Local-structure-preserving discontinuous Galerkin methods with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations

Transcription

1 Local-structure-preserving discontinuous Galerkin methods with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations Wei Guo, Fengyan Li and Jianxian Qiu 3 Abstract: In this paper, a family of high order numerical methods are designed to solve the Hamilton-Jacobi equation for the viscosity solution. In particular, the methods start with a hyperbolic conservation law system closely related to the Hamilton-Jacobi equation. The compact one-step one-stage Lax-Wendroff type time discretization is then applied together with the local-structure-preserving discontinuous Galerkin spatial discretization. The resulting methods have lower computational complexity and memory usage on both structured and unstructured meshes compared with some standard numerical methods, while they are capable of capturing the viscosity solutions of Hamilton-Jacobi equations accurately and reliably. A collection of numerical experiments is presented to illustrate the performance of the methods. ey Words: local-structure-preserving, discontinuous Galerkin method, Lax-Wendroff type time discretization, limiter, WENO scheme, high order accuracy, Hamilton-Jacobi equation AMS subject classification: 65M6, 35F Department of Mathematics, Nanjing University, Nanjing, Jiangsu 93, P.R. China. wguo6@gmail.com. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, N 8, USA. lif@rpi.edu. The research was supported in part by the NSF grant DMS-6548, NSF CAREER award DMS-8474 and an Alfred P. Sloan Research Fellowship. 3 Department of Mathematics, Nanjing University, Nanjing, Jiangsu 93, P.R. China and School of Mathematical Sciences, iamen University, iamen, Fujian 365, P.R. China. jxqiu@nju.edu.cn. The research was supported in part by NSFC grants 934, 88 and 8793.

2 Introduction We consider the Hamilton-Jacobi (HJ) equation { t + H( ) =, in Ω [, T], (x, ) = (x), in Ω, (.) with suitable boundary conditions and H( ) is the Hamiltonian. Such equations arise in various applications, such as optimal control, differential games, image processing and computer vision. Viscosity solutions were introduced and studied [, ] to single out physically relevant solutions of (.), and these solutions are Lipschitz continuous but may have discontinuous derivatives even for smooth initial and boundary conditions. The goal of this paper is to develop a family of high order numerical methods to solve the HJ equation for the viscosity solution accurately and reliably. In particular, we start with rewriting the HJ equation (.) into a conservation law system for w =, { w t + H(w) =, in Ω [, T], w(x, y, ) = (x, y), in Ω, (.) then the compact one-step one-stage Lax-Wendroff type time discretization is applied together with the local-structure-preserving discontinuous Galerkin spatial discretization. The missing constant in is further recovered according to the original equation (.). The Lax-Wendroff type time discretization, used here as an alternative to the widely used one-step multi-stage Runge-utta methods when solving (.) or hyperbolic conservation laws, is a one-step one-stage and therefore compact and low-storage time discretization procedure. Also called Taylor type or Cauchy-owalewski type time discretization in the literature, this procedure is based on the classical Lax-Wendroff scheme [], and it relies on converting all (or partial, when approximations with certain accuracy are expected) time derivatives in a temporal Taylor expansion of the solution into spatial derivatives by repeatedly using the underlying differential equation and its differentiated forms.

3 As a class of finite element methods with the distinct feature of using discontinuous piecewise functions as approximations, discontinuous Galerkin (DG) methods are chosen in this paper as the spatial discretization due to their many attractive properties, such as the ease in handling complicated geometries and boundary conditions, the flexibility in using various local approximating functions (polynomials, divergence-free functions, plane-wave functions) [5, 36, 5] or approximations with different local accuracy, the local conservation property when solving the conservation laws, and the compactness to achieve high order accuracy and hence efficient parallel implementation. These properties also contribute to the growing popularity of DG methods in many applications, e.g. [3, 8, 7,, 35]. When DG methods are applied to HJ equations, the difficulty comes from that such equations generally are not in the divergence form. This prevents one from using the standard procedure to design DG methods for these equations. Our proposed methods are closely related to the pioneering work by Hu and Shu [6] which was later reinterpreted and simplified in [3]. This work is based on the connection between the HJ equation and the conservation law system, that is, the gradient w = of the solution to the HJ equation satisfies (.), a weakly hyperbolic conservation law system. The new feature of the system (.) is the interdependence of the components of w when (.) is multi-dimensional in space, due to the fact w = =. In [3], locally curl-free polynomials are used to approximate w = in the standard DG framework for the hyperbolic system, the missing constant in is then recovered according to the original equation (.). Such DG methods are one example of the local-structure-preserving DG methods [5, 3, 4, 36, 5], designed for certain applications in order to result better cost efficiency. And this efficiency is achieved by incorporating the a priori knowledge of the exact solutions into the choice of local approximating functions in the DG formulations, largely due to the flexibility of DG methods in using various local approximations especially when compared with the classical finite element methods. Other DG methods to directly solve HJ equations (.) for include the one in [3], a method with 3

