A discontinuous Galerkin finite element method for directly solving the Hamilton Jacobi equations

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1 Journal of Computational Physics 3 (7) A discontinuous Galerkin finite element method for directly solving the Hamilton Jacobi equations Yingda Cheng, Chi-Wang Shu * Division of Applied Mathematics, Brown University, Bo F, Providence, RI 91, United States Received 1 May 6; received in revised form 14 September 6; accepted 19 September 6 Available online November 6 Abstract In this paper, we propose a new discontinuous Galerkin finite element method to solve the Hamilton Jacobi equations. Unlike the discontinuous Galerkin method of [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton Jacobi equations, SIAM Journal on Scientific Computing 1 (1999) ] which applies the discontinuous Galerkin framework on the conservation law system satisfied by the derivatives of the solution, the method in this paper applies directly to the solution of the Hamilton Jacobi equations. For the linear case, this method is equivalent to the traditional discontinuous Galerkin method for conservation laws with source terms. Thus, stability and error estimates are straightforward. For the nonlinear conve Hamiltonians, numerical eperiments demonstrate that the method is stable and provides the optimal (k + 1)th order of accuracy for smooth solutions when using piecewise kth degree polynomials. Singularities in derivatives can also be resolved sharply if the entropy condition is not violated. Special treatment is needed for the entropy violating cases. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme. Ó 6 Elsevier Inc. All rights reserved. Keywords: Hamilton Jacobi equations; Discontinuous Galerkin; High order accuracy; Conve Hamiltonian 1. Introduction In this paper, we consider the numerical solution of the Hamilton Jacobi equation u t þ Hðu 1 ;...; u d ; 1 ;...; d Þ¼; uð; Þ ¼u ðþ ð1:1þ Here d is the space dimension. In this paper, we will only consider the linear or conve Hamilton Jacobi equations, namely the Hamiltonian H is a linear or conve function of u i. The solutions to the above equations are Lipschitz continuous but may admit discontinuous derivatives. For linear case with discontinuous coefficients or the nonlinear case, this is true even if the initial condition is smooth, and the solutions are also non-unique. We are only interested in the viscosity solution [6], which is the unique practically relevant solution and * Corresponding author. Tel.: ; fa: addresses: ycheng@dam.brown.edu (Y. Cheng), shu@dam.brown.edu (C.-W. Shu) /$ - see front matter Ó 6 Elsevier Inc. All rights reserved. doi:1.116/j.jcp.6.9.1

2 satisfies the entropy condition. Fortunately, this entropy condition can be epressed in a simple form when the Hamiltonian is linear or conve. Essentially non-oscillatory (ENO) or weighted ENO (WENO) finite difference schemes have been developed to solve the Hamilton Jacobi equation (1.1), see, e.g. [16,1,19]. These finite difference methods work quite efficiently for Cartesian meshes, however, on unstructured meshes the scheme is quite complicated [19]. Alternatively, the Runge Kutta discontinuous Galerkin (DG) finite element method, originally devised to solve the conservation laws [3,4,,1,5], has the advantage of fleibility for arbitrarily unstructured meshes, with a compact stencil, and with the ability to easily achieve arbitrary order of accuracy. In [9], Hu and Shu proposed a discontinuous Galerkin method to solve the Hamilton Jacobi equation (1.1). They use the fact that the derivatives of the solution u satisfy a conservation law system, and apply the usual discontinuous Galerkin method on this system to advance the derivatives of u. The solution u itself is then recovered from these derivatives by a least square procedure for multi-dimensional cases and with an independent evolution of the cell averages of u. Later, Li and Shu [14] reinterpreted the method of Hu and Shu by using a curl-free subspace for the discontinuous Galerkin method. The algorithm in [14] is mathematically equivalent to that in [9], but the least square procedure is avoided and the computational cost is reduced for multi-dimensional calculations. The method in [9,14] works well numerically, with provable stability results for certain special cases [9,1]. However, since this method is based on the conservation law system satisfied by the derivatives of u, a scalar problem (1.1) is converted to a system for the multi-dimensional case, which is moreover only weakly hyperbolic at some points. This seems to have made the algorithm indirect and complicated. It is, therefore, desirable to design a discontinuous Galerkin method which solves directly the solution u to the Hamilton Jacobi equation (1.1). In this paper, we develop such a discontinuous Galerkin method. This paper is organized as follows: in Section, we describe the formulation of our scheme for the onedimensional case. Theoretical analysis for the linear case is provided. In Section 3, we generalize the scheme to two space dimensions. Numerical results of both one and two dimensions are presented in Section 4. Finally, in Section 5, concluding remarks are given.. One-dimensional case Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) For the simple one-dimensional case, (1.1) becomes u t þ Hðu ; Þ ¼ uð; Þ ¼u ðþ ð:1þ and we consider only the case where H(u,) is a linear or conve function of u. If we want to solve this equation on the interval [a,b], first we divide it into N cells as follows: a ¼ 1 < 3 <...< Nþ 1 ¼ b ð:þ We denote ¼ð j 1 ; jþ 1 Þ; j ¼ 1 j þ 1 jþ ; I 1 jþ1= ¼ j ; jþ1 ð:3þ and D j ¼ jþ 1 j 1 ; h ¼ ma D j ð:4þ j Now, we define the approimation space as V k h ¼ft : tj P k ð Þ; j ¼ 1;...; Ng ð:5þ where P k ( ) denotes all polynomials of degree at most k on. We now formulate our scheme for (.1) and give theoretical analysis for the linear case..1. Formulation of the scheme Let us denote H 1 ¼ oh ou.ifh 1 is always non-negative, then we can define the upwind version of our scheme as: find u h ð; tþ V k h, such that

