme with the volume conservation procedure. As would be expected, when the volume conservation procedure is imposed, the numerical deviations of the third-order WENO scheme are substantially reduced. In the orthogonal grid system, compared to the first-order scheme, the third-order WENO scheme with the volume conservation procedure yields the nearly perfect conformity with the initial level-set function and the much lower error(0.31e-03). Figure 5. Construction of the distance function using WENO scheme with volume conservation treatment 4. Discretization of Hamilton-Jacobi Equation in non-orthogonal coordinate and numerical Results

614 Minyoung Yun For the practical applications of the level set approach, we often have to numerically deal with the Hamilton-Jacobi equation in the non-orthogonal grid system. This study extends numerical schemes of the Hamilton-Jacobi equation to the first-order and higher-order scheme in the non-orthogonal grid system. The detailed formulations are given below. First-order upwind scheme In order to discretize the Hamilton-Jacobi equations in non-orthogonal generalized coordinate, the coordinate of mathematical formulation must be transformed from physical domain to computational domain expressed in terms of the following metrics and Jacobian. ξ x = Jy η, ξ y = Jx η, η x = Jy ξ, η y = Jx ξ, (23) 1 J =. x ξ y η y ξ x η The metric and Jacobians at computational faces and nodes are discretized as (1) East-West Face: x ξ = x i+1,j x i,j ξ = x i+1,j x i,j ( ξ = η =1), x η = x i+1,j+1 + x i,j+1 x i+1,j 1 x i,j 1 4 η = 0.25(x i+1,j+1 + x i,j+1 x i+1,j 1 x i,j 1 ), y ξ = y i+1,j y i,j, y η =0.25(y i+1,j+1 + y i+1,j y i 1,j+1 y i 1,j ), y η = y i,j+1 y i,j. (2) North-South Face: x ξ =0.25(x i+1,j+1 + x i+1,j x i 1,j+1 x i 1,j ), x η = x i,j+1 x i,j, y ξ =0.25(y i+1,j+1 + y i+1,j y i 1,j+1 y i 1,j ), y η = y i,j+1 y i,j. (3) Nodal Point: x ξ =0.5(x i+1,j x i 1,j ),

Numerical studies on Hamilton-Jacobi equation 615 x η =0.5(x i,j+1 x i,j 1 ), y ξ =0.5(y i+1,j y i 1,j ), y η =0.5(y i,j+1 y i,j 1 ). The coordinate of Hamilton-Jacobi equation can be transformed from the physical domain to the computational domain by utilizing the chain rule. x = ξ x ξ + η x η + ζ x ζ, y = ξ y ξ + η y η + ζ y ζ, (24) z = ξ z ξ + η z η + ζ z ζ. Substituting these relations into equation (4), φ t = t = sgn(φ 0)(1 φ ) = { [ ( ) 2 ( ) 2 ( ) 2 τ = sgn(φ 0) 1 α + β + γ ξ η ζ ( ) ( ) ( )] 1/2 } +l + m + n, ξ η ξ ζ η ζ where φ(ξ,η,ζ;0)=φ 0 (ξ,η,ζ; τ = 0) (25) α = ξx 2 + ξy 2 + ξz, 2 l =2(ξ x η x + ξ y η y + ξ z η z ), β = ηx 2 + η2 y + η2 z, m =2(ξ xζ x + ξ y ζ y + ξ z ζ z ), γ = ζx 2 + ζ2 y + ζ2 z, n =2(η xζ x + η y ζ y + η z ζ z ). In two-dimensional case, equation (25) is expressed as { [ ( ) 2 ( ) 2 ( t = sgn(φ ) ] } 1/2 0) 1 α + β + l, (26) ξ η ξ η where φ(ξ,η;0)=φ 0 (ξ,η) α = ξ 2 x + ξ 2 y, β = η 2 x + η 2 y, l =2(ξ x η x + ξ y η y ).