4 the lowest computational complexity among all available DG methods for HJ equations in multi-dimensional cases, with its performance for nonconvex Hamiltonians H( ) yet to be investigated. Another one is the central DG method recently developed in []. This work suggests a general procedure to design DG methods for problems in the non-divergence form, and the method relies on two sets of meshes and hence has certain complexity when being implemented on non-rectangular meshes. Besides DG methods, other high order numerical methods for HJ equations include finite difference schemes [6, 7, 8, 3, 8] which are simple to implement yet generally require structured meshes, and the methods as in [, 37] which use nodal based approximations on unstructured meshes yet involve complicated reconstruction procedure to achieve high order accuracy. In order for the proposed methods to converge to the viscosity solutions, nonlinear limiters are necessary as in [6,, ] especially for some examples with the nonconvex Hamiltonian. In this paper, we adopt the WENO limiter strategies developed in [3, 38]. Such limiting procedures, based on the WENO reconstruction used in high order finite volume methods [5, 9, 7, 37], are very robust, and they can simultaneously achieve the uniform high order accuracy in numerical solutions and the reliable transition around non-smooth features such as shocks or around sharp gradient structures. Compared with using the standard piecewise polynomial functions in DG methods, advantages in using the locally curl-free approximations to solve HJ equations include the reduction of the computational cost and memory usage, as well as a more natural framework for analysis. One can refer to [3] for more discussions. Though it is generally more involved to formulate and to code DG methods with the Lax-Wendroff type time discretization (LWDG) [3, 4, 9] compared with DG methods using the Runge-utta time-discretization (RDG) [8, 7, 6, 4, 9], the LWDG method is more compact and requires lower storage. It is demonstrated in [9] that the LWDG method is more cost effective and produces sharper discontinuity transition than the RDG method for certain problems including two-dimensional 4

5 Euler systems of compressible gas dynamics. Moreover, nonlinear limiters, if used, only need to be applied once during each time step in LWDG methods yet they need to be applied for each inner stage during one Runge-utta time step in RDG methods. This leads to additional cost reduction in computation. The overall cost efficiency of the proposed methods is illustrated through the two dimensional Riemann problem in section 3.. The rest of this paper is organized as follows. In section, we formulate the localstructure-preserving DG methods with the Lax-Wendroff type time discretization for HJ equations, and some implementation details are also discussed. In section 3, a collection of numerical examples is presented to demonstrate the performance of the proposed methods. The concluding remarks are given in section 4. Description of the numerical method In this section, we present the formulation and some implementation details of a family of local-structure-preserving discontinuous Galerkin methods with the Lax-Wendroff type time discretization for the two dimensional Hamilton-Jacobi (HJ) equation (.). The methods can be similarly formulated for higher dimensional cases.. Lax-Wendroff type time discretization Similar as in [6], we first rewrite the HJ equation (.) into a weakly hyperbolic conservation law system (.) for w = (w, w ) =. Based on the idea of the classical Lax-Wendroff scheme [], the Lax-Wendroff type time discretization for (.) starts with a temporal Taylor expansion of w i, i =,, w i (x, y, t + t) = w i (x, y, t) + t(w i ) t + t (w i) tt + t3 6 (w i) ttt +..., (.) here t is the time step. The time derivatives in (.) are then converted into spatial derivatives by repeatedly using the equation system (.) and its differentiated versions. In 5

