A new type of Hermite WENO schemes for Hamilton-Jacobi equations

Transcription

1 A new type of Hermite WENO schemes for Hamilton-Jacobi equations Jun Zhu and Jianxian Qiu Abstract In this paper, a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes is designed for solving the Hamilton-Jacobi equations. The crucial idea is borrowed from the classical HWENO schemes of Qiu and Shu [Hermite WENO schemes for Hamilton- Jacobi equations, Journal of Computational Physics, 4(5), 8-99], in which the function and its first derivative values are also evolved in time and used in the spatial reconstructions. New HWENO reconstructions with three unequal-sized spatial stencils are designed and thus serve crucial role in this paper, which are basically different from any existing HWENO schemes. These reconstructions result in an important innovation that we perform only spatial HWENO reconstructions for numerical fluxes of function φ, and high-order linear reconstructions for numerical fluxes of derivatives u and v. So such new HWENO schemes are more simple and efficient than any other HWENO schemes. And the new HWENO schemes could obtain less truncation errors in smooth regions and keep sharp transitions near strong discontinuities. Extensive benchmark examples are performed to illustrate the good performance of these new schemes. Key Words: HWENO scheme, unequal-sized spatial stencil, high-order linear reconstruction, HWENO reconstruction, Hamilton-Jacobi equation. AMS(MOS) subject classification: 65M6, 5L65 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 6, P.R. China. zhujun@nuaa.edu.cn. The research of J. Zhu is partly supported by NSFC grant 87. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, iamen University, iamen, Fujian 65, P.R. China. jxqiu@xmu.edu.cn. The research of J. Qiu is partly supported by NSFC grant 579 and NSAF grant U647.

2 Introduction In this paper, a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes is presented for solving the Hamilton-Jacobi equations { φt H(x,,x n,t,φ,φ x,...,φ xn ) =, (x,...,x n,t) Ω [, ), φ(x,...,x n,) = φ (x,...,x n ), (x,...,x n ) Ω. (.) Such Hamilton-Jacobi equations often take into effect in the applications of image processing, geometric optics, material science, computer vision, and so on [7, 9, 9]. Although the exact solutions φ(x,...,x n,t) for simulating (.) are continuous, their derivatives might contain discontinuities. And it is well known that many Hamilton-Jacobi equations are multi-solution equations unless we apply the physical implications and then could obtain their viscosity solutions uniquely []. Since the Hamilton-Jacobi equations are of close relationship with the conservation laws, We often resolve the solutions of the Hamilton-Jacobi equations by the application of the solutions of the conservation laws. And the classical numerical schemes for simulating the conservation laws can be directly adapted or with minor modifications for solving the Hamilton- Jacobi equations. Following such principles, Osher et al. [, ] designed some high-order accurate essentially non-oscillatory (ENO) schemes for solving the Hamilton-Jacobi equations. Lafon and Osher [5] constructed ENO schemes for solving the Hamilton-Jacobi equations on unstructured meshes. Then Jiang and Peng [] proposed classical high-order finite difference weighted ENO (WENO) schemes for solving the Hamilton-Jacobi equations on structured meshes, which are excellent expanding application of the basic fifth-order finite difference WENO schemes presented by Jiang and Shu [] for the conservation laws. Some high-order WENO schemes [, 8, 4, 5] were proposed for two-dimensional Hamilton- Jacobi equations with the application of the nodal WENO-type algebraic polynomial interpolations on unstructured meshes. After that, a new methodology of discontinuous Galerkin (DG) method for the conservation laws with modification to solve for the Hamilton-Jacobi equations was proposed by Hu and Shu []. Then Cheng and Shu [8] presented direct

