Finite difference Hermite WENO schemes for the Hamilton-Jacobi equations Feng Zheng, Chi-Wang Shu 3 and Jianian Qiu 4 Abstract In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton-Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical flues to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Etensive numerical eperiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme. Keywords: HWENO method; Hamilton-Jacobi equation; finite difference method. AMS(MOS) subject classification: 65M6, 35L65 The research of F. Zheng and J. Qiu is partially supported by NSFC grants 579 and NSAF grant U6347. The research of C.-W. Shu is partially supported by ARO grant W9NF-5--6 and NSF grant DMS-4875. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High- Performance Scientific Computing, Xiamen University, Xiamen, Fujian 365, P. R. China. E-mail: fzbz88-3@63.com. 3 Division of Applied Mathematics, Brown University, Providence, RI 9, USA. E-mail: shu@dam.brown.edu 4 School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling & High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 365, P.R. China. E-mail: jqiu@mu.edu.cn.
Introduction In this paper, we design a new finite difference Hermite weighted essentially non-oscillatory (HWENO) method to solve the following Hamilton Jacobi (HJ) equation t +H( ) =, (,) = () Ω R d where = (,,..., n ) T. We consider only up to two dimensions for simplicity and hence use and y instead of and in the sequel. The HJ equations come from many applications, from control theory and geometric optics, to image processingand level set method andso on. It is well known that solutions of the nonlinear HJ equation are typically continuous. However, the derivatives of the solutions could be discontinuous even though the initial condition () is smooth enough. The (weak) solution may not be unique unless suitable assumptions (viscosity solutions) are made, see, e.g. [4]. It is well known that HJ equations are closely related to conservation laws, so many successful numerical methods for solving conservation laws can be easily adapted to solve HJ equations. Hu and Shu [6] proposed a discontinuous Galerkin (DG) method to solve the HJ equation. Li and Shu [] reinterpreted and simplified the two dimensional method of Hu and Shu. Zhu and Qiu [, 3] used HWENO method for the HJ equations on both structured and unstructured meshes. Tao and Qiu [7] made use of the central HWENO method to solve the HJ equation successfully. As most of the finite volume methods resort to solving the conservation laws for the derivatives of the solution that seems to be not direct, Cheng and Shu [] designed DG methods to directly solve the HJ equation. Later, Yan and Osher [8] proposed a local DG (LDG) method to solve the HJ equation directly. In [3] Cheng and Wang improved the work in [] by utilizing the Roe speed and entropy fi at the cell interface, and based on [3], Zheng and Qiu [] developed HWENO schemes to directly solve the HJ equations. Comparing to the finite volume or DG methods, which evolve cell averages or a complete polynomial, the finite difference methods, which evolve only point values, may be easier to implement
and more efficient. Now, let us briefly review the early works. Crandall and Lion [5] introduced first order monotone schemes which can converge to the viscosity solution. Osher and Shu [] presented higher order finite difference ENO schemes, and Jiang and Peng [7] developed higher order finite difference weighted ENO (WENO) schemes. Qiu et al. [4, ] put forward Hermite WENO (HWENO) schemes for HJ equations with Runge-Kutta or La-Wendroff time discretization. Finite difference schemes on unstructured meshes have also been designed. Abgrall [] proposed first order monotone schemes on triangular meshes. Lafon and Osher [9] designed second order schemes. Higher order WENO schemes were developed by Zhang and Shu [9], and higher order HWENO schemes were proposed by Zhu and Qiu []. For a detailed review of high order HJ equations about both the finite difference and finite volume method, we recommend [5]. In this paper, following the methods developed in [3, 4, ], both and its first derivative (e.g. in one dimension) at the grid points are used in the HWENO method to reconstruct point values of the derivatives, and both of them are evolved by