High-Order Central WENO Schemes for 1D Hamilton-Jacobi Equations

Transcription

1 High-Order Central WENO Schemes for D Hamilton-Jacobi Equations Steve Bryson and Doron Levy Program in Scientific Computing/Computational Mathematics, Stanford University and the NASA Advanced Supercomputing Division, NASA Ames Research Center, Moffett Field, CA ; bryson@nas.nasa.gov Department of Mathematics, Stanford University, Stanford, CA ; dlevy@math.stanford.edu Introduction We consider high-order central approximations for solutions of one-dimensional Hamilton-Jacobi HJ) equations of the form t φx, t)+h φ x,x)=0, x R, ) subject to the initial data φx, t=0) = φ 0 x). Solutions for ) with smooth initial data typically remain continuous but develop discontinuous derivatives in finite time. Such solutions are not unique; the physically relevant solution is known as the viscosity solution see[,3,4,5,8,5]andthereferences therein). Various numerical methods were proposed in order to approximate the solutions of ). Examples for such methods are the high-order Godunov-type schemes that were introduced in [0, ], and were based on an Essentially Non-Oscillatory ENO) reconstruction step [7] that was evolved in time with a first-order monotone flux. The least dissipative monotone flux, the Godunov flux, requires solving Riemann problems at cell interfaces. A fifth-order Weighted ENO WENO) scheme, based on [0, 8], was introduced by Jiang and Peng [9]. Recently, Lin and Tadmor introduced in [6, 7] central schemes for approximating solutions of the HJ equation. These schemes are based on the Nessyahu-Tadmor scheme for approximating solutions of hyperbolic conservation laws [9]. Unlike upwind schemes, central schemes do not require Riemann solvers, which makes them attractive for solving systems of equations and for multi-dimensional problems. A second-order semi-discrete version of these schemes was introduced by Kurganov and Tadmor in []. While less dissipative, the semi-discrete scheme requires the estimation of the local speed of propagation, which is computationally intensive in particular in multi-dimensional problems. In a later work [], the numerical viscosity was further reduced by computing more precise information about local speed of propagation. To address the problem of schemes that are too computationally intensive, we introduced in [] efficient first- and second-order central schemes for approximating the solutions of multi-dimensional versions of ).

2 Steve Bryson and Doron Levy Unlike the previous attempts, our schemes in [] scale well with increasing dimension. In this paper we derive fully-discrete Central WENO CWENO) schemes for approximating solutions of ), which combine our previous works [, 3, 4]. We introduce third- and fifth-order accurate schemes, which are the first central schemes for the HJ equations of order higher than two. The core ingredient in the derivation of our schemes is a high-order CWENO reconstructions in space. Acknowledgment: We would like to thank Volker Elling for helpful discussions. CWENO Schemes for HJ Equations We are interested in approximating solutions of ) subject to the initial data φx, t =0) = φ 0 x). For simplicity we assume a uniform grid grid in space and time with mesh spacings, h := x and t. We denote the grid points by x i = i x, t n = n t, and the fixed mesh ratio by λ = t/ x. Let ϕ n i denote the approximate value of φ x i,t n ), and ϕ x ) n i denote the approximate value of the derivative φ x x i,t n ). We define ϕ n i := ϕn i+ ϕn i, ϕ n i := ϕn i ϕn i and 0 ϕ n i := ϕn i+ ϕn i. We assume that the approximate solution at time t n, ϕ n i is given. In order to approximate the solution at the next time step t n+, ϕ n+ i, we start by reconstructing a continuous piecewise-polynomial from the data, ϕ n i,and sample it at the half-integer points, {x i+/ }, in order to obtain the pointvalues of the interpolant at these points ϕ n i+/ as well as the derivative, ϕ i+/. We then evolve ϕn from time t n to time t n+ according to ), i+ t n+ )) ϕ x i+,tn+) = ϕ x i+,tn) H ϕ x x i+,t dt. ) t n This evolution is done at the half-integer grid points where the reconstruction is smooth as long as the CFL condition λ H ϕ x ) / is satisfied). Finally, in order to return to the original grid, we project ϕ n+ i+/ back onto the integer grid points {x i } toendupwithϕ n+ i. Since the evolution step ) is done at points where the solution is smooth, we can approximate the time integral at the RHS of ) using a sufficiently accurate quadrature rule. For example, for a third- and fourth-order method, this integral can be replaced by a Simpson s quadrature, t n+ t n )) H ϕ x x i+,t dt t [ H ϕ x x 6 i+,tn)) 3) )) +4H ϕ x x i+,tn+ +H ϕ x x i+,tn+))].