6 order to achieve kth order accuracy in time, one only needs to work with the first k time derivatives in (.), namely, w i t,..., k w i t k, i =,. In this paper, we will proceed up to third order accuracy in time, although the procedure described here can be naturally extended to higher order accuracy. By using the governing equation system (.) and its differentiated forms, and with the notations H i (w, w ) = H(w,w ) w i, H ij (w, w ) = H(w,w ), i, j =,, one gets w i w j (w ) t = (H(w, w )) x = H (w, w )(w ) x H (w, w )(w ) x, (w ) t = (H(w, w )) y = H (w, w )(w ) y H (w, w )(w ) y, (w ) tt = (H(w, w )) tx = (H (w, w )(w ) t + H (w, w )(w ) t ) x = (H (w, w )(w ) x + H (w, w )(w ) x )(w ) t H (w, w )(w ) xt (H (w, w )(w ) x + H (w, w )(w ) x )(w ) t H (w, w )(w ) xt, (w ) tt = (H(w, w )) ty = (H (w, w )(w ) t + H (w, w )(w ) t ) y = (H (w, w )(w ) y + H (w, w )(w ) y )(w ) t H (w, w )(w ) yt (H (w, w )(w ) y + H (w, w )(w ) y )(w ) t H (w, w )(w ) yt, (w ) xt = (H (w, w )(w ) x + H (w, w )(w ) x ) x = (H (w, w )(w ) x + H (w, w )(w ) x (w ) x + H (w, w )(w ) x ) H (w, w )(w ) xx H (w, w )(w ) xx, (w ) yt = (H (w, w )(w ) x + H (w, w )(w ) x ) y = (H (w, w )(w ) x (w ) y + H (w, w )((w ) x (w ) y + (w ) y (w ) x ) + H (w, w )(w ) x (w ) y ) H (w, w )(w ) xy H (w, w )(w ) xy, (w ) xt = (H(w, w )) yx = (w ) yt, 6

7 (w ) yt = (H (w, w )(w ) y + H (w, w )(w ) y ) y = (H (w, w )(w ) y + H (w, w )(w ) y (w ) y + H (w, w )(w ) y ) H (w, w )(w ) yy H (w, w )(w ) yy. (w ) ttt = (H(w, w )) xtt = G x, (w ) ttt = (H(w, w )) ytt = G y, with G =(H(w, w )) tt = H (w, w )(w ) t + H (w, w )(w ) t (w ) t +H (w, w )(w ) t + H (w, w )(w ) tt + H (w, w )(w ) tt. Now one can replace the time derivatives of w i with its spatial derivatives according to the formulas derived above, and approximate (.) and therefore (.) with { w (x, y, t + t) = w (x, y, t) th x, w (x, y, t + t) = w (x, y, t) th y. (.) In particular, (.) provides a second order approximation when and a third order approximation when H = H + t (H (w, w )(w ) t + H (w, w )(w ) t ) (.3) H = H + t (H (w, w )(w ) t + H (w, w )(w ) t ) + t 6 G (.4) Next, local-structure-preserving discontinuous Galerkin methods will be formulated to discretize H x and H y in (.).. Local-structure-preserving discontinuous Galerkin spatial discretization Let T h = {} denote a regular triangulation of the computational domain Ω, with the element being a triangle, a rectangle or a more general polygon, and the edge of as e. And h = max Th diam(). Let V h denote a finite dimensional discrete space consisting 7