3 4 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) ðo t u h ð; tþþhðo u h ð; tþ; ÞÞv h ðþd þ ma 1= H 1 ðo u h ; j 1= Þ½u h Š j 1 ðv h Þ þ j 1 ¼ ; j ¼ 1;...; N ð:6þ holds for any v h V k h. Here ½u hš j 1 ¼ u h ð þ ; tþ u j 1 h ð tþ denotes the jump of u j 1 ; h at the cell interface j 1. Our definition of the scheme is motivated by the usual discontinuous Galerkin method for the linear case, see the net subsection.. For general H, our scheme is formulated as follows: find u h ð; tþ V k h, such that ðo t u h ð; tþþhðo u h ð; tþ; ÞÞv h ðþd þ 1 min H 1 ðo u h ; jþ1= Þ min H 1 ðo u h ; jþ1= Þ þ1= þ1= ½u h Š jþ 1ðv h Þ jþ 1 þ 1 ma H 1 ðo u h ; j 1= Þþ ma H 1 ðo u h ; j 1= Þ 1= 1= ½u h Š j 1ðv h Þ þ j 1 ¼ ; j ¼ 1;...; N ð:7þ holds for any v h V k h. This is a Roe type generalization of the upwind scheme (.6). In the schemes (.6) and (.7), we need the reconstructed information of o u h on the cells 1/ and +1/. Notice that these cells include the points j 1 and jþ 1, respectively, where the numerical solution u h (,t) is discontinuous. We use an L reconstruction technique as follows. We define a polynomial w jþ 1 ðþ P kþ1 on [ +1, such that u h vd ¼ w jþ 1 vd ð:þ for any v P k on, and u h vd ¼ w jþ 1 vd ð:9þ þ1 þ1 for any v P k on +1. Then we use o u h ¼ o w jþ 1 on +1/ when taking the maimum or minimum in (.6) and (.7). In the case of a uniform mesh and piecewise constant polynomials (k = ), the reconstructed derivative becomes o u h ¼ u jþ1 u j ð:1þ h on +1/. This agrees with our intuitive definition of o u h. For a practical implementation, once a local basis is chosen, the coefficients of w jþ 1 ðþ are linear combinations of the coefficients of u h j Ij and of u h j Ijþ1. These linear combination coefficients can be pre-computed to save computational cost. We would like to remark that in (.7), the last two terms involving the jumps of u h are added for stability, whereas the first integral term guarantees the accuracy of our scheme. The purpose of taking the maimum and minimum is to obtain better stability by adding more viscosity, while still maintaining accuracy since these maimum and minimum values are a O(h) perturbation from H 1 (o u h ( j+1/,t), j+1/ ), which guarantees accuracy according to truncation error analysis and numerical tests. For linear Hamiltonians with discontinuous coefficients or nonlinear Hamiltonians, since our scheme is of Roe type, it may generate entropy violating solutions. We have, therefore, adopted the following entropy correction procedure: 1. For each cell, determine if it is a potentially entropy violating cell. We will provide the criteria for this determination in the numerical Section 4. If the cell is marked as a potentially entropy violating cell, then use Step below to update u h in this cell; otherwise, update u h by (.6) or (.7).