3 DG methods for solving the Hamilton-Jacobi equations for φ(x,...,x n,t) with some remedy procedures originated from [] in extreme cases. an together with Osher [] designed a local DG method for directly solving the Hamilton-Jacobi equations precisely with sufficient applications of auxiliary variables. And Qiu et al. [5, 6, 7, 8, 6, 8] devoted their efforts to construct the compact Hermite WENO schemes whose solutions and firstorder derivative values are applicable via time approaching for computing the conservation laws and the Hamilton-Jacobi equations successfully. Some classical HWENO schemes et al. [4, 5,,,, 4] were also designed for solving the hyperbolic conservation laws effectively. One major advantage of HWENO schemes superior to WENO schemes is their compactness in reconstruction procedures. We do remark that all existing HWENO schemes are more costly in computation and storage than that of the same order accurate WENO schemes on the same computational meshes, since the solution and its derivatives need to be stored and evolved in time. So we will propose new spatial reconstruction methodologies to remedy such drawback in this paper. This continuing paper based on [5, 8] is of interest to study the high-order simulation methods for the Hamilton-Jacobi equations with some significant improvements, obeying the basic reconstruction principles of HWENO schemes specified in [6, 7], and is a direct extension of [4] from the finite difference WENO framework to the mixture of the finite difference and finite volume HWENO frameworks. The reconstruction procedures are briefly narrated as follows, which will extremely decrease the computing time than any other existing HWENO schemes. For example, in one-dimensional case, for the purpose of obtaining high-order approximations of φ x at the midpoint of the target cell, we construct a quintic polynomial depending on four nodal values and two cell averages of the x-derivative values of φ, and similarly construct two quadratic polynomials depending on three nodal values of φ. Then we get first-order derivative functions of such three unequal degree polynomials and obtain their values at the midpoint of the target cell, then modify the derivative function of the quintic polynomial by the other two derivative functions with suitable positive param-

4 eters, and reassemble such three unequal first-order derivative polynomials in a HWENO type manner including artificially setting positive linear weights, computing smoothness indicators, and redesigning new formula of the nonlinear weights [, 6, 9]. Hereafter, in order to get high-order approximations of φ x at the left and right edge points of the target cell, we construct another quintic polynomial depending on four nodal values and two cell averages of the derivative values of φ and obtain its first-order derivative to get the values at associated points of the target cell. After performing these procedures, we solve the ODEs with a third-order TVD Runge-Kutta time discretization method [] to obtain a new HWENO scheme both in space and time. Here we would like to emphasize that such new HWENO schemes are very efficient and could keep good convergence property than the classical HWENO schemes [6, 7, 8] for obtaining the same order accuracy on the same computational meshes, since spatial HWENO reconstructions and high-order linear reconstructions (i.e., reconstruction with constant coefficients) are also used in this paper, while only the spatial HWENO reconstructions are used all the way in any other HWENO schemes et al. [4, 5,,, 5, 6, 7, 8,, 4, 6, 8]. And it is also shown that the new HWENO schemes could efficiently get less absolute truncation errors in all accuracy test cases in comparison with that specified in [8]. Generally speaking, let us mention a few features and advantages of these new HWENO schemes: the first is the linear weights can be any positive numbers on condition that their sum is one, the second is their simplicity and easy extension to multi-dimensions, the third is the HWENO spatial reconstructions with three unequal-sized spatial stencils and high-order linear reconstructions are used, different from all the other existing HWENO schemes which perform the HWENO spatial reconstructions for all variables and are more costly in coding, computation, storage, and so on. The organization of this paper is as follows. In Section, we review and construct the new HWENO schemes in detail for solving the Hamilton-Jacobi equations in one and two dimensions and present extensive numerical results in Section to illustrate the simplicity, 4

5 accuracy, and efficiency of these new approaches. Concluding remarks are given in Section 4. HWENO schemes for Hamilton-Jacobi equations In this section, we give the framework for solving the Hamilton-Jacobi equations briefly and then develop the procedures of the new fifth-order accurate HWENO schemes for both one- and two-dimensional Hamilton-Jacobi equations in detail.. The framework for one-dimensional case The control equations (.) are studied in one dimension. For simplicity, the computational domain Ω has been divided as a uniform mesh with intervals I i = [x i /,x i/ ], i =,...,N. We denote x i = (x i / x i/ ) to be the interval center and I i = h is the length of I i, respectively. (.) can be rewritten in one dimension as { φt H(x,t,φ,φ x ) =, φ(x,) = φ (x). (.) Define u(x,t) = φ x (x,t) and take into account the first-order derivative of (.) in one dimension, the conservation laws are obtained as { ut H(u) x =, u(x,) = u (x). (.) We define φ i = φ i (t) = φ(x i,t) and the cell average ū i (t) = h integrate (.) over the target cell I i and get { dφi (t) = H(u(x dt i,t)), dū i (t) = H(u(x i/,t)) H(u(x i /,t )). dt h xi/ x i / u(x,t)dx. Then we (.) Hereafter, we approximate (.) by such formula { dφi (t) dt = H i, dū i (t) dt = Ĥi/ Ĥ i / h, (.4) 5