8 of piecewise smooth functions with respect to T h, which will be specified later. The discontinuous Galerkin (DG) discretization for (.) is defined as follows: find w n+ h (x, y) = (w n+,h (x, y), wn+,h (x, y)) V h, such that v = (v, v ) V h, T h, w n+,h v dxdy = w,h n v dxdy (.5) ( + t n H(w,h, n w,h)(v n ) x dxdy ) Ĥ,e, (w,h, n w,h)v n ds e and e w n+,h v dxdy = w,h n v dxdy (.6) ( + t n H(w,h n, wn,h )(v ) y dxdy ) Ĥ,e, (w,h n, wn,h )v ds e e for n =,,,. In (.5)-(.6), w n h approximates the solution of (.) (and therefore (.)) at t n, and t n = t n+ t n. The function Ĥi,e, (i =, ) is a consistent and conservative numerical flux [6], and it is related to the so-called generalized Riemann solvers [34] and depends on the numerical solution w h and its spatial derivatives from both sides of the element interface e. In this paper, the following Lax-Friedrichs flux is used for i =,, Ĥ i,e, (w, w ) = ( H(w, w ) + H(w+, w+ )) n i α i (w+ i w i ). Here n = (n, n ) is the unit outward normal of along its edge e =, with w = w and w + = w. And α i = max w,w H i (w, w ), with the maximum taken in the computational domain for i =,. In addition, with Ĥ e, = (Ĥ,e,, Ĥ,e,), the DG discretization (.5)-(.6) can be rewritten more compactly: find w n+ h v V h, T h, w n+ h vdxdy = wh n vdxdy + t n ( V h, such that ) H(wh n ) vdxdy Ĥ e, (wh n ) vds. e e (.7) 8

9 And wh V h can be initialized through the L projection, namely, wh vdxdy = vdxdy, v V h, T h. (.8) To finalize the DG method with the Lax-Wendroff type time discretization defined in (.7), (.8), and (.3) (or (.4)), one would need to specify the discrete space V h. By following [3], we take V h = V k h = {(v, v ) : v =, P k (), T h }. (.9) Note the function v V h satisfies v = on each T h, a property of the exact solution w of (.) due to the relation w = =. With such a choice, the method described above is termed as the local-structure-preserving DG (LSP-DG) method with the Lax-Wendroff type time discretization. Similar LSP-DG methods were also investigated in [5, 4, 36, 5], with the objective to develop DG methods with better cost efficiency when solving certain differential equations. Such efficiency is achieved by incorporating the a priori knowledge of the exact solutions into the choice of local approximating functions in the DG formulations, largely due to the flexibility of DG methods in using various local approximations especially when compared with the classical finite element methods. Some advantages in using this locally curl-free discrete space V k h as (.9) compared with using the standard piecewise polynomial space in DG methods include the reduction of the computational cost and memory usage, as well as a more natural framework for analysis. One can refer to [3] for more details. Though it is generally more involved to formulate and to code DG methods with the Lax-Wendroff type time discretization (LWDG) [3, 4, 9] compared with DG methods using the Runge-utta time-discretization (RDG) [8, 7, 6, 4, 9], the LWDG method is more compact and requires lower storage. To implement the LSP-DG method with the Lax- Wendroff type time discretization defined by (.7)-(.9) and (.3) (or (.4)), when the mesh 9

10 T h consists of rectangular elements, a local orthogonal basis can be adopted for Vh k over each element. For example, for a rectangular element T h with the center (x, y ) and the size x y, the basis functions of V k h can be for k = : ϕ = (, ), ϕ = (, ), for k = : add ϕ = (ξ, ),ϕ 3 = (, η),ϕ 4 = (λη, ξ), ( for k = 3 : add ϕ 5 = ξ ), ϕ 7 = (, η, ϕ 6 = ( λξη, ξ ), ), ϕ 8 = ( η, λ ξη ). Here ξ = x x, η = y y y, and λ =. Once the local basis functions are chosen, the x y x numerical solution w n h to (.6) restricted to can be expressed as L k wh(x, n y) = ω n,(l) ϕ l (x, y), for (x, y), (.) l= where L k = k(k+3) is the dimension of V k h. And the degrees of freedom ω n,(l) moments defined by ω n,(l) = a l w n h ϕ l dxdy, l =,,, L k, are the with the normalization constant a l = ϕ l (x, y) dxdy. Now ω n,(l) for T h, l =,, L k can be evolved based on (.7)-(.8) as follows, ( ω n+,(l) = ω n,(l) + t n H(w a h) n ϕ l dxdy l e e ω,(l) = ϕ l dxdy. a l Ĥ e, (w n h) ϕ l ds ), n =,, With these {ω n,(l) },l,n, one obtains w h V h which approximates the solution w of (.) and therefore for the HJ equation (.). The strategies to recover the missing constant in proposed in [6] are based on Runge-utta time discretizations. In this paper, we adopt a similar recovering strategy based on the Lax-Wendroff procedure. Denote q as the