3 CWENO Schemes for HJ Equations 3 The intermediate values of the derivative in time, ϕ x xi+/,t n+/),and ϕ x xi+/,t n+), which are required in the quadrature 3), can be predicted using a Taylor expansion or with a Runge-Kutta RK) method. For details we refer the reader to [3, 9] and the references therein. The remaining ingredient is the piecewise-polynomial reconstruction in space. A careful study of the above procedure reveals that there are actually three different quantities that should be recovered in every time step. First, given ϕ i at time t n we need to reconstruct the point-values at the half-integer grid points, ϕ i+/, at the same time t n. This is the first term on the RHS of ). The second term on the RHS of ) requires evaluating the Hamiltonian H at the derivative ϕ i+/. Hence, the second quantity we should recover is ϕ i+/ from ϕ i. Finally, the predictor step that provides the values at the quadrature nodes in 3), require us to estimate ϕ i+/ from ϕ i+/ at every step of the RK method. In the next two sections we will focus on the reconstruction of these three quantities, first for a third-order method and then for a fifth-order method. The projection from ϕ n+ i+/ onto the original grid points to get ϕn+ i is accomplished using the same reconstruction used to approximate ϕ n i+/ from ϕ n i.. A Third-Order Scheme Following the above procedure, a third-order scheme can be generated by combining a third-order accurate ODE solver in time with a sufficiently high-order reconstruction in space. Here we present fourth-order CWENO reconstructions of the point values of ϕ i+/ and its derivative ϕ i+/. The reconstruction of ϕ i+/ from ϕ i. In order to obtain a fourth-order reconstruction of ϕ i+/ we will write a convex combination of two quadratic polynomials, ϕ [] constructed on a stencil which is left-biased with respect to x i+/, and the right-biased ϕ [] +, ϕ [] x) =ϕ i + ) ϕ i x xi )+ h ϕ [] + x) =ϕ i + h h ) ϕ i x xi )x x i+ )+O h 3), ) ϕ i x xi )+ h ) ϕ i x xi )x x i+ )+O h 3). An evaluation of these approximations at {x i+ } reads ) ϕ [] = ) 8 ϕ i +6ϕ i +3ϕ i+ ), ϕ [] + = 8 3ϕ i+6ϕ i+ ϕ i+ ). x i+ A straightforward computation shows that x i+ ϕ[] x i+ )+ ϕ[] + x i+ )=ϕ i+ + O h 4).

4 4 Steve Bryson and Doron Levy The fourth-order WENO estimate of ϕ i+/ is therefore given by the convex combination ) ) ) ϕ [4] w x i+ = w ϕ [] i+ x i+ + w + ϕ [] i+ + x i+, where the weights satisfy w i+/ + w+ i+/ =,w± i+/ 0, i. Insmooth regions we would like to satisfy wi w i + to attain an O ) h 4) error, while when the stencil {x i,x i,x i+,x i+ } supporting ϕ w x i+ contains a discontinuity, the weight of the more oscillatory polynomial should vanish. Following [0, 8], we meet these requirements by setting w k i+ = αk i+ l αl i+, α k i+ = c k ) p 4) ɛ + S k i+ where k, l {+, } k and l will range over a larger space of symbols when we use more interpolants). The constants c ± =/ and are independent of the grid-point. We choose ɛ as 0 6 to prevents the denominator in 4) from vanishing, and set p = see [0]). The smoothness measures S i ± should be large when ϕ is nearly singular. Following the standard practice with WENOtype schemes [0], we take S ± i to be the sum of the L -norms of the first and second derivatives on the stencil supporting ϕ [] ±. If we approximate the first derivative at x i+/ by h + ϕ i+/, the second derivative by h ϕ i+/, and define the smoothness measure S i+ [r, s] =h s j=r h + ϕ i+j+ ) + h s j=r+ then for the fourth-order interpolation of ϕ w x i+ S i+/ [, 0] and S + i+/ = S i+/ [0, ]. ) h + ϕ i+j+, 5) ) we have S i+/ = The reconstruction of ϕ i+/ from ϕ i. To obtain a fourth-order estimate of the derivative ϕ x i+/ ) from ϕx i ), we start from the cubic interpolants x) =ϕ i + ) ϕ i x xi )+ h h ) ϕ i x xi )x x i+ ) + 6h 3 ) ϕ i x xi )x x i+ )x x i )+O h 4), + x) =ϕ i + ) ϕ i x xi )+ h h ) ϕ i x xi )x x i+ ) + 6h 3 ) ϕ i x xi )x x i+ )x x i+ )+O h 4). Differentiating ± at x i+