6 where H i and Ĥi±/ are the numerical fluxes satisfying common conditions of monotonicity, Lipschitz continuity and consistency with the physical flux H(u) et al. The simple Lax- Friedrichs fluxes are defined as and H i = H( u i u i ) α (u i u i ), (.5) Ĥ i/ = (H(u i/ )H(u i/ ) α(u i/ u i/ )), (.6) where α = max u H (u), u ± i, andu± i/ arethenumerical approximations to thenodalvalues of u(x i,t) and u(x i/,t), respectively. For simplicity, the variable t is omitted if it does not cause confusion in the following. The method of lines ODE (.4) is rewritten as the formula d w(t) = L(w). (.7) dt Then we use a third-order TVD Runge-Kutta time discretization method [] w () = w n tl(w n ), w () = 4 wn 4 w() 4 tl(w() ), w n = wn w() tl(w() ), (.8) to solve for (.7) and obtain fully discrete scheme both in space and time. The key component of the HWENO schemes is the reconstruction, from point values φ i and the cell averages ū i to the points values {u ± i,u± i/ }. This reconstruction should be both high-order accurate and essentially non-oscillatory. We outline the procedure of this reconstruction for the fifth-order case in the following. Algorithm. The procedure to reconstruct u i. Step. Select a big spatial stencil T = {x i,x i,x i,x i,i i,i i }, which is a combination of four points and two intervals. Then we construct a Hermite quintic polynomial p (x) satisfying p (x j ) = φ j,j = i,i,i,i, (.9) 6

7 and p h (x)dx = ū j, j = i,i. (.) I j Its first-order derivative function at x i is explicitly formulated as p (x i) = 6φ i φ i 5φ i 7φ i hū i 8hū i. (.) 5h Step. Select two smaller spatial stencils T = {x i,x i,x i } and T = {x i,x i,x i }, which are combinations of three points, respectively. Then we construct two quadratic polynomials p (x) and p (x) satisfying and p (x j ) = φ j,j = i,i,i, (.) p (x j ) = φ j,j = i,i,i. (.) Their first-order derivative functions at x i are explicitly narrated as and p (x i ) = 4φ i φ i φ i, (.4) h p (x i) = φ i φ i. (.5) h Remark : Comparing with [8], we select the same big spatial stencil T and different two smaller stencils T and T for constructing three polynomials. Only three nodal point values of φ are used to obtain biased quadratic polynomials p (x) and p (x) with respect to thetargetpointx i,whichmeansnocellaveragesofthederivativevaluesofφareintroducedin comparison with that specified in [5, 8]. By performing such new HWENO reconstruction procedures, it is the first time that we can apply the information of φ and ū to get high-order approximation in smooth regions and only apply the information of φ without introducing any ū to keep essentially non-oscillatory property in nonsmooth regions. Step. With the similar idea by Levy, Puppo, and Russo [6, 7] for central WENO methods, we rewrite rewrite p (x i) as ( p (x i ) = γ p γ (x i ) l= γ l γ p l(x i ) ) γ l p l(x i ). (.6) l= 7

8 Clearly, the equality (.6) holds for arbitrarily chosen positive linear weights γ, γ, and γ satisfying l= γ l=. Following the practice in [, 7, 9, 4, 4], for example, we set one type of the linear weights as γ =.98 and γ =γ =. in this paper. Step 4. Compute the smoothness indicators β n, which measure how smooth the functions p n(x)arenearthetargetpointx i. Thesmaller thesesmoothnessindicators, thesmoother the functions are near the target point. We use the similar recipe for the smoothness indicators as in [,, 5]: β n = r α= I i h α ( ) d α p n (x) dx,n =,,. (.7) dx α r equals five for n = and r equals two for n =,, respectively. The associated smoothness indicators are explicitly written and the their expansions in Taylor series at x i are obtained as β = (4487φ i 8φ i 8φ i φ i φ i 84459φ i φ i 945φ i φ i 955φ i 8h5 (7φ i φ i 5φ i 5φ i ) 4φ i (5φ i 455φ i 46465φ i ) 6h 6 (5ū i ū i ) 4h(8865φ i ū i 4985φ i ū i 5575φ i ū i 675φ i ū i 749φ i ū i 764φ i ū i 4565φ i ū i 6895φ i ū i )h ( 47675ū i 977ū i ū i 677ū i ))/(5h ) 4(φ() = h (φ () i ) i ) φ () i φ (4) i h O(h 6 ), (.8) β = ( φ i φ i φ i ) /h = h (φ () i ) h φ () i φ () i O(h 4 ), (.9) i φ (4) i 4φ() β = (φ i φ i φ i ) /h = h (φ () i ) h 6 O(h 6 ). (.) Step 5. Compute the nonlinear weights based on the linear weights and the smoothness indicators. For instance, as shown in [, 6, 9], we use new τ which is simply defined as the absolute difference between β, β, and β. So the associated difference expansions in Taylor 8