11 center of an element (or the barycenter for general elements), we can update (q, t+ t) according to the Taylor expansion (q, t + t) = (q, t) + t t (q, t) + t tt(q, t) + t3 6 ttt(q, t) +..., where time derivatives are obtained from the spatial derivatives by repeatedly using the governing equations (.) and (.). This gives the approximation to at q (q, t n+ ) = (q, t n ) t n H(w n h(q )), (.) with the second and the third order accuracy when H is defined in (.3) and (.4), respectively. In practice, this procedure only needs to be applied to a few elements, e.g., the corner elements, during the time evolution in the simulation. And the missing constant of at an arbitrary location q = (x, y) can be recovered through integration at the final time step T = t N. h (q, T) = h (q, T) + q q w N,hdx + w N,hdy. (.) When the mesh T h also consists of other shapes of elements such as triangles, the following local basis functions are used for V k h over : k = : ϕ = (, ), ϕ = (, ), k = : add ϕ = (ξ, ),ϕ 3 = (, η),ϕ 4 = (η, ξ), k = 3 : add ϕ 5 = ( ξ, ), ϕ 6 = ( ξη, ξ ), ϕ 7 = (, η ), ϕ 8 = ( η, ξη ). Here ξ = x x, η = y y, and (x, y ) and are the barycenter (or other suitable reference points) and the area of respectively. Note such basis functions, though simple, are not orthogonal, and a local mass matrix of the dimension L k L k needs to be inverted in each element T h. Fortunately these small matrices only need to be computed once and they can be stored for later use during the time evolution. For V k h with higher k, the orthogonal basis is preferred for better numerical performance.

12 In order for the proposed methods to reliably capture the viscosity solution, nonlinear limiters are necessary especially when the Hamiltonian H is nonconvex. Such phenomena was also observed in [6,, ]. In this paper, we adopt the WENO limiter strategies developed in [3] for rectangular meshes and in [38] for triangular meshes. Such limiting procedures are based on the WENO reconstruction used in high order finite volume methods [5, 9, 7, 37]. And when they are applied to an element T h which is identified as a troubled cell, all moments of the numerical solutions in this element will be abandoned except the cell average. The abandoned moments will be reconstructed based on the cell averages of the solution in neighboring elements using the WENO methodology. Not like some limiters which may degenerate the accuracy especially of high order schemes when mistakenly used in the smooth region of the solution [7, 4, 9, ], WENO limiters are very robust, and they can simultaneously achieve the uniform high order accuracy in numerical solutions and the reliable transition around non-smooth features such as shocks or around sharp gradient structures. In our numerical experiments, WENO methods are applied to reconstruct the polynomial of wh n, in which the degree of reconstructed polynomial in each small stencil is the same as that of w n h to maintain the order of accuracy of the DG schemes. We use the optimal linear weights to get higher order reconstructions for rectangular meshes, that is we use the third and fifth order WENO reconstructions for P and P 3 cases, respectively. For triangular meshes a fixed set of average linear weights is used in each small stencil for the ease of implementation, that is we use the second and third order WENO type reconstructions for P and P 3 cases, respectively. Though the magnitude of error by this implementation will be a little bigger than those by the implementation with optimal linear weights, the orders of accuracy of the underlying DG methods are maintained by both the implementations. Such implementation strategy also saves computational cost as we do not compute the more complex optimal linear weights, and the negative linear weights can be avoided as well. Note that nonlinear limiters, if used, only need to be applied once during each time step in LWDG