5 ϕ [3],i+ ϕ [3] +,i+ CWENO Schemes for HJ Equations 5 = 4h ϕ i 3ϕ i ϕ i +3ϕ i+ ), = 4h 3ϕ i +ϕ i+ +3ϕ i+ ϕ i+3 ). Again, ϕ [3] +,i+ ϕ [3] = ϕ +,i+ i+ + O h 4), ) and a fourth-order WENO reconstruction of ϕ x i+ is ϕ [4] i+/ = w ϕ [3] + w + ϕ [3] i+,i+ i+ +,i+ where the weights are of the form 4) with c ± = / and S i+/ = S i+/ [, 0] and S + i+/ = S i+/ [0, ]. The reconstruction of ϕ i+/ from ϕ i+/. Repeating the above procedure, this time with three quadratic interpolants ϕ [] x) =ϕ i+ + ) ) ϕ h i+ x x i+ + ) ) ) h ϕ i+ x x i+ x x i+ 3 + O h 3), ϕ [] 0 x) =ϕ i+ + ) ) 0 ϕ h i+ x x i+ + ) ) ) h ϕ i+ x x i x x i+ 3 + O h 3), ϕ [] + x) =ϕ i+ + ) ) ϕ h i+ x x i+ + ) ) ) h ϕ i+ x x i+ x x i+ 3 + O h 3), results with 6 ϕ [] +,i+ 3 ϕ [] + 0,i+ 6 ϕ [] = ϕ +,i+ i+ + O h 4), where ϕ [],i+ ϕ [] +,i+ = h ϕ i 3 4ϕ i +3ϕ i+ ), ϕ [] 0,i+ = h 3ϕ i+ +4ϕ i+ 3 ϕ i+ 5 ). The fourth-order WENO estimate of ϕ i+/ is = h ϕ i+ 3 ϕ i ), ϕ [4] i+/ = w ϕ [] + w 0 i+,i+ i+ ϕ [] + w + ϕ [] 0,i+ i+ +,i+ where the weights w are of the form 4) with c = c + =/6,c 0 =/3, and the oscillatory indicators S i+/ = S i+/ [, ], S i+/ = S i+/ [, 0], and S + i+/ = S i+/ [0, ].

6 6 Steve Bryson and Doron Levy. A Fifth-Order Scheme Once again, similarly to the third-order scheme, we need to reconstruct the point-values of ϕ and ϕ. We start with the reconstruction of ϕ i+/ and ϕ i+/ from ϕ i. We write sixth-order interpolants as a convex combination of cubic interpolants, x) and + x) introduced above and 0 x) =ϕ i + ) ϕ i x xi )+ h h ) ϕ i x xi )x x i+ ) + 6h 3 ) ϕ i x xi )x x i+ )x x i+ )+O h 4). In this case where In a similar way, where 3 6 ϕ[3] + 5,i+ 8 ϕ[3] + 3 0,i+ 6 ϕ[3] +,i+,i+ 0,i+ +,i+ = ϕ i+ + O h 6), = 6 ϕ i 5ϕ i +5ϕ i +5ϕ i+ ), = 6 ϕ i +9ϕ i +9ϕ i+ ϕ i+ ), = 6 5ϕ i +5ϕ i+ 5ϕ i+ + ϕ i+3 ) ϕ [3] + 49,i+ 40 ϕ [3] 9 0,i+ 80 ϕ [3] = ϕ +,i+ i+/ + O h 6), ϕ [3],i+ ϕ [3] 0,i+ ϕ [3] +,i+ = 4h ϕ i 3ϕ i ϕ i +3ϕ i+ ), = 4h ϕ i 7ϕ i +7ϕ i+ ϕ i+ ), = 4h 3ϕ i +ϕ i+ +3ϕ i+ ϕ i+3 ). The sixth-order WENO estimates for ϕ i+/ and ϕ i+/ are ϕ [6] i+ ϕ [6] i+ = w + w 0 i+,i+ i+ + w +, 0,i+ i+ +,i+ = w ϕ [3] + w 0 i+,i+ i+ ϕ [3] + w + ϕ [3], 0,i+ i+ +,i+ where the weights for ϕ are given by 4), with c = c + =3/6,c 0 =5/8 and the oscillatory indicators are S i+/ = S i+/ [, 0], Si+/ 0 = S i+/ [, ] and S + i+/ = S i+/ [0, ]. The negative weights for ϕ require special treatment see [] for details). Following [] we split the positive and negative