time marching. Comparing with the classical WENO method developed by Jiang and Shu in [8], the HWENO method requires etra work and storage but it is much more compact with the same order of accuracy. Comparing with DG methods, the HWENO method could achieve high order accuracy by adaptive stencils and hence could maintain the essentially non-oscillatory property. Notice that the HJ equations cannot be written in a conservation form, hence it would seem more natural to use finite difference methods based on point values instead of finite volume methods based on cell averages. In this way, we could avoid the costly multi-dimensional reconstructions and could use dimension-by-dimension interpolations, thereby reducing the computational cost and improving efficiency. It is important to use numerical flues(monotone Hamiltonians) to evolve the solution itself for HJ equations to ensure convergence to weak solutions. Since the derivatives of the HJ solution satisfy conservation laws, previous HWENO schemes tend to use conservative approimations to evolve them as well, hence causing complications and etra computational cost in multi-dimensional reconstruction. In this paper, we design a new HWENO scheme which evolves the solution itself through classical numerical 3
flues to ensure convergence to weak solutions, and evolves the derivatives of the solution in a nonconservative fashion, thereby allowing efficient dimension-by-dimension interpolation resulting in a gain of simplicity and efficiency. Etensive numerical eperiments are performed to illustrate the good performance of our schemes. This paper is organized as follows. In Section, we describe the detailed steps about construction and implementation of the finite difference HWENO schemes in both one and two dimensions for the HJ equations. In Section 3, we present etensive numerical results to verify the accuracy, stability and resolution of our method. Finally, a conclusion is given in Section 4. The numerical method for the Hamilton-Jacobi equations In this section, we will give the framework of the schemes first and then the detailed steps of the HWENO reconstructions for both one and two dimensional Hamilton-Jacobi equations.. One dimensional Hamilton-Jacobi equations The Hamilton-Jacobi equations in the one dimensional case can be written as: t +H( ) = (,) = () [a,b]. (.) For simplicity, we consider a uniform mesh that is defined as a = < < < N < N = b. However, thisassumptionisnotneededforourschemes. Wedenote j = ( j,t)asthenumerical approimation to the viscosity solution and u j = ( j,t) as the numerical approimation to its first derivative. Then, by taking the spatial derivative on both sides of (.), we obtain the following system of equations: d j dt = H( ) =j du j dt = H (u j )u =j where the H (u) = H u. We replace H( ) =j with a suitable monotone numerical flu, denoted by Ĥ(u j,u+ j ), and we will use the simple local La-Friedrichs flu as an eample in the sequel. For stability, we also split H (u j ) into a positive part and a negative part locally, denoted by H + j and (.) 4
H j respectively, and approimate u by upwind methods. Then, we have the following scheme: where d j dt = Ĥ(u j,u+ j ) du j dt = H+ j u j H j u+ j (.3) Ĥ(u j,u+ j ) = H(u j +u+ j ) α(u+ j u j ) (.4) with α = ma H (u), D = [min(u u D j,u+ j ),ma(u j,u+ j )], and H j = (H ( u j +u+ j ) H ( u j +u+ j ) ), H + j = (H ( u j +u+ j )+ H ( u j +u+ j ) ). (.5) u ± j andu± j,theleftandrightlimitsofthepointvaluesofu( j,t)andu ( j,t), willbeapproimated by the HWENO method which we will describe in detail in the net subsection. Notice that, even though we already have u j as part of our numerical solution, we would still want to obtain upwindbiased approimations u ± j through the point values of and u nearby (of course without using u j itself) in order to use the monotone flu to ensure stability and convergence to weak solutions. We observe that either H j or H+ j is equal to zero. In practical, we need to compute u j only when H + j is not equal to zero, and compute u+ j only when H j is not equal to zero. This will reduce the computational cost because only one of the u j and u+ j needs to be computed. After the spatial discretization, we can rewrite the scheme as U t = L(U), where L denotes the operator of the spatial discretization, and then use the third-order total variation diminishing (TVD) Runge-Kutta time discretization [6] to solve the semi-discrete form (.3): U () = U n + tl(u n ) U () = 3 4 Un + 4 (U() + tl(u () )) (.6) U n+ = 3 Un + 3 (U() + tl(u () )). HWENO reconstruction in one dimension u ± j. In this subsection, we will describe the HWENO reconstruction procedure for u ± j = ± j and 5