13 methods yet they need to be applied for each inner stage during one Runge-utta time step in RDG methods. This leads to additional cost reduction in computation. 3 Numerical results In this section, a collection of two-dimensional numerical experiments is presented to illustrate the high order accuracy and the reliability of the proposed methods when solving the viscosity solutions of the Hamilton-Jacobi equations. Both uniform rectangular meshes and unstructured triangular meshes are used in the computation. An example of the triangular meshes with the characteristic meshsize h = / is given in Figure 3.. WENO limiters are applied to all examples, though they are not needed for smooth examples and for some nonsmooth examples. In order to decide whether limiters are needed in each mesh element during the simulation, the TVB limiters with the constant M =. [33, 6, 9] are used as the troubled cell indicator. We want to remark that a good choice of the TVB constant M is problem dependent for the reliability and cost efficiency of the limiting procedures. Generally, if M is chosen too large, some cells with spurious oscillation may not be recognized. If M is chosen too small, some good cells will be identified as troubled cells. In this case, the implementation of WENO limiters on these cells will introduce extra computational cost without degenerating the accuracy of the methods. 3. Accuracy tests In this section, the accuracy of the proposed methods will be examined. Example 3.. Consider the Burgers equation t + ( x + y + ) = (3.) with the initial condition (x, y) = cos(π(x+y)/) and the 4-periodic boundary condition. This example involves a convex Hamiltonian. The computation is first carried out on the 3

14 Figure 3.: An unstructured triangular mesh with the characteristic meshsize h = / uniform rectangular meshes. When t =.5/π, the solution is still smooth, and the errors and numerical orders of accuracy of the proposed methods are shown in Table 3.. The results confirm the designed accuracy of the schemes, namely, the method is kth order accurate when V h = V k h in (.9) and the first k time derivatives in (.) are used. The L and L errors are computed by using a (6 6)-point Gaussian quadrature formula in each cell. Though the limiter strategy is not needed for this smooth example, we still report the numerical results with the limiter procedure turned on, and these results show that the WENO limiter keeps the designed order of accuracy uniformly. Next, the computation is done on unstructured triangular meshes. The errors and numerical orders of accuracy of the proposed methods are reported in Table 3.. The L and L errors are computed by a 7-point Gaussian quadrature formula in each cell. Similar accuracy pattern is observed as in the cases with rectangular meshes. Example 3.. Consider t cos( x + y + ) = (3.) with the initial condition (x, y) = cos(π(x+y)/) and the 4-periodic boundary condition. 4

15 Table 3.: Burgers equation. L and L errors and numerical orders of accuracy on uniform rectangular meshes with N x N y cells. t =.5/π. With the WENO limiter Without the limiter N x N y L error order L error order L error order L error order.46e- 9.E-.3E- 5.99E- 4.73E E-.7.74E E-.6 P E E E E E-4.4.9E-3.7.5E E E E E-5.4.E-4..E- 3.64E- 3.3E E E E E E-3 3. P E E E-5.37.E E E E E E E E E-6.84 Table 3.: Burgers equation. L and L errors and numerical orders of accuracy on unstructured triangular meshes with the characteristic meshsize h. t =.5/π. With the WENO limiter Without the limiter h L error order L error order L error order L error order / 3.95E-.7E- 7.3E-3 4.9E- /4.4E E-..45E E-3.5 P /8.6E-3.7.3E-.9.8E E-3.54 /6 4.E E E E-4.6 /3 8.E E-4..35E E-5.4 / 6.36E E- 6.3E E-3 /4 3.48E E E E-4.69 P 3 /8.4E E E E /6.74E E E E /3.8E E E E

16 Table 3.3: The example of t cos( x + y +) =. L and L errors and numerical orders of accuracy on uniform rectangular meshes with N x N y cells. t =.5/π. With the WENO limiter Without the limiter N x N y L error order L error order L error order L error order.33e- 6.34E-.5E- 4.74E-.4E E E E-3.3 P E-4..4E E-4.5.4E E E-4..9E E E-5.5.8E E-5..6E E- 4.79E-.47E-.5E- 9.54E E E E-3.85 P E E-4.4.4E E E E-5.9.8E E E E E E-6.76 This example involves a nonconvex Hamiltonian. The computation is first carried out on the uniform rectangular meshes. When t =.5/π, the solution is still smooth, and the errors and numerical orders of accuracy of the proposed methods are shown in Table 3.3. The L and L errors are computed by using a (6 6)-point Gaussian quadrature formula in each cell. The results confirm the designed accuracy of the proposed methods. Next, we use unstructured triangular meshes. The errors and numerical orders of accuracy are shown in Table 3.4. A 7-point Gaussian quadrature formula is used to compute the L and L errors. The expected orders of accuracy are obtained. WENO limiter procedures again are turned on for this example to demonstrate the non-degeneracy in the accuracy of the proposed high order methods when such limiting strategies are applied. 3. Examples with discontinuous derivatives In this subsection, we further examine the performance of the proposed methods when applied to examples with discontinuous derivatives. The computation is done with both P and P 3 approximations. That is, V h = V k h and the first k time derivatives in (.) are 6