9 series at x i are ω n = ( ) β β β β τ = = O(h 6 ), (.) ( ω n l= ω, ω n = γ n τ ), n =,,. (.) l εβ n Here ε is a small positive number to avoid the denominator to become zero. It is directly to verify from (.8) to (.) and the parameter τ in the smooth region of the numerical solution satisfying τ εβ n = O(h 4 ),n =,,, (.) on condition that ε β n. Therefore, the nonlinear weights ω n obviously satisfy the order accuracy conditions ω n = γ n O(h 4 ) [, 6], providing the formal fifth-order accuracy to the WENO scheme in [,, ]. We take ε = 6 in our computation. Step 6. The final nonlinear HWENO reconstruction of φ x,i is defined by a convex combination of three (modified) reconstructed polynomial approximations ( ) u i ω p γ l γ (x i ) p γ l(x i ) ω l p l(x i ). (.4) l= The reconstruction to u i is mirror symmetric with respect to x i of the above procedure. Remark : The terms on the right hand side of (.4) serve a very crucial role in this paper. Our primary objective is to use a high-degree polynomial to obtain a high-order approximation in smooth regions and use the HWENO procedures to rely more on at least one linear polynomial to keep the essentially non-oscillatory property near discontinuities. The total number of polynomials is thus no more than other finite difference/volume HWENO schemes in the literature for the same order of accuracy, and the choice of the linear weights has considerable flexibility, thus leading to efficiency of the new HWENO reconstructions. Thereafter, we reconstruct the approximations of u ± i/ with the high-order linear reconstructions instead of the HWENO reconstructions from the left and right sides of the point x i/. These procedures are vastly different to any other existing HWENO schemes [5, 8] with less truncation errors in smooth regions. l= 9

10 Algorithm. The procedure to reconstruct u i/ is elaborated as follows. We select a big spatial stencil Q = {x i,x i,x i,x i,i i,i i }, which is a combination of four points and two intervals. Then we construct a Hermite quintic polynomial q(x) satisfying and q(x j ) = φ j,j = i,...,i, (.5) q (x)dx = ū j, j = i,i. (.6) h I j Its first-order derivative function at x i/ is explicitly formulated as q (x i/ ) = 4φ i 4φ i 6φ i 6φ i hū i 4hū i. (.7) h The fifth-order linear reconstruction of u i/ is defined by u i/ q (x i/ ). (.8) The fifth-order linear reconstruction to u i / is mirror symmetric with respect to x i of the above procedure. Remark : Comparing with [8], we only select a different big spatial stencil Q for constructing a high-degree polynomial and do not select smaller stencils any more. Since we only use cell averages of the derivative values of φ to construct p (x), it is not very important to apply a HWENO reconstruction and Algorithm is good enough to maintain a high-order spatial approximation in smooth regions. If this procedure arouses strong oscillations and causes ū unacceptable via time approaching, it does not matter too much when approximating u ± i because of Algorithm can abandon the contribution of ū and only use φ to suppress spurious oscillations.. The framework for two-dimensional case The control equations (.) are extended to two-dimensional case. For simplicity, we also assume Ω has been divided as a uniform mesh with two-dimensional intervals I i,j =