616 Minyoung Yun Since equation (4) is a hyperbolic equation, the upwinding flux in the ζ-direction is expressed as a = D ξ φ i,j = φ i,j φ i 1,j ξ, b = D + ξ φ i,j = φ i+1,j φ i,j, (27) ξ { ξ = max(a, b, 0), if φ 0 > 0 max( a, b, 0), if φ 0 < 0. (28) Figure 6. Non-orthogonal grid, 16 16 mesh and Ω. =[ 6, 6] [ 6, 6] Figure 7. Construction of the distance function using 1 st upwind scheme without volume conservation treatment in the non-orthogonal grid system; Dashed(160), Dash Dot(320), Dotted(480), Long Dash(640), Solid(800)

Numerical studies on Hamilton-Jacobi equation 617 Similarly, the upwinding fluxes in the η-direction and ζ-direction are obtained. To investigate the reinitialization process in non-orthogonal generalized coordinate, a circle with radius 4 and center location(0,0) has been chosen as the initial level-set function. Computations are based on 16 16 grid arrangement and the third-order Runge-Kutta method is used for the temporal integration. Figure 6 presents the computational mesh arrangement for the non-orthogonal grid system. Figure 7 shows numerical result obtained by the first-order upwind scheme utilizing equations (27) and (28). It can be clearly shown that there exist the considerable level of the circle shrinks and grid anisotropy. Compared to the solution of the orthogonal coordinate system, numerical results indicate that the circle shrinks and the anisotropy are much higher. In the first-order scheme in this non-orthogonal grid system, numerical error (1.93e+00) in Table 1 is much higher than numerical error (6.57e- 01) in the orthogonal grid system. This increased error could be mainly caused by the grid skewness of the non-orthogonal system. Figure 8 Construction of the distance function using 1 st upwind scheme with volume conservation treatment in the non-orthogonal grid system Figure 8 shows the construction of the distance function using the first order scheme with volume conservation in the non-orthogonal grid system. Similar to the orthogonal grid system, the volume conservation treatment greatly improves the solution quality of the first-order scheme and drastically reduces the departure from the initial level-set function. Compared to the first-order scheme without any treatment, the volume conservation procedure substantially reduces numerical error up to 6.29e-02. As would be expected, this numerical error is slightly higher than error (3.47e-02) in orthogonal grid arrangement.

618 Minyoung Yun High-order WENO scheme In order to minimize the numerical diffusion as well as to improve the numerical stability, this study adopts the second-order Lax-Friedrichs splitting scheme. Equation (4) can be expressed as the generalized form of the two-dimensional Hamilton-Jacobi equation described below. Here, the Hamiltonian, H can be written as t + H(φ x,φ y ) = 0 (29) H(φ x,φ y )=sgn(φ 0 )( φ 2 x + φ2 y 1). (30) Transforming the coordinate, equation (29) can be expressed as where t + H(φ ξ,φ η ) = 0 (31) H(φ ξ,φ η )=H(ξ x φ ξ + η x φ η,ξ y φ ξ + η y φ η ). (32) By applying the Lax-Friedrichs Scheme to equation (31), ( 1 φ n+1 i,j = φ n i,j t n H 2 (u+ i,j + u i,j ), 1 ) 2 (v+ i,j + v i,j ) + t n 2 A(u± i,j ; v± i,j )(u+ i,j u i,j ) + t n 2 B(u± i,j ; v± i,j )(v+ i,j v i,j ) (33) Here the notations of the abbreviated variables are explained below. u ± i,j = ± ξ φn i,j, v ± i,j = ± η φ n i,j. A(u ± i,j ; v± i,j { )=max H } 1 (u, v) u I(u,u + ),v [C, D] B(u ± i,j ; v± i,j { )=max H } 2 (u, v) v I(v,v + ),u [A, B] H 1 = ξ x H 1 + ξ y H 2, H2 = η x H 1 + η y H 2, (34)

Numerical studies on Hamilton-Jacobi equation 619 H 1 = H(φ x,φ y ) = sgn(φ 0 φ ) x, x φ 2 x + φ2 y H 2 = H(φ x,φ y ) = sgn(φ 0 φ ) y, y φ 2 x + φ2 y [A, B] = value range for (φ ξ ) i, [C, D] = value range for (φ η ) j. Here all the fluxes at face upwinded by the third-order WENO upwind scheme. Figure 9 shows numerical result obtained by the third-order WENO scheme without the volume conservation treatment. Compared to the first-order scheme without the volume conservation treatment, the circle shrinks and grid anisotropy are nearly disappeared. However, unlike the orthogonal grid system, there exists the noticeable departure from the initial level-set function. In the first-order scheme in this non-orthogonal grid system, numerical error (7.28e-02) in Table 1 is almost 10 times higher than numerical error (7.30e-03) in the orthogonal grid system. Figure 9. Construction of the distance function using WENO scheme without volume conservation treatment Figure 10 shows the construction of the distance function predicted by the third-order WENO scheme with volume conservation in the non-orthogonal grid system. Numerical results of the third-order WENO scheme in the nonorthogonal grid system clearly indicate that the departures from the initial level set function are nearly disappeared by the volume conservation treatment. In the non-orthogonal grid system, compared to the first-order scheme, the third-order WENO scheme with the volume conservation procedure yields the quite good