17 Table 3.4: The example of t cos( x + y + ) =. L and L errors and numerical orders of accuracy on unstructured triangular meshes with the characteristic meshsize h. t =.5/π. With the WENO limiter Without the limiter h L error order L error order L error order L error order / 3.4E-.3E- 8.45E-3 3.5E- /4 6.8E E-.8.45E E-3.9 P /8.46E E-3..5E E-3.3 /6 3.6E E E E-4. /3 7.3E-5..84E-4.34.E E-5.7 / 6.9E E- 8.54E-4 8.8E-3 /4 3.8E E E E-3.39 P 3 /8.4E E E E /6.E E E E /3.3E E E E used with k = and k = 3 respectively. In addition, once a mesh element is indicated as a troubled cell, the WENO limiter strategy will be applied in this element. Though the computation is done for both uniform rectangular meshes and unstructured triangular meshes for Examples , there is no visible difference in the plots, and therefore only the solutions on triangular meshes are reported in this paper. Example 3.3. Consider the Burgers equation (3.) with the same initial and boundary conditions as in Example 3.. At t =.5/π, the solution has developed discontinuous derivatives. The numerical solutions on the unstructured triangular mesh with h = /8 are shown in Figure 3.. The solution is well resolved for this example. We remark that the proposed method without limiter can also produce non-oscillatory reliable results for this example, and such limiter is more crucial for the cases with some nonconvex Hamiltonians, e.g. Example 3.5. Example 3.4. Consider the example of (3.) with the same initial and boundary condition as in Example 3.. We report the results at t =.5/π when the discontinuous derivative 7

18 Figure 3.: Burgers equation. The numerical solution at t =.5/π on the unstructured triangular mesh with h = /6. P (left) and P 3 (right). has developed in the solution. The numerical solutions on the unstructured triangular mesh with h = /8 are shown in Figure 3.3. Both P and P 3 results approximate the viscosity solution very well. And the proposed methods without the limiter procedure also produce non-oscillatory reliable results for this example. Example 3.5. Consider a two dimensional Riemann problem { t + sin( x + y ) =, < x, y <, (x, y, ) = π( y x ), (3.3) with a nonconvex Hamiltonian and the outflow boundary condition. The results at t = on the unstructured triangular mesh with h = / are reported in Figure 3.4. Similar to what was observed in [6, ], without the limiter procedure, the numerical solutions of the proposed methods do not converge to the viscosity solution for this example, see Figure 3.5. In fact, the rarefaction wave is missing and the numerical solution has serious oscillations. On the other hand, the reliable WENO limiter strategies ensure the non-oscillatory results and the convergence of the methods to the correct viscosity solution. 8

19 Figure 3.3: The example of t cos( x + y + ) =. The numerical solution at t =.5/π on the unstructured triangular mesh with h = /6. P (left) and P 3 (right) Figure 3.4: Two dimensional Riemann problem. The numerical solution at t = on the unstructured triangular mesh with h = /. P (left) and P 3 (right). 9

20 Figure 3.5: Two dimensional Riemann problem. The numerical solution without limiter at t = on the unstructured triangular mesh with h = /. P (left) and P 3 (right). Finally, we want to use this example to compare the computation efficiency of our proposed LW-LSP-DG method and the RDG method in [6]. In Table 3.5, the CPU time is reported for these two methods on rectangular meshes. To ensure the linear stability, the CFL number for the proposed method is taken to be /3 of that for the RDG method, see [9]. The time step t is determined by t = CFL min(dx, dy) max i=, (max w,w H i (w, w ) ) with the maximum taken in the computational domain. In our implementation, the CF L number is chosen to be. and.5 for P and P 3 approximations in the LW-LSP-DG methods, and.8 and.75 for P and P 3 approximations in the RDG methods. The simulation is performed on a computer with Intel(R) Core(TM) Duo CPU P87 and 3 GB RAM. It is observed that the LW-LSP-DG methods show better computational efficiency than the RDG methods, especially when the high order polynomial approximation is adopted, even though the CFL number for the RDG methods is.5 times of that for