11 [x i /,x i/ ] [y j /,y j/ ], i =,...,N, j =,...,M, the interval center (x i,y j ) = ( (x i /x i/ ), (y j /y j/ )) and I i,j = (x i/ x i / ) (y j/ y j / ) = h to be theareaofi i,j, respectively. Wedenotexdirectionlinesegment intervalsii,j x = [x i /,x i/ ] with y = y j, and y direction line segment intervals I y i,j = [y j /,y j/ ] with x = x i. Then (.) can be reformulated as { φt H(x,y,t,φ,φ x,φ y ) =, φ(x,y,) = φ (x,y). (.9) We define u = φ x, v = φ y and these first-order derivatives of (.9) are taken as { ut H x =, u(x,y,) = φ (x,y) x, and { vt H y =, v(x,y,) = φ (x,y). y We define φ i,j (t) = φ(x i,y j,t), ū i,j (t) = u(x,y,t)dx, v h Ii,j x i,j (t) = h I y i,j (.) (.) v(x,y,t)dy, and integrate (.) and (.) over the intervals [x i /,x i/ ] and [y j /,y j/ ], respectively. The semidiscrete form in two dimensions is dφ i,j (t) = H(u(x dt i,y j,t),v(x i,y j,t)), dū i,j (t) = dt (H(u(x h i/,y j,t),v(x i/,y j,t)) H(u(x i /,y j,t),v(x i /,y j,t))), d v i,j (t) = (H(u(x dt h i,y j/,t),v(x i,y j/,t)) H(u(x i,y j /,t),v(x i,y j /,t))). (.) Then (.) is approximated by below formula dφ i,j (t) = H dt i,j, dū i,j (t) = dt h (Ĥi/,j Ĥi /,j), = h (Ĥi,j/ Ĥi,j /). d v i,j (t) dt (.) Where H and Ĥ are Lipschitz continuous monotone fluxes consistent with H(u,v). The simple Lax-Friedrichs fluxes are applied in two dimensions H i,j = H( u i,j u i,j, v i,j v i,j ) α (u i,j u i,j ) β (v i,j v i,j ), Ĥ i/,j = (H(u i/,j,v i/,j )H(u i/,j,v i/,j ) α(u i/,j u i/,j )), (.4) Ĥ i,j/ = (H(u i,j/,v i,j/ )H(u i,j/,v i,j/ ) β(v i,j/ v i,j/ )). Where α = max u H (u,v), β = max v H (u,v), H stands for the partial derivative of H with respect to φ x and H stands for the partial derivative of H with respect to φ y, and u ± i,j,

12 v i,j ±, u± i/,j, v± i/,j, u± i,j/, and v± i,j/ are the fifth-order accurate approximations from the left to right and top to bottom directions to the nodal values of u(x i,y j,t), v(x i,y j,t), u(x i/,y j,t), v(x i/,y j,t), u(x i,y j/,t), and v(x i,y j/,t), respectively. The values of u ± i,j and v± i,j, and u± i/,j and v± i,j/ are designed by the one-dimensional procedures of Algorithm and Algorithm, respectively, with the index for the other dimension frozen. The reconstruction procedures of u ± i,j/ and v± i/,j are very different from [5, 8] and are narrated in detail as follows. Algorithm. The procedure to reconstruct u i,j/ and v i/,j. Select a big spatial stencil T = {(x i,y j ), (x i,y j ), (x i,y j ), (x i,y j ), (x i,y j ), (x i,y j ), (x i,y j ), (x i,y j ), (x i,y j ), I x i,j, Ix i,j, Ix i,j, Ix i,j, Ix i,j, Iy i,j, Iy i,j, Iy i,j, I y i,j, Iy i,j }, which is a combination of nine points and ten intervals. Then we construct an incomplete quintic polynomial p(x,y) span{, x x i h, y y j h, (x x i) h, (x x i)(y y j ) h, (y y j) h, (x x i ) h, (x x i) (y y j ) h, (x x i)(y y j ) h, (y y j) h, (x x i) 4 h 4, (x x i) (y y j ) h 4, (x x i) (y y j ) h 4, (x x i)(y y j ) h 4, (y y j) 4 h 4, (x x i ) 5, (x x i) (y y j ), (x x i) (y y j ), (y y j) 5 h 5 h 5 h 5 h 5 } satisfying and p(x k,y l ) = φ k,l, (k,l) = (i,j ),(i,j ),(i,j ),(i,j),(i,j),(i,j), (i,j ),(i,j ),(i,j ), h Ik,l x p(x,y)dx = ū x k,l, (k,l) = (i,j ),(i,j),(i,j),(i,j),(i,j ), h I y k,l p(x,y)dy = v y k,l, (k,l) = (i,j ),(i,j),(i,j),(i,j),(i,j ). Then the fifth-order linear reconstruction of u i,j/ is defined by (.5) (.6) (.7) u i,j/ x p(x i,y j/ ) = ( 5φ i,j 5φ i,j 6φ i,j 6φ i,j 45φ i,j 45φ i,j 4hū i,j 8hū i,j 6h v i,j 88hū i,j 8hū i,j 6h v i,j hū i,j )/(4h), (.8)