Step. Reconstruction of j by the HWENO method from the point values { i,u i }.. GiventhesmallstencilsS = { j, j, j },S = { j, j, j+ }, S = { j, j, j, j+ }, and the big stencil T = {S,S,S }, we construct Hermite cubic polynomials p (), p (), p (), and a fifth-degree polynomial q() such that: p ( j+i ) = j+i, i =,,, p ( j ) = u j p ( j+i ) = j+i, i =,,, p ( j+ ) = u j+ p ( j+i ) = j+i, i =,,,, q( j+i ) = j+i, i =,,,, q ( j± ) = u j± In fact, we only need the derivative values of these polynomials at the point = j, which have the following epressions: p ( j ) = j +4 j 5 i +4u j p ( j ) = j +4 j 5 j+ +u j+ 4 p ( j ) = j 6 j +3 j + j+ 6 q ( j ) = j +8 j 9 j j+ +9u j +3u j+ 8.. For each small stencil S m,m =,,, we compute the smoothness indicator, which measures the smoothness of the polynomials in each stencil: the smaller the indicator is, the smoother the polynomial is in the stencil. We use the formula similar to [8] to figure out the indicators, denoted by β m,m =,,: β m = 3 l= l ( l lp m( j )) m =,, (.7).3. We compute the linear weights, denoted by γ m,m =,,, satisfying: q ( j ) = γ m p m ( j ) m= for all the point values of { i } and {u i } in the big stencil T, which leads to γ = 4, γ = 3, γ = 5 6
.4. We compute the nonlinear weights based on the linear weights and the smoothness indicators by: ω m = ω m k= ω, ω m = k γ m (β m +ε), m =,, (.8) where ε is a small positive number to avoid the denominator becoming zero. In our numerical tests, we use ε = 6. The final HWENO approimation epression is: j = ω m p m ( j ), The reconstruction of u + j is mirror symmetric with respect to j of the above procedure. m= Step. Reconstruction of u j by the HWENO method from the point values { i,u i }.. GiventhesmallstencilsS = { j, j, j },S = { j, j, j+ }, S = { j, j, j, j+ }, and the big stencil T = {S,S,S }, we construct Hermite quartic polynomials p (), p (), p (), and a sith-degree polynomial q() such that: p ( j+i ) = j+i, i =,,, p ( j+i) = u j+i, i =, p ( j+i ) = j+i, i =,,, p ( j+i ) = u j+i, i =, p ( j+i ) = j+i, i =,,,, q( j+i ) = j+i, i =,,,, p ( j ) = u j q ( j+i ) = u j+, i =,, Again, we only need the second order derivative values of these polynomials at the point j, which have the following epressions: p ( j ) = u j +8u j + j +6 j 7 j ( ) p ( j ) = 4u j u j+ + j 8 j +7 j+ ( ) p ( j ) = 6u j j + j 5 j +4 j+ 6( ) q ( j ) = 8u j +8u j 6u j+ + j +54 j 8 j +6 j+ 8( ).. For each small stencil S m,m =,,, we compute the smoothness indicators respectively: β m = 4 l=3 l ( l lp m( j )), m =,, (.9) 7
.3. We compute the linear weights, denoted by γ m,m =,,, satisfying: q = γ m p m ( j ) m= for all the point values of { i } and {u i } in the big stencil T, which leads to γ = 4, γ = 3, γ = 5.4. We compute the nonlinear weights ω m as in (.8), the final HWENO approimation epression is: u j = ω m p m ( j ), m= The reconstruction of u + j is mirror symmetric with respect to j of the above procedure..3 Two dimensional Hamilton-Jacobi equations The Hamilton-Jacobi equation in the two dimensional case is written as t +H(, y ) = (,y,) = (,y) (,y) [a,b] [c,d] (.) For simplicity, we also assume the computational domain has been uniformly meshed as a = < < < N < N = b and c = y < y < < y N < y N = d. Furthermore, we define ij = ( i,y j,t) as the numerical approimation to the viscosity solution, and u ij = ( i,y j,t) and v ij = y ( i,y j,t) as the numerical approimations to its first order partial derivatives with respect to the variables and y, respectively. Then, by taking spatial derivatives on both sides of (.), we have the following system of equations: d ij = H(, y ) dt du ij = H (u,v)u H (u,v)v dt dv ij = H (u,v)u y H (u,v)v y dt (.) 8