620 Minyoung Yun conformity with the initial level-set function and the much lower error (3.47e- 02). Due the grid skewness, this error in the non-orthogonal grid is almost 3 times higher than numerical error (1.31e-03) in orthogonal grid. Figure 10. Construction of the distance function using WENO scheme with volume conservation treatment 5. Conclusions In the reconstruction of the signed distance function in the context with the level set method, the present study has numerically investigated the effects of numerical schemes together with the volume conservation treatment as well as the level-set formulations based on the non-orthogonal grids. Based on numerical results, the following conclusion can be drawn. 1. Numerical results indicate that the present procedure together with the volume conservation treatment and the higher-order scheme successfully demonstrated the capability to reconstruct the signed distance function. Because of its simplicity, accuracy, and efficiency, the present scheme can be effectively used as a tool for the computation of a signed distance function, either for problems where this function is required or as an intermediate step in level set calculations. 2. Without the proper volume conservation treatment, the first-order results show the substantial degree of the circle shrinks and grid anisotropy. These numerical errors are increased with the number of iterations. Numerical diffusion associated with the first order scheme is mainly responsible for these deviations. When the third-order WENO scheme is used, these deviations are considerably reduced.

Numerical studies on Hamilton-Jacobi equation 621 3. When the proper volume conservation treatment is employed to correctly capture the interface, without disturbing the distance function property away from the interface, the volume shrinks and grid anisotropy are greatly reduced. Compared to the first-order scheme, the third-order WENO scheme predicts the much closer agreement with the initial level set. 4. In order to handle the complex geometry encountered in the real engineering problems, the level-set formulations based on the non-orthogonal grids have been successfully implemented. Numerical results of the third-order WENO scheme in the non-orthogonal grid system clearly indicate that the departures from the initial level set function are nearly disappeared by the volume conservation treatment. Compared to the first-order scheme, the third-order WENO scheme with the volume conservation procedure yields the much better conformity with the initial level-set function. Due the grid skewness, the numerical error in the non-orthogonal grid is times higher than error in the orthogonal grid. References 1. S. Chen, B. Merriman, S. Osher, and P. Smereka, A simple level set method for solving Stefan problems, J. Comput. Phys. 135(8)(1997). 2. M. Sussman, P. Smereka, and S. Osher, A level set method for computing solutions to incompressible two phase flow, J. Comput. Phys. 135(1994). 3. M. Sussman and P. Smereka, Axisymmetric free boundary problems, J. Fluid Mech. 341(1997). 4. D. Adalsteinsson and J.A. Sethian, An overview of level set methods for etching, deposition, and lithography development, IEEE Transactions on Semiconductor Devices 10(1)(1997). 5. F. Alvarez, F. Guichard, J.-M. Morel, and P.-L. Lions, Axioms and fundamental equations of image processing, Arch. Rat. Mech. and Analysis 123(1993), 199-257. 6. Mark L. Green, Statistics of images, the TV algorithm of Rudin-Osher-Fatemi for image denoising and an improved denoising algorithm, UCLA CAM Report 02(55)(2002) 7. Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79(1)(1988), 12-49. 8. Mark Sussman and Emad Fatemi, An Efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM Journal Scientific Computing 20(4), 1165-1191. 9. Guang-Shan Jiang and Chi-Wang Shu, Efficient Implementation of weighted ENO schemes, Journal of Computational physics 126(1996), 202-228. 10. Jianliang Qian and William W. Symes, An Adaptive Finite-difference method for travel times and amplitudes, Geophysics 67(1)(2002), 167-176. 11. J.A. Sethian, Level Set Methods and Fast Marching Methods: Evolving interface in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge Univ. Press (1999). 12. M.G. Crandall and P.L. Lions, Viscosity Solution of Hamilton-Jacobi Equations, Transaction of AMS 277(1983), 1-43.

622 Minyoung Yun 13. S. Osher and J.A. Sethian, Front Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, Journal of Computational Physics 79(1988), 12-49. Minyoung Yun received her BS from Chonbuk National University and Ph.D in Computer Science from the University of Alabama in Huntsville. In 1994 she was an assistant professor at the Alabama A & M University. Since 1995, she has been an associate professor at the Sungkyul University. Her research interests include algorithm analysis, ant colony optimization and numerical algorithm in image processing. Division of Computer and Communications Engineering, Sungkyul University, Anyang 430-742, Korea e-mail: alabama@sungkyul.edu