13 and v i/,j is defined by v i/,j y p(x i/,y j ) = ( 5φ i,j 6φ i,j 45φ i,j 5φ i,j 6φ i,j 45φ i,j 6hū i,j 8h v i,j 4h v i,j 88h v i,j h v i,j 6hū i,j 8h v i,j )/(4h). The reconstruction for u i,j/ is mirror symmetric of that for u i,j/ y j/, and reconstruction for v i/,j is mirror symmetric of that for v i/,j (.9) with respect to with respect to x i/. Remark 4: We select a big spatial stencil T and construct an incomplete quintic polynomial p(x, y) to approximate its x- and y-directional derivatives, respectively, without introducing any smaller spatial stencils and associated lower degree polynomials once again, because of the cross terms u ± i,j/ and v± i/,j play a lesser role towards spurious oscillations and a linear reconstruction for those terms might be good enough [8]. It is also verified by our two-dimensional numerical examples in the following section. Numerical tests In this section, we set CFL number as.6, denote new HWENO schemes as HWENO-ZQ schemes which are specified in the previous section both in one and two dimensions, and denote classical HWENO schemes as HWENO-QS schemes which are narrated in [8]. For the purpose of testing whether the random choice of the positive linear weights would degrade the theoretical fifth-order accuracy of HWENO-ZQ schemes or not, we set different types of the linear weights in one- and two-dimensional accuracy test cases as: () γ =.98 and γ =γ =.; () γ =γ =γ =./.; () γ =. and γ =γ =.495. And all linear weights in one- and two-dimensional test cases are set as () in other examples, respectively, unless specified otherwise. Example.. We solve the following nonlinear one-dimensional Hamilton-Jacobi equation: φ t cos(φ x ) =, < x <, (.)

14 Table.: φ t cos(φ x ) =. φ(x,) = cos(πx). Periodic boundary conditions. T =.5/π. HWENO-ZQ () scheme HWENO-QS scheme grid points L error order L error order L error order L error order 4.6E-8 5.9E-7 9.7E-8.8E-6.59E E E E E-9 5..E E E E E E E E E E E HWENO-ZQ () scheme HWENO-ZQ () scheme grid points L error order L error order L error order L error order 5.86E-8 5.9E E-8 6.4E-7.9E E E E E E E E E E E E E E E E with the initial condition φ(x, ) = cos(πx) and periodic boundary conditions. We compute the result up to t =.5/π. The errors and numerical orders of accuracy are shown in Table.. We can see HWENO-ZQ scheme achieves its optimal order of accuracy. For comparison, errors and numerical orders of accuracy by the classical HWENO-QS scheme are shown in the same table. We can see that both HWENO-ZQ and HWENO-QS schemes achieve their designed order of accuracy. Figure. shows that HWENO-ZQ scheme needs less CPU time than HWENO-QS scheme does to obtain the same quantities of L and L errors, so HWENO-ZQ scheme is more efficient than HWENO-QS scheme in this one-dimensional test case. Example.. We solve the following nonlinear scalar two-dimensional Burgers equation: φ t (φ x φ y ) =, x,y <, (.) with the initial condition φ(x, y, ) = cos(π(x y)/) and periodic boundary conditions. We compute the result to t =.5/π and the solution is still smooth at that time. The errors and numerical orders of accuracy by the HWENO-ZQ scheme are shown in Table. and the numerical error against CPU time graphs are presented in Figure.. We can observe 4

15 Log(L error) -7-8 Log(L infinity error) Log(CPU time (sec)) Log(CPU time (sec)) Figure.: φ t cos(φ x ) =. φ(x,) = cos(πx). Computing time and error. Number signs and a solid line denote the results of HWENO-ZQ scheme with different linear weights (), (), and (); squares and a solid line denote the results of HWENO-QS scheme Log(L error) Log(L infinity error) Log(CPU time (sec)) Log(CPU time (sec)) Figure.: φ t (φxφy) =. φ(x,y,) = cos(π(xy)/). Computing time and error. Number signs and a solid line denote the results of HWENO-ZQ scheme with different linear weights (), (), and (); squares and a solid line denote the results of HWENO-QS scheme. that the fifth-order accuracy is actually achieved and HWENO-ZQ scheme can get better results and is more efficient than HWENO-QS scheme in this two-dimensional test case. Example.. We solve the following two-dimensional Hamilton-Jacobi equation: φ t cos(φ x φ y ) =, x,y <, (.) with the initial condition φ(x, y, ) = cos(π(x y)/) and periodic boundary conditions. We also compute the result until t =.5/π. The errors and numerical orders of accuracy by the HWENO-ZQ scheme with different linear weights in comparison with HWENO-QS scheme areshownintable.andthenumerical erroragainstcputimegraphsareinfigure 5