where the H (u) = H u and H (u) = H v. As u y is equal to v in the smooth case, we can also rewrite the system of equations as the following: d ij = H(, y ) dt du ij = H (u,v)u H (u,v)u y dt dv ij = H (u,v)v H (u,v)v y dt (.) In the same way as before, we replace H(, y ) with a monotone numerical flu, and split H (u,v) and H (u,v) into a positive part and a negative part respectively. Then we discretize (.) into the following d ij dt du ij dt dv ij dt = Ĥ(u ij,u+ ij,v ij,v+ ij ) = H + ij u ij H ij u+ ij H iju y ij = H ij v ij H + ij v y ij H ij v+ y ij (.3) where Ĥ(u ij,u+ ij,v ij,v+ ij ) refers to a two-dimensional monotone flu, such as the simple local La- Friedrichs flu defined as Ĥ(u ij,u+ ij,v ij,v+ ij ) = H(u ij +u+ ij, v ij +v+ ij ) α(u+ ij u ij ) β(v+ ij v ij ), where α = ma u D,v E H (u,v) and β = ma u D,v E H (u,v). With regard to α, we take D as a local region and E as a global region, namely D = [min(u ij,u+ ij ),ma(u ij,u+ ij )] and E = [min(v,v + ),ma(v,v + )] [c,d]. We compute the coefficient β similarly, ecept that we then take D globally and E locally. This way of computing the local La-Friedrichs flu is needed to ensure monotonicity, see []. H ij, H ij are the average values of H and H at ( i,y j ) defined as the following: H ij = H ( u ij +u+ ij, v ij +v+ ij ), H ij = H ( u ij +u+ ij, v ij +v+ ij ), and H, H+, H, H+ as the negative and positive parts of H and H respectively, defined as follows: H ij = (H ij H ij ), H + ij = (H ij + H ij ), 9
H ij = (H ij H ij ), H + ij = (H ij + H ij ). ± ij, u± ij and v± ij are the left and right limits of the point values u( i,y j,t), u ( i,y j,t) and v ( i,y j,t) with respect to the variable, and ± y ij, u ± y ij and v y ± ij are the left and right limits of the point values v( i,y j,t), u y ( i,y j,t) and v y ( i,y j,t) with respect to the variable y. The values of ± ij, ± y ij, u ± ij and v± y ij are reconstructed with the one dimensional method in each direction with the other direction fied. As to the mied derivatives v,u y, we simply use the following fourth order central approimations in the and y directions, respectively: v ij = v i+,j +8v i+,j 8v i,j +v i,j, u yij = u i,j+ +8u i,j+ 8u i,j +u i,j. y In practice, we only need to compute either u ij or u+ ij but not both. v y ij and v + y ij can be computed similarly. Then, we can rewrite the scheme (.3) as U t = L(U), where L denotes the operator of spatial discretization, and use the third-order total variation diminishing (TVD) Runge-Kutta time discretization (.6) to solve the semi-discrete form (.3). 3 Numerical results In this section, we provide numerical eperiments for the fifth order HWENO method in one and two dimensional cases. In all the accuracy tests, we set t =.6 5 3/α in the one dimensional case and t = α.6 5 3 + β.6 y 5 3 in the two dimensional case, in order to guarantee that spatial numerical errors dominate. In other tests, we simply take t =.6 /α and t = α.6 + β.6 y and two dimensional cases respectively, unless otherwise indicated. 3. One dimensional case: accuracy test in the one Eample 3.: We solve the following linear scalar equation: t + =,
with the initial condition (, ) = sin(π), and periodic boundary condition. We compute the solution upto t =, thenumerical errorsandnumerical ordersof accuracy for thehweno method are shown in Table. We can see that the scheme achieves or eceeds fifth order accuracy. Table : t + =,(,) = sin(π). Periodic boundary conditions. t = N L error L order L error L order L error L order.58e-.44e-.6e- 9.4E-4 4.77 5.67E-4 4.67 4.4E-4 4.59 4.75E-5 5..85E-5 4.93.56E-5 4.8 8 8.34E-7 5.5 5.7E-7 5. 4.95E-7 4.98 6.94E-8 5.43.3E-8 5.54.E-8 5.6 3.67E- 6.86.3E- 6.9 8.59E- 6.89 Eample 3.: We solve the Burgers equation: t + ( +) =, with the initial condition (, ) = cos(π), and periodic boundary condition. We compute the solution up to t =.5 π. At this time, the solution is still smooth. The numerical results are shown in Table. Again, we can see that the scheme can reach its designed order of accuracy. Table : t + ( +) =,(,) = cos(π). Periodic boundary conditions. t =.5/π N L error L order L error L order L error L order.e-3 9.78E-4 8.4E-4.64E-4 3.75 5.7E-5 4.7 3.4E-5 4.63 4.7E-5 3.69 3.E-6 4.3.88E-6 4. 8 5.E-7 4.63.7E-7 4.85 5.95E-8 4.98 6.7E-8 4.9.9E-9 5..5E-9 5.8 3.87E- 5.89 5.8E- 5.8.93E- 5.76 Eample 3.3: We solve the nonlinear scalar equation: t cos( +) =, < < with the initial condition (,) = cos(π), and periodic boundary condition. When t =.5 π, the solution is still smooth. The numerical results are shown in Table 3. Again, we observe that the scheme can achieve its designed accuracy.