4 . Update u h by the DG method of Hu and Shu [9], namely, recover o u h by taking the derivative of u h, then compute o t (o u h ) by the usual DG method for the conservation law satisfied by o u h. This will determine u h up to a constant. The missing constant is obtained by requiring ðo t u h ð; tþþhðo u h ð; tþ; ÞÞd ¼ : ð:11þ The entropy correction in Step bears comparable computational cost as our scheme (.6) or (.7) for this one-dimensional case. It is our eperience that such entropy corrections are needed only in very few isolated cells, see Section 4... Theoretical analysis We first consider a linear Hamilton Jacobi equation u t þ aðþu ¼ ð:1þ and assume, for the time being, that a() is a smooth function. If a() >, our scheme (.6), after replacing the maimum by the point value at j 1, becomes finding u h ð; tþ V k h, such that ðo t u h ð; tþþaðþðo u h ð; tþþþv h ðþd þ að j 1 Þ½u h Š j 1ðv h Þ þ j ¼ ; 1 j ¼ 1;...; N ð:13þ holds for any v h V k h. After integration by parts, this is equivalent to o t u h ð; tþv h ðþd aðþu h ð; tþo v h ðþd þðau h Þ jþ 1 ðv h Þ jþ ðau 1 h Þ j ðv 1 h Þ þ j 1 ¼ a ðþu h ð; tþv h ðþd ð:14þ We observe that the scheme (.14) is the standard DG scheme for conservation laws with source terms using upwind flues [4] for the equation u t þðaðþuþ ¼ a ðþu ð:15þ which is equivalent to (.1). Similarly, for general a(), our scheme (.7), after replacing the maimum and minimum by the point value at j 1 and jþ 1, respectively, becomes finding u h ð; tþ V k h, such that ðo t u h ð; tþþaðþðo u h ð; tþþþv h ðþd þ 1 ðað jþ 1 Þ jað jþ 1 ÞjÞ½u hš jþ 1 ðv h Þ jþ 1 þ 1 ðað j 1 Þþjað j 1 ÞjÞ½u hš j 1 ðv h Þ þ j 1 ¼ ð:16þ holds for any v h V k h. After integration by parts, this is equivalent to o t u h ð; tþv h ðþd aðþu h ð; tþo v h ðþd þ dau h jþ 1 ðv h Þ jþ dau 1 h j 1 ðv h Þ þ j ¼ 1 a ðþu h ð; tþv h ðþd ð:17þ where Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) dau h jþ 1 ¼ 1 ðað jþ 1Þþjað jþ 1ÞjÞðu hþ jþ þ 1 1 ðað jþ 1Þ jað jþ 1ÞjÞðu hþ þ jþ 1 denotes the Roe flu. This is the standard DG scheme for conservation laws with source terms using Roe flues [4] for Eq. (.15) which is equivalent to (.1). We can, therefore, use the standard techniques in the analysis for the DG schemes to obtain the following theoretical results.

5 4 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) Proposition.1. Suppose there is a constant b such that the derivative of a() satisfies a () < b for [a,b], then we have the following L stability for our scheme (.16): ku h ðtþk L 6 e bt= ku h ðþk L Proof. This follows from the standard proof of the cell entropy inequality for DG schemes applied to scalar conservation laws [11]. On each cell, we can prove as in [11] u h ðu h Þ t u h d a d þ ^F jþ 1 ^F j 1 þ H j ¼ for some entropy flu ^F jþ 1 and H j P. Summing over j, we have d dt b a u h d 6 Since a < b, we have b b a a u h d d b u h dt d 6 b u h d a a Integrating over t finishes the proof. h Proposition.. If a() and the solution u of (.1) are smooth and the scheme (.16) with the finite element space (.5) is used, then we have the following optimal L error estimate ku h ðtþ uðtþjj L 6 Ch kþ1 Proof. The proof is similar to that for standard DG schemes. The optimal (k + 1)th order of convergence is obtained through a special projection in the proof, see for eample [1] for the details. h For nonlinear problems, we notice that our scheme is consistent only when k P 1. For eample, for the Burgers equation u t þ u ¼ ; ð:1þ with u P, our upwind scheme (.6) with piecewise constant space (the space (.5) with k = ) gives ðu j Þ t þ u j u j 1 ¼ ð:19þ D j where u = u j on cell. This is clearly consistent with a different equation u t þ u ¼ and is inconsistent with the Burgers Eq. (.1). Ifk P 1, we can prove that our scheme is consistent, which is also verified by numerical eperiments in Section 4. For eample, the P 1 upwind scheme (.6) to solve the equation u t + H(u )=is! u j þ H u1 ½u j h Š j 1 þ a j ¼ ð:þ t D j D j where u ¼ u j þ u1 j j D j.3. Time discretization on cell. Since ½u h Š j 1 ¼ Oðh Þ for smooth functions, the scheme is consistent. Up to now, we have taken the method of lines approach and have left t continuous. We can use total variation diminishing (TVD) high-order Runge Kutta methods [17] to solve the method of lines ODE u t ¼ LðuÞ: ð:1þ