16 Table.: φ t (φxφy) =. φ(x,y,) = cos(π(xy)/). Periodic boundary conditions. T =.5/π. HWENO-ZQ () scheme HWENO-QS scheme grid points L error order L error order L error order L error order.9e E-8 5.8E E-7 5.E E E-8 5..E E E E E E E E E E E E E HWENO-ZQ () scheme HWENO-ZQ () scheme grid points L error order L error order L error order L error order.46e-8.6e-7.e-8.e-7 7.E E E E E E E E E E E E E E E E HWENO-ZQ scheme with different types of linear weights is better than HWENO-QS scheme in this two-dimensional test case. Example.4. We solve the linear equation: φ t φ x =, (.4) with the initial condition φ(x,) = φ (x.5) together with the periodic boundary conditions, where: φ (x) = ( 9 π )(x) cos( πx), x <, cos(πx), x <, 5 cos(πx), x <, 84πcos(πx) 6πx(x ), x <. (.5) We plot the results with cells at t = and t = in Figure.4. We can observe that the results by the HWENO-ZQ scheme have good resolution at the right corner singularity and HWENO-QS scheme gives a bigger offset at the left corner singularity especially at t =. Example.5. We solve the one-dimensional nonlinear Burgers equation: φ t (φ x ) =, (.6) 6

17 Table.: φ t cos(φ x φ y ) =. φ(x,y,) = cos(π(x y)/). Periodic boundary conditions. T =.5/π. HWENO-ZQ () scheme HWENO-QS scheme grid points L error order L error order L error order L error order 7.8E-8.46E-6.5E-7.87E-6.97E E E E E E E E E E E E E E E E HWENO-ZQ () scheme HWENO-ZQ () scheme grid points L error order L error order L error order L error order 7.9E-8.46E-6 7.4E-8.46E-6.E E E E E E E E E E E E E E E E Log(L error) Log(L infinity error) Log(CPU time (sec)) Log(CPU time (sec)) Figure.: φ t cos(φ x φ y ) =. φ(x,y,) = cos(π(xy)/). Computing time and error. Number signs and a solid line denote the results of HWENO-ZQ scheme with different linear weights (), (), and (); squares and a solid line denote the results of HWENO-QS scheme. 7

18 - - φ φ Figure.4: One-dimensional linear equation. grid points. Left: t = ; right: t =. Solid line: the exact solution; cross symbols: HWENO-ZQ scheme; square symbols: HWENO-QS scheme. with the initial condition φ(x, ) = cos(πx) and the periodic boundary conditions. We plot the results at t =.5/π when discontinuous derivative appears. The solutions of the HWENO-ZQ scheme are given in Figure.5. We can see the scheme gives good results for this problem. Example.6. We solve the nonlinear equation with a non-convex flux: φ t cos(φ x ) =, (.7) with the initial data φ(x,) = cos(πx) and the periodic boundary conditions. Then we plot the results at t =.5/π in Figure.6 when the discontinuous derivative appears in the solution. We can see that the HWENO-ZQ scheme and HWENO-QS scheme give good results for this problem. Example.7. We solve the one-dimensional Riemann problem with a nonconvex flux: { φt 4 (φ x )(φ x 4) =, < x <, (.8) φ(x,) = x. This is a demanding test case, for many schemes have poor resolutions or could even converge to a non-viscosity solution for this case. We plot the results at t= by the scheme with 4 8

19 φ φ Figure.5: One-dimensional Burgers equation. Left: 4 grid points; right: 8 grid points. T =.5/π. Solid line: the exact solution; cross symbols: HWENO-ZQ scheme; square symbols: HWENO-QS scheme. φ φ Figure.6: Problem with the non-convex flux H(φ x ) = cos(φ x ). Left: 4 grid points; right: 8gridpoints. T =.5/π. Solidline: theexact solution; crosssymbols: HWENO-ZQ scheme; square symbols: HWENO-QS scheme. 9