Table 3: t cos( +) =,(,) = cos(π). Periodic boundary conditions. t =.5/π N L error L order L error L order L error L order.84e-3 9.56E-4 7.7E-4 3.87E-4.5.E-4 3.9 5.6E-5 3.78 4 6.53E-5.57.7E-5 3.6 3.9E-6 3.84 8 5.7E-6 3.5 7.8E-7 4..97E-7 4.3 6.49E-7 4.5.96E-8 4.6 7.3E-9 4.75 3 6.4E-9 5.8 6.8E- 5.44.8E- 5.33 3. One dimensional case: discontinuous derivatives Eample 3.4: We solve the linear equation: t + = with the initial condition (,) = (.5) together with the periodic boundary condition, where () = ( 3 + 9 + π 3 )(+)+ cos( 3π ) 3 < 3, 3 +3cos(π) <, 3 5 3cos(π) < 3, 8+4π +cos(3π) 3 +6π( ) <. 3 We plot the result at t =. and t = 8. in Figure (a) and Figure (b) respectively, and observe that the HWENO schemes have good resolution for the corner singularity. Eample 3.5: We solve one dimensional Burgers equation: t + ( +) = with (,) = cos(π) and periodic boundary conditions. We compute the solution up to t = 3.5/π. Atthistime, thediscontinuousderivativehasalreadyappearedinthesolution. Weshowthe numerical results with the meshes N = 4 and N = 8 in Figure (a) and Figure (b) respectively. Again, we can see that the scheme gives high resolution for this problem.
- - T= T=8 - HWENO Eact - HWENO Eact -3-3 -4-4 -5-5 - -.5.5 (a) - -.5.5 (b) Figure : One dimensional linear equation. N = cells. (a) t =., (b) t = 8.. Solid line: the eact solution; Square symbol: the HWENO scheme. N=4 N=8 -. HWENO Eact -. HWENO Eact -.4 -.4 -.6 -.6 -.8 -.8 - - -. - -.5.5 (a) -. - -.5.5 (b) Figure : One dimensional Burgers equation. t = 3.5/π. (a) N = 4 and (b) N = 8. Solid line: the eact solution; Square symbol: the HWENO scheme. 3
Eample 3.6: We solve the nonlinear equation with a non-conve / non-concave flu t cos( +) = with the initial data (, ) = cos(π) and periodic boundary conditions. This time, we compute the solution up to t =.5/π. We observe the results with N = 4 and N = 8, which are shown in Figure 3, and find that the method gives high resolution in this case. N=4 N=8.5 HWENO Eact.5 HWENO Eact -.5 -.5 - - -.5.5 (a) - -.5.5 (b) Figure 3: H(u) = cos(u + ). t =.5/π. (a) N = 4 and (b) N = 8. Solid line: the eact solution; Square symbol: the HWENO scheme. Eample 3.7: We solve the problem t + 4 ( )( 4) = < < with the initial data (,) =. As the derivative of (,) is discontinuous, the initial value of u is undefinedat = which is a grid point. We simply take u(,) = in our code, and observe that the value of u(,) makes little influence on the final result in our numerical eperiment. We plot the results at t = with N = 4 and N = 8 cells in Figure 4, and can see that the scheme gives good results for this problem. 4
- - -. N=4 -. N=8 -.4 HWENO Eact -.4 HWENO Eact -.6 -.6 -.8 -.8 - - -.5.5 (a) - - -.5.5 (b) Figure 4: H(u) = (/4)(u )(u 4). t =. (a) N = 4 and (b) N = 8. Solid line: the eact solution; Square Symbol: the HWENO scheme. 3.3 Two dimensional case: accuracy test Eample 3.8: We solve the two dimensional linear scalar equation t + + y =,y with the initial data (,y,) = sin(π( + y)/) and periodic condition. The result at t = is listed in Table 4. We see that the scheme can achieve the designed accuracy. Table 4: t + + y =,(,y,) = sin(π(+y)/). Periodic boundary conditions. t = N L error L order L error L order L error L order.85e-.75e-.45e-.6e-3 4.6 7.38E-4 4.57 6.3E-4 4.57 4 4 3.6E-5 5..5E-5 4.88.8E-5 4.8 8 8.E-6 5. 7.8E-7 5. 6.88E-7 4.98 6 6.87E-8 5.9.89E-8 5.37.6E-8 5.4 3 3 4.6E- 5.96 3.E- 5.93.73E- 5.89 Eample 3.9: We solve the two dimensional Burgers equation t + ( + y +) =,y 5