6 The third-order TVD Runge Kutta method that we use in this paper is given by u ð1þ ¼ u n þ DtLðu n Þ Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) u ðþ ¼ 3 4 un þ 1 4 uð1þ þ 1 4 DtLðuð1Þ Þ ð:þ u nþ1 ¼ 1 3 un þ 3 uðþ þ 3 DtLðuðÞ Þ Detailed description of the TVD Runge Kutta method can be found in [17], see also [7,]. 3. Two-dimensional case In this section, we consider the case of two spatial dimensions. The equation is given by u t þ Hðu ; u y ; ; yþ ¼; uð; y; Þ ¼u ð; yþ ð3:1þ and we again only consider the case where H(u,u y,,y) is a linear or conve function of u and u y. For simplicity of presentation, we consider in this paper only rectangular domains and cells, although our method can be easily defined on general triangulations as other DG methods. Suppose Eq. (3.1) is solved on the domain [a, b] [c, d]. We use rectangular meshes defined as a ¼ 1 < 3 <...< ¼ b; c ¼ < <...< y N þ 1 y1 y3 N ¼ d ð3:þ yþ 1 and I i;j ¼½ i 1 ; iþ 1 Š½y j 1 ; y jþ 1 Š; J i ¼½ i 1= ; iþ1= Š; K j ¼½y j 1= ; y jþ1= Š J iþ1= ¼½ i ; iþ1 Š; K jþ1= ¼½y j ; y jþ1 Š; i ¼ 1;...; N ; j ¼ 1;...; N y ð3:3þ We define the approimation space as V k h ¼ft : tj I i;j P k ði i;j Þ; i ¼ 1;...; N ; j ¼ 1;...; N y g ð3:4þ where P k (I i,j ) denotes all polynomials of degree at most k on I i,j. Let us denote H 1 ¼ oh ou and H ¼ oh ou y. We define our scheme as: find u h ð; tþ V k h, such that ðo t u h ð; y; tþþhðo u h ð; y; tþ; o y u h ð; y; tþ; ; yþþv h ð; yþddy I i;j þ 1 min H 1 ðo u h ; o y u h ; iþ1= ; yþ min H 1 ðo u h ; o y u h ; iþ1= ; yþ K J j iþ1= J iþ1= ½u h Šð iþ 1 ; yþv h ð yþdy iþ; 1 þ 1 ma H 1 ðo u h ; o y u h ; i 1= ; yþþ ma H 1 ðo u h ; o y u h ; i 1= ; yþ K J j i 1= J i 1= ½u h Šð i 1 ; yþv h ð þ yþdy i ; 1 þ 1 min H ðo u h ; o y u h ; ; y jþ1= Þ min H ðo u h ; o y u h ; ; y jþ1= Þ J yk i jþ1= yk jþ1= ½u h Šð; y jþ 1 Þv h ð; y jþþd 1 þ 1 ma H ðo u h ; o y u h ; ; y j 1= Þþ ma H ðo u h ; o y u h ; ; y j 1= Þ J yk i j 1= yk j 1= ½u h Šð; y j 1 Þv h ð; y þ ¼ j Þd 1 ð3:5þ holds for any v h V k h. In the above formula, we define o u h ¼ 1 ððo u h Þ þ þðo u h Þ Þ; o y u h ¼ 1 ððo yu h Þ þ þðo y u h Þ Þ The main idea is that, on the interfaces of cells, along the normal direction we would use the reconstructed information of the partial derivatives as in the one-dimensional case. Tangential to the interface, the average of the partial derivatives from the two neighboring cells is used. The reconstruction process is the same as that