21 Table 3.5: The CPU time (in seconds) of the proposed LW-LSP-DG method and the RDG method in [6] to compute a two-dimensional Riemann problem. Methods LW-LSP-DG RDG in [6] P P 3 P P 3 M Table 3.6: The percentage of troubled cells subject to WENO limiters for the proposed LW- LSP-DG method and the RDG method in [6] for a two-dimensional Riemann problem. Methods LW-LSP-DG RDG in [6] P P 3 P P 3 M LW-LSP-DG methods. This observation is consistent with the facts that the locally curlfree polynomial space has lower dimension and the WENO limiter only needs to be applied once during each time step. In Table 3.6, we further document the percentages of troubled cells for both methods, which are computed based on the averaging over time. One can see that the larger TVB constant M renders a smaller percentage of troubled cells, and this will reduce the computational cost of the methods when the limiter is involved. In this example, reasonably good numerical results are produced for both M=. and M=. Example 3.6. Consider the following example from optimal control, { t + sin(y) x + (sin x + sign( y )) y sin y ( cosx) =, π < x, y < π, (x, y, ) =, (3.4) with the periodic condition, see [7]. And the Hamiltonian is nonsmooth. In Figures , we report the numerical results approximating the solution and the optimal control

22 Figure 3.6: The optimal control problem. The numerical solution at t = on the unstructured triangular mesh with h = π/3. P (left) and P 3 (right). ω = sign( y ) on the unstructured triangular mesh with h = π/3. Note the proposed method computes w =, and this is very desirable for those problems in which the most interesting information is contained in the first derivative of, as in this optimal control example. Example 3.7. Consider the following problem involving a propagating surface [6] { t ( ε) x + y + =, < x, y <, (x, y, ) = (cos(πx) )(cos(πy) ), (3.5) 4 where is the mean curvature defined by = xx( + y) xy x y + yy ( + x) ( + x +, y )3 and ε is a small constant. The boundary condition is periodic. The local discontinuous Galerkin (LDG) method is adopted to deal with the second derivative terms [, 6]. This is done in the similar framework as the proposed methods, and the details are not included here for brevity. The numerical solutions for ε = (pure convection) and for ε =. are

23 ω.5 ω Figure 3.7: The optimal control problem. The computed optimal control ω = sign( y ) at t = on the unstructured triangular mesh with h = π/3. P (left) and P 3 (right). plotted in Figure 3.8 and Figure 3.9 respectively when an unstructured triangular mesh with h = /4 is used. 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 times. Example 3.8. We consider the same equation of a propagating surface as in Example 3.7 with the initial condition (x, y) = sin( π x +y ), yet defined in a unit disk {(x, y) : x +y < }. A Neumann-type boundary condition = is applied. It is difficult to use rectangular meshes for this problem. We adopt an unstructured triangular mesh instead, which consists of 46 triangles and 84 nodes. We plot the results of ε = (pure convection) and ε =. by the proposed method in Figures And the surfaces at t = for ε = and for ε =., and at t =.3 for ε =., are shifted downward in order to show the detail of the solution at later times. 3

24 t=.9 t=.9 t=.6 t= t=.3 t= t=..35 t= Figure 3.8: Propagating surface. The numerical solution on the unstructured triangular mesh with h = /4. ε =. P (left) and P 3 (right)..5.5 t=.6 t=.6 t=.3 t=.3 t=.. t= t=..85 t= Figure 3.9: Propagating surface. The numerical solution on the unstructured triangular mesh with h = /4. ε =.. P (left) and P 3 (right). 4

25 3 3 t=.8 t= t=. t=..5.5 t=.6 t=.6 t=...5 t= Figure 3.: Propagating surface on a disk. The numerical solution on the unstructured triangular mesh. ε =. P (left) and P 3 (right). t=. t=. t=.6.5 t=.6.5 t=.3..5 t=.3..5 t=..7 t= Figure 3.: Propagating surface on a disk. The numerical solution on the unstructured triangular mesh. ε =.. P (left) and P 3 (right). 5

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