with the initial data (,y,) = cos( π ( + y)) and periodic boundary condition. We compute the result up to t =.5/π and the solution is still smooth at that time. Again, from Table 5, we can see that the scheme achieves its designed order. Table 5: t + ( + y + ) =, (,y,) = cos( π ( + y)). Periodic boundary conditions. t =.5/π N L error L order L error L order L error L order.78e-3 9.6E-4 8.3E-4 3.64E-4.9 9.8E-5 3.9 5.49E-5 3.9 4 4 3.E-5 3.5 6.7E-6 3.97.97E-6 4. 8 8.36E-6 4.56.5E-7 4.64.6E-7 4.8 6 6 4.5E-8 4.9 8.44E-9 4.9 3.7E-9 5.6 3 3 9.86E- 5.5.9E- 5.33 8.7E- 5.9 Eample 3.: We solve t cos( + y +) =,y with the initial data (,) = cos( π (+y)) and periodic boundary condition. We compute the result up to t =.5/π and the solution is still smooth at that time. Again, from Table 6, we can see the scheme achieves the designed order of accuracy. Table 6: t cos( + y + ) =, (,y,) = cos( π ( + y)). Periodic boundary conditions. t =.5/π N L error L order L error L order L error L order.35e-3.5e-3 7.63E-4 3.E-4.87 9.7E-5 3.43 5.39E-5 3.8 4 4 5.59E-5.53.6E-5 3.9 4.7E-6 3.7 8 8 5.E-6 3.48 6.E-7 4.4.7E-7 4.6 6 6.8E-7 4.46.67E-8 4.49 7.77E-9 4.47 3 3 6.8E-9 5.8 8.4E- 5.5.83E- 4.78 3.4 Two Dimensional case: discontinuous derivatives Eample 3.: We solve the two dimensional Burgers equation t + ( + y +) =,y 6
with the initial data (,y,) = cos( π ( + y)) and periodic boundary condition. We compute the result up to t =.5/π and the derivative discontinuity has appeared in the solution. We plot the results with 4 4 cells in Figure 5 and observe high resolution in this eample. Z Y X y.5 - -.5 y - - - - - - - - (a) (b) Figure 5: Two dimensional Burgers equation. t =.5/π with the HWENO schemes with 4 4 cells. Contours of the solution in Figure 5(a) and the surface of the solution in Figure 5(b) Eample 3.: We solve a problem from optimal control: t +sin(y) +(sin()+sign( y )) y sin (y)+cos() =, π <,y < π with (,y,) = and periodic boundary conditions. In this case, the equation can be denoted as t +H(, y,,y) =, and our scheme can be obtained through the following system of equations as before: d ij = H(, y,,y) dt du ij = H (u,v,,y)u H (u,v,,y)u y H (u,v,,y) dt dv ij = H (u,v,,y)v H (u,v,,y)v y H y (u,v,,y) dt Following (.3), we can easily deal with the terms H(, y,,y),h (u,v,,y),h (u,v,,y). The additional terms H (u,v,,y) and H y (u,v,,y) can be approimated as H (u,v,,y) =i,y=y j H ( u ij +u+ ij 7, v ij +v+ ij, i,y j )
H y (u,v,,y) =i,y=y j H y ( u ij +u+ ij, v ij +v+ ij, i,y j ) The time step is still taken as t = α. The solution and optimal control ω = sign() at.6 + β.6 y t = are plotted in Figure 6(a) and Figure 6(b), respectively. Again, we can observe that high resolution is achieved by our method. Z Z Y Y X X.5.5 3 ω -.5.5 - -3 - - 3-3 - - y 3 y - -3 - - (a) (b) Figure 6: Theoptimal control problem. t = with the HWENO schemes with 6 6 cells. Surface of the solution in Figure 6(a) and of the optimal control ω = sign( y ) in Figure 6(b) Eample 3.3: We solve the problem t cos( + y +) =, <,y < with (,y,) = cos( π (+y)) and periodic boundary conditions. We compute the solution up to t =.5/π and the solution has already developed discontinuous derivatives. The results are shown in Figure 7. We observe high resolutions for this eample. Eample 3.4: We solve the problem with another neither conve nor concave Hamiltonian t +sin( + y ) =, <,y < with (,y,) = π( y ). In this case, u(,y,) is undefined along = and v(,y,) is undefined along y =, and we simply take u(,y,) = and v(,,) = respectively. We 8