7 44 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) in the one-dimensional case, ecept that we need to fi or y, then perform the reconstruction on the other spatial variable. If H 1 >orh >, we can apply the corresponding upwind scheme in that direction. A similar entropy correction procedure as in the one-dimensional case is adopted here as well for the cases with linear Hamiltonians with discontinuous coefficients or nonlinear Hamiltonians: 1. For each cell I i,j, determine if it is a potentially entropy violating cell. We will again provide the criteria for this determination in the numerical Section 4. If the cell I i,j is marked as a potentially entropy violating cell, then use Step below to update u h in this cell; otherwise, update u h by (3.5).. Update u h by the DG method of Hu and Shu [9] as reinterpreted by Li and Shu [14], namely, recover o u h and o y u h by taking the derivatives of u h, then compute o t (o u h ) and o t (o y u h ) by the usual DG method for the conservation laws satisfied by o u h and o y u h in a locally curl-free discontinuous Galerkin space. This will determine u h up to a constant. The missing constant is obtained by requiring ðo t u h þ Hðo u h ; o y u h ; ; yþþddy ¼ : ð3:6þ I i;j 4. Numerical results In this section, we provide numerical eperimental results to demonstrate the behavior of our schemes Linear smooth problems In this subsection, linear smooth problems are computed using our scheme. In this case, our scheme is equivalent to the standard DG scheme for conservation laws with source terms. Eample We solve the one-dimensional problem >< u t þ sinðþu ¼ uð; Þ ¼sinðÞ ð4:1þ >: uð; tþ ¼uðp; tþ The eact solution is uð; tþ ¼sin tan 1 e t tan ð4:þ We use the general scheme (.16) and list the results in Tables for P, P 1, P and P 3, respectively. We clearly observe (k + 1)th order of accuracy for P k polynomials. Eample We solve the two-dimensional linear Hamilton Jacobi equation with variable coefficients u t yu þ u y ¼ : ð4:3þ Table 4.1 Errors and numerical orders of accuracy for Eample when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.49E 1.6E 1.9E+.5E E E E E E E.9.4E.9.4E E.99.4E.99.1E 1 1. Final time t = 1. CFL =.9.

8 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) Table 4. Errors and numerical orders of accuracy for Eample when using P 1 polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.1E.5E.15E 1.31E E E E E E E E E E E E Final time t = 1. CFL =.3. Table 4.3 Errors and numerical orders of accuracy for Eample when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.4E 4.1E 3.5E 3.6E E 4..E E E E E E 6.93.E E E E 6.5 Final time t = 1. CFL =.1. Table 4.4 Errors and numerical orders of accuracy for Eample when using P 3 polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.1E 5.51E 5.9E 4.14E E 6 3..E E E E E E E E E E Final time t = 1. CFL =.5. The computational domain is [ 1,1]. The initial condition is given by >< :3 6 r u ð; yþ ¼ :3 r :1 < r < :3 ð4:4þ >: : r 6 :1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ð :4Þ þðy :4Þ. We also impose periodic boundary condition on the domain. This is a solid body rotation around the origin. The eact solution can be epressed as uð; y; tþ ¼u ð cosðtþþysinðtþ; sinðtþþycosðtþþ ð4:5þ For this problem, the derivatives of u are not continuous. Therefore, we do not epect to obtain (k + 1)th order of accuracy for P k polynomials, see Table 4.5. At t =p, i.e. the period of rotation, we take a snapshot at the line = y in Fig We can see that a higher order scheme can yield better results for this nonsmooth initial condition. Eample We solve the same Eq. (4.3) as that in Eample 4.1., but with a different initial condition as! u ð; yþ ¼ep ð :4Þ þðy :4Þ ð4:6þ r

9 46 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) Table 4.5 Errors and numerical orders of accuracy for Eample 4.1. when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N N cells N N L 1 error Order L error Order L 1 error Order.41E 3.13E.11E E E E..47E E E E E E.1 Final time t = 1. CFL = Fig Eample uniform mesh. t = p. Solid line: the eact solution; rectangles: the numerical solution. One-dimensional cut of 45 with the ais. Left: P 1 polynomial; right: P polynomial. We take r =.5 such that at the domain boundary, u is very small, hence imposing a periodic boundary condition will lead to small non-smoothness errors. We then observe the desired order of accuracy in Table 4.6. Eample We solve the two-dimensional linear Hamilton Jacobi equation with variable coefficients u t þ f ð; y; tþu þ gð; y; tþu y ¼ ð4:7þ The computational domain is still [ 1,1], and the advection coefficients are f ð; y; tþ ¼sin t ðpþ sinðpyþ cos T p ; gð; y; tþ ¼ sin t ðpyþ sinðpþ cos T p where T is the period of deformation. The initial condition is given by Table 4.6 Errors and numerical orders of accuracy for Eample when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N N cells N N L 1 error Order L error Order L 1 error Order.14E.1E 1.E E E.1.53E E E E E E E 3.69 Final time t = 1. CFL =.1.