20 φ φ Figure.7: Problemwiththenon-convexfluxH(φ x ) = 4 (φ x )(φ x 4). Left: 4gridpoints; right: 8 grid points. T =. Solid line: the exact solution; cross symbols: HWENO-ZQ scheme; square symbols: HWENO-QS scheme. and 8 grid points, respectively, to verify the numerical solution in Figure.7. We can see that two HWENO schemes give good results for this problem. Example.8. We solve the same two-dimensional nonlinear Burgers equation (.) as in Example. with the same initial condition φ(x,y,) = cos(π(xy)/), except that we nowplot theresults att =.5/π infigure.8when thediscontinuous derivative hasalready appeared in the solution. We observe HWENO-ZQ scheme could give good resolutions for this example. Example.9. The two-dimensional Riemann problem with a nonconvex flux: { φt sin(φ x φ y ) =, x,y <, φ(x,y,) = π( y x ). (.9) The solution of the HWENO-ZQ scheme is plotted at t = in Figure.9. We can also observe good resolutions for this numerical simulation. Example.. A problem from optimal control: { φt sin(y)φ x (sin(x)sign(φ y ))φ y sin(y) ( cos(x)) =, π x,y < π, φ(x,y,) =, (.)

21 Z.5 - Φ Figure.8: Two-dimensional Burgers equation. 4 4 grid points. T =.5/π. HWENO- ZQ scheme. Left: contours of the solution; right: the surface of the solution. Z Φ Figure.9: Two-dimensional Riemann problem with a non-convex flux H(φ x,φ y ) = sin(φ x φ y ). 4 4 grid points. T =. HWENO-ZQ scheme. Left: contours of the solution; right: the surface of the solution.

22 Z Z Φ.5 ω ω Figure.: The optimal control problem. 6 6 grid points. T =. HWENO-ZQ scheme. Left: the surface of the solution; right: the optimal control ω =sign(φ y ). with periodic conditions []. The solutions of HWENO-ZQ scheme are plotted at t = and the optimal control ω =sign(φ y ) is shown in Figure.. Example.. A two-dimensional Eikonal equation with a nonconvex Hamiltonian, which arises in geometric optics [4] is given by: { φt φ x φ y =, x,y <, φ(x,y,) = (cos(πx) )(cos(πy) ), (.) 4 The solutions of the HWENO-ZQ scheme are plotted at t =.6 in Figure.. Good resolutions are observed with the proposed HWENO-ZQ scheme. Example.. The problem of a propagating surface []: { φt ( εk) φ x φ y =, x,y <, φ(x,y,) = (cos(πx) )(cos(πy) ), (.) 4 where K is the mean curvature defined by: K = φ xx(φ y ) φ xy φ x φ y φ yy (φ x) (φ x φ y) /, and ε is a small constant. A periodic boundary conditions areused. The approximation of K is constructed by the methods similar to the first-order derivative terms and three different

23 Z.8.6 Φ Figure.: Eikonal equation with a non-convex Hamiltonian. 8 8 grid points. T =.6. HWENO-ZQ scheme. Left: contours of the solution; right: the surface of the solution. second order derivatives of associated Hermite reconstruction polynomials are needed. The results of ε = (pure convection) and ε =. by HWENO-ZQ scheme are presented in Figure.. The surfaces at t = for ε = and for ε =., and at t =. for ε =. are shifted downward in order to show the detail of the solution at later time. 4 Concluding remarks In this paper, we investigate designing a new type of Hermite WENO schemes for simulating the Hamilton-Jacobi equations in one and two dimensions on structured meshes. The constructions of such HWENO schemes are based on new Hermite interpolation with a series of unequal-sized spatial stencils and a third-order TVD Runge-Kutta time discretization method []. The original idea of the reconstruction for the HWENO schemes comes from [5, 6, 7, 8]. In such new HWENO schemes, both the solution and its first derivatives are also evolved via time approaching and used in the HWENO reconstructions and high-order linear reconstructions, respectively, in contrast to the regular HWENO schemes [5, 6, 7, 8, 8] where only the HWENO reconstructions are used all the time. Comparing

24 t=.9 t=.6 t=.6 t=. t=. t=. φ-. Φ Φ t= φ-.5 t= φ Figure.: Propagating surface. 6 6 grid points. Left: ε = ; right: ε =.. HWENO- ZQ scheme. 4

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