Z Y X y.5 - -.5 y - - - - - - - - (a) (b) Figure 7: Two dimensional equation with a neither conve nor concave Hamiltonian. t =.5/π with the HWENO schemes with 4 4 cells. Contours of the solution in Figure 7(a) and surface of the solution in Figure 7(b) compute the solution up to t =. The solution is shown in Figure 8. Again, we observe our schemes can achieve high resolution. Z Y X.5 y -.5 - -.5.5 - - -.5.5 y -.5 - - -.5 (a) (b) Figure 8: Two dimensional equation with a neither conve nor concave Hamiltonian. t = by the HWENO schemes with 4 4 cells. Contours of the solution in Figure 8(a) and surface of the solution in Figure 8(b) 9
Eample 3.5: We solve the two dimensional Eikonal equation t + + y + =,,y < with the initial data (,y,) = 4 (cos(π) )(cos(πy) ). We compute the solution up to t =.6. The solution is shown in Figure 9. High resolutions are observed with our scheme. Z X Y.8 -.4.6 -.45 y.4 -.5. -.55..4.6.8 (a). -.6.4..4 y.6.8.8.6 (b) Figure 9: Eikonal equation. t =.6 by the HWENO schemes with 6 6 cells. Contours of the solution in Figure 9(a) and surface of the solution in Figure 9(b) Eample 3.6: We solve t ( εk) + y + =,,y < where K is the mean curvature defined by: (,y,) = 4 (cos(π) )(cos(πy) ) K = (+ y ) y y + yy (+ ) (+ + y )3/ andεisasmallconstant, withtheinitial data(,y,) = 4 (cos(π) )(cos(πy) ) andperiodicboundarycondition. Whenε, theequationcanbedenotedas t +H(, y,, y, yy ) =
. The system of equations to be approimated then becomes d ij = H(, y,, y, yy ) dt du ij = H u H y u y H u H y u y H yy u yy dt dv ij = H v H y v y H v H y v y H yy v yy dt The procedure to deal with the terms H, H and H y are the same as before. Here, we simply use fourth order central differences in each direction with the other direction fied to approimate the terms u,u yy : u ij = u i+,j 6u i+,j +3u i,j 6u i,j +u i,j u yyij = u i,j+ 6u i,j+ +3u i,j 6u i,j +u i,j y As to the term u y, we take the following fourth order approimation using the values {u kl,k = i,,i+,l = j,,j +}: u y ij = 6 y (74u i+,j +63u i+,j +63u i,j+ +74u i,j+ 74u i+,j+ 63u i+,j+ 44u i,j+ +63u i,j+ 63u i+,j+ +44u i+,j+ +44u i,j 63u i,j +63u i+,j 44u i+,j 63u i,j 74u i,j ). The approimation to the terms v,v y,v yy can be obtained in a similar way. The time step is taken as t = α, where γ.6 + β.6 y + γ.3 + γ.3 y + γ 3 = ma H, γ = ma H y,.3 y y γ 3 = ma H yy and α, β are defined the same as before. The results of ε = (pure convection) yy and ε =. by the HWENO method with 6 6 cells are presented in Figure (a) and Figure (b) respectively. The surfaces at t = for ε = and for ε =., and at t =. for ε =., are shifted downward in order to show the details of the solution at later time. 4 Conclusion In this paper, we present a high order scheme based on the finite difference framework for the Hamilton-Jacobi equations in one and two dimensions. The main advantage of this scheme is
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