10 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) >< :3 6 r u ð; yþ ¼ :3 r :1 < r < :3 ð4:þ >: : r 6 :1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ð :4Þ þðy :4Þ. This is a numerical test for incompressible flow first introduced by LeVeque in [13]. During the evolution, the initial data are severely deformed, then it returns to the original shape after one period. At t = 1.5, i.e. the period of rotation, we take a snapshot at the line = y in Fig. 4.. We can clearly observe that a higher order scheme yields better results for this nonsmooth initial condition. 4.. Linear nonsmooth problems In this subsection, the Hamiltonian H is a linear function of $u with nonsmooth coefficients. Eample We solve the model problem >< u t þ signðcosðþþu ¼ uð; Þ ¼sinðÞ >: uð; tþ ¼uðp; tþ ð4:9þ The eact solution is given by if 6 t 6 p/ sinð tþ if 6 6 p >< sinð þ tþ if p uð; tþ ¼ < 6 3p t 1 if 3p t < 6 3p þ t >: sinð tþ if 3p þ t < 6 p ð4:1þ Fig. 4.. Eample uniform mesh. t = 1.5. Solid line: the eact solution; rectangles: the numerical solution. One-dimensional cut of 45 with the ais. Left: P 1 polynomial; right: P polynomial.

11 4 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) Fig Eample t = 1, CFL =.1, using P polynomials. Solid line: the eact solution; rectangles: the numerical solution. Left: N = ; right: N = 1. if p/ 6 t 6 p 1 if 6 6 t p >< sinð tþ if t p uð; tþ ¼ < 6 p sinð þ tþ if p < 6 3p t >: 1 t < 6 p if t P p uð; tþ ¼ 1: if 3p ð4:11þ ð4:1þ For the viscosity solution, at ¼ p, there will be a shock forming in u, and at ¼ 3p, there is a rarefaction wave. We first test the scheme without any entropy correction. If we take N to be a multiple of 4, then the discontinuity of a() is eactly located at a cell interface. In this case, the entropy condition is violated by our scheme at the two cells neighboring 3p, and the numerical solution obtained is not close to the viscosity solution, see Fig. 4.3, left. If instead, we take other values of N such that the discontinuity of a() is not at the cell interface, then the entropy condition is not violated and the numerical solution obtained approimates the viscosity solution very well, see Fig. 4.3, right. The test above indicates the necessity of an entropy correction in this case. The criteria for the entropy correction is as follows. For the cell =( j-1/, j+1/ ), if a ð j 1= Þ < < a þ ð j 1= Þ ð4:13þ or a ð jþ1= Þ < < a þ ð jþ1= Þ ð4:14þ is satisfied, we will compute (u ) t on the cell by solving the conservation law for u = u as u t þðsignðcosðþþuþ ¼ using the standard DG method with polynomials in P k-1, and then recover u by requiring ðu t þ signðcosðþþu Þd ¼ : ð4:15þ ð4:16þ

12 For this one-dimensional eample it does not increase the computational cost. We can see in Fig. 4.4 that, after this entropy correction, the numerical solution approimates the viscosity solution very well. The numerical errors and order of accuracy are shown in Table 4.7. Since the eact solution is not smooth, we do not epect the full (k + 1)th order accuracy Nonlinear smooth problems Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) In this subsection, the Hamiltonian H is a nonlinear smooth function of $u. Eample One-dimensional Burgers equation u t þ u >< ¼ uð; Þ ¼sinðÞ >: uð; tþ ¼uðp; tþ ð4:17þ The eact solution when u is still smooth is obtained by the characteristics methods. First solve from ¼ þ cosð Þt ð4:1þ then get u as Fig Eample t = 1, CFL =.1, N =, using P polynomials. Solid line: the eact solution; rectangles: the numerical solution. Left: without entropy correction; right: with entropy correction. Table 4.7 Errors and numerical orders of accuracy for Eample 4..1 when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.64E 3.15E.41E.16E E E E E E E 4..5E E E 5..64E 5..16E 4. Final time t = 1. CFL =.1.

13 41 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) uð; tþ ¼sinð Þþ cosð Þ t ð4:19þ When t =.5, the solution is still smooth, and the epected third order accuracy is obtained for P polynomials, see Table 4.. After t = 1, a shock will form in u, our scheme can resolve the derivative singularity sharply, see Fig Eample One-dimensional Burgers equation with a nonsmooth initial condition u t þ u ¼ >< p if 6 6 p uð; Þ ¼ p else where in ½; pš; >: uð; tþ ¼uðp; tþ ð4:þ For the viscosity solution, the sharp corner at p will be smoothed out, and a rarefaction wave will form in the derivative. Since the entropy condition is violated by our Roe type scheme, we need to apply the entropy correction procedure. Fig. 4.6 shows the comparison of the numerical solution with and without the entropy correction. Clearly the entropy correction is needed to obtain a good approimation to the entropy solution. The criteria for the entropy correction is as follows. For the cell =( j-1/, j+1/ ), if either u ð j 1=Þ < < u þ ð j 1=Þ ð4:1þ Table 4. Errors and numerical orders of accuracy for Eample when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.13E 4.E 4.4E 4.17E E E E E E E E 7.97.E E.99.59E.99.5E 7.97 Final time t =.5. CFL = Fig Eample Numerical solution. Solid line: N = 5; rectangles: N = 4. Final time t = 1.5, CFL =.5, P polynomials.

14 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) Fig Eample t = 1, CFL =.5, N =, using P polynomials. Solid line: the eact solution; rectangles: the numerical solution. Left: without entropy correction; right: with entropy correction. or u ð jþ1=þ < < u þ ð jþ1=þ is satisfied, then the entropy correction is needed. ð4:þ Eample Two-dimensional Burgers equation. ( u t þ ðu þu y þ1þ ¼ ð4:3þ uð; y; Þ ¼ cosð þ yþ with periodic boundary condition on the domain [,p]. We use a uniform rectangular mesh. At t =.1, the solution is still smooth. Numerical errors and order of accuracy are listed in Table 4.9, demonstrating the epected order of accuracy. At t = 1, the solution is no longer smooth. We plot the numerical solution in Fig We observe good resolution of the kinks in the solution Nonlinear nonsmooth problems In this subsection, the Hamiltonian H is a nonlinear nonsmooth function of $u. Table 4.9 Errors and numerical orders of accuracy for Eample when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N N cells N N L 1 error Order L error Order L 1 error Order 1 1.3E.43E.35E 1.3E 3.9.5E E E E 4.91.E E E E 3.55 Final time t =.1. CFL =.1.

15 41 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) y 4 6 Fig Eample Numerical solution when t = 1, CFL =.1, 4 4 uniform mesh, using P polynomials. 6 t Fig. 4.. Eample t = 1, CFL =.1, N = 16 uniform mesh, using P polynomials. e =1 1. Rectangular symbols mark the cells in which the entropy correction is performed. Those cells are plotted every five time steps. Eample We solve the Eikonal equation given by >< u t þju j¼ uð; Þ ¼sinðÞ >: uð; tþ ¼uðp; tþ ð4:4þ The eact solution is the same as the eact solution of Eample 4..1, given by (4.1) (4.1). Because the entropy condition is violated by our scheme in some cells, we need to apply the entropy correction technique. The criteria are as follows. If we denote u = u, then for the cell =( j-1/, j+1/ ), if u ð j 1= Þ < e and e < u þ ð j 1= Þ ð4:5þ or u ð jþ1= Þ < e and e < u þ ð jþ1= Þ ð4:6þ are satisfied, we use the entropy correction on. We take the parameter e =1 1 in the calculation, which is introduced to avoid unnecessary entropy corrections due to small numerical errors in the computation.

16 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) Table 4.1 Errors and numerical orders of accuracy for Eample when using P polynomials and Runge Kutta third order time discretization on a uniform mesh of N cells N L 1 error Order L error Order L 1 error Order 4.7E 3.15E.E.3E E E E E E E E E Final time t = 1. CFL = y y Fig Eample Steady state solution with 4 4 uniform mesh, using P polynomials. Left: three-dimensional plot; right: contour plot. Fig. 4. shows the space time location where the entropy correction is applied. We observe that the correction is mostly applied at a few cells neighboring the boundary of the rarefaction wave. The number of cells in which the correction is performed is relatively small compared to the total number of cells. The numerical errors and the order of accuracy are listed in Table 4.1. Since the solution is not smooth, we do not epect the full (k + 1)th order accuracy. Eample We solve the two-dimensional Eikonal equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u t þ u þ u y ¼ 1 ð4:7þ First of all, we consider the case of the computational domain being [, 1] n[.4,.6]. For the inner boundary along [.4,.6], we impose the boundary condition u =. On the other hand, we impose free outflow boundary conditions on the outer boundary. The initial condition is taken as u (,y) = ma{j.5j, jy.5j}.1. The steady state solution should give us a function that is equal to the distance of the point to the inner boundary. For the outer boundary cells, we use the upwind version of our scheme according to the direction of the local wind. For all other cells, the general scheme (3.5) is used. We plot the numerical steady state solution in Fig Net, we consider this eample with a point source condition; namely, we take the inner boundary to be the center point (.5,.5). In this case, we would need u in the center cell to be the L projection of the eact distance function. For all other cells, the computation is the same as for the previous case. The initial

17 414 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) y y Fig Eample Steady state solution with uniform mesh, using P polynomials. Left: three-dimensional plot; right: contour plot. condition is taken as u (,y) = ma{j.5j, jy.5j}. We plot the numerical steady state solution in Fig We can see that in both cases we obtain very good resolution to the viscosity solution. 5. Concluding remarks We have developed a discontinuous Galerkin finite element method for solving Hamilton Jacobi equations approimating directly the solution variable rather than its derivatives as in the earlier work in [9,14]. Both linear and conve nonlinear Hamiltonians are considered in this paper, while the case for non-conve Hamiltonians is left for future study. One and two-dimensional numerical results demonstrate that the method approimates the viscosity solutions very well. In the future, we will also eplore more direct entropy correction techniques without resorting to the techniques in [9,14]. We remark that Osher and Yan [15] has recently developed another class of discontinuous Galerkin type scheme for solving Hamilton Jacobi equations, which also approimates directly the solution variable rather than its derivatives. A comparison of these two different approaches would be interesting. Acknowledgments Research supported by ARO Grant W911NF , NSF Grant DMS and AFOSR Grant FA References [1] B. Cockburn, S. Hou, C.-W. Shu, The Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Mathematics of Computation 54 (199) [] B. Cockburn, S.-Y. Lin, C.-W. Shu, TVB Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, Journal of Computational Physics 4 (199) [3] B. Cockburn, C.-W. Shu, The Runge Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Mathematical Modelling and Numerical Analysis 5 (1991) [4] B. Cockburn, C.-W. Shu, TVB Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation 5 (199) [5] B. Cockburn, C.-W. Shu, The Runge Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics 141 (199)

18 Y. Cheng, C.-W. Shu / Journal of Computational Physics 3 (7) [6] M. Crandall, P.L. Lions, Viscosity solutions of Hamilton Jacobi equations, Transactions of the American Mathematical Society 77 (193) 1 4. [7] S. Gottlieb, C.-W. Shu, Total variation diminishing Runge Kutta schemes, Mathematics of Computation 67 (199) [] S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability preserving high order time discretization methods, SIAM Review 43 (1) [9] C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton Jacobi equations, SIAM Journal on Scientific Computing 1 (1999) [1] G. Jiang, D. Peng, Weighted ENO schemes for Hamilton Jacobi equations, SIAM Journal on Scientific Computing 1 (1999) [11] G. Jiang, C.-W. Shu, On cell entropy inequality for discontinuous Galerkin methods, Mathematics of Computation 6 (1994) [1] O. Lepsky, C. Hu, C.-W. Shu, The Analysis of the discontinuous Galerkin method for Hamilton Jacobi equations, Applied Numerical Mathematics 33 () [13] R. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SIAM Journal on Numerical Analysis 33 (1996) [14] F. Li, C.-W. Shu, Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton Jacobi equations, Applied Mathematics Letters 1 (5) [15] S. Osher, J. Yan, A new discontinuous Galerkin method for the Hamilton Jacobi equation, in preparation. [16] S. Osher, C.-W. Shu, High order essentially non-oscillatory schemes for Hamilton Jacobi equations, SIAM Journal on Numerical Analysis (1991) [17] C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics 77 (19) [1] J. Yan, C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis 4 () [19] Y.-T. hang, C.-W. Shu, High order WENO schemes for Hamilton Jacobi equations on triangular meshes, SIAM Journal on Scientific Computing 4 (3)