7 CWENO Schemes for HJ Equations 7 weights in the following way: first, we set γ = γ+ =9/40, γ0 =49/40 and γ+ = γ+ + =9/80, γ0 + =49/0. Then, For k, l {, 0, +}, setσ ± = k γk ± so that similarly to 4), α k ±,i+ = γ k ± σ ± ɛ + S k i+ ) p and w k i+ α k +,i+ = σ + l αl +,i+ α k,i+ σ. l αl,i+ Because i+/ and ϕ [3] i+/ are defined on the same stencils, they use the same smoothness measures S i+/. All that is left is the reconstruction of ϕ i+/ from ϕ i+/. In this case a sixth-order approximation to ϕ i+/ requires a weighted sum of four cubic interpolants. This reconstruction is similar to the previous ones. We skip the details and summarize the result: where ϕ [6] = w ϕ [3] + w 0 ϕ [3] + w 0+ ϕ [3] + w + ϕ [3], i+ i+,i+ i+ 0,i+ i+ 0+,i+ i+ +,i+ ϕ [3],i+ ϕ [3] 0,i+ ϕ [3] 0+,i+ ϕ [3] +,i+ = 6h ϕ i 5 +9ϕ i 3 8ϕ i +ϕ i+ ), = 6h ϕ i 3 6ϕ i +3ϕ i+ +ϕ i+ 3 ), = 6h ϕ i 3ϕ i+ +6ϕ i+ 3 ϕ i+ 5 ), = 6h ϕ i+ +8ϕ i+ 3 9ϕ i+ 5 +ϕ i+ 7 ). Here, c = c + =/0,c 0 = c 0 +=9/0, S i+/ = S i+/ [ 3, ], S 0 i+/ = S i+/ [, 0], S 0+ i+/ = S i+/ [, ] and S + i+/ = S i+/ [0, ]. 3 Numerical Examples In all our numerical simulations, the ODE solvers we use are the non-linear fourth-order Strong-Stability Preserving Runge-Kutta SSP-RK) methods of [6]. We start by testing the accuracy of our new CWENO methods when approximating the solution of the linear advection equation, ϕ t + ϕ x =0. The initial data is taken as ϕ x, 0) = sin 4 πx), the mesh ratio λ =0.9 and the time T = 4. The results obtained with the fifth-order method of. are shown in Table.

8 8 Steve Bryson and Doron Levy Table. Error and convergence rate for linear advection with initial condition ϕ x, 0) = sin 4 πx) N L error L order Next, we test the CWENO methods with two nonlinear Hamiltonians: a convex Hamiltonian ϕ t + ϕ x +) = 0 and a non-convex Hamiltonian ϕ t cos ϕ x + ) = 0. The interval is [0, ], the boundary conditions are periodic and the initial conditions for both Hamiltonians are taken as ϕ x, 0) = cos πx). The exact solution to both problems is smooth until t /π, after which a singularity forms. A second singularity forms in the non-convex H example at t.9/π. The results of the accuracy test with the fifth-order method are shown in Table, and the solution at time T =.5/π is plotted in Figure. Following [9] the errors in Table after the formation of the singularity are computed at a distance of 0. away from any singularities. Table. L Error and convergence rate estimates for convex and non-convex Hamiltonians. top: T =0.5/π, bottom: T =.5/π. λ =0.3 N convex convex non-convex non-convex L error L order L error L order N convex convex non-convex non-convex L error L order L error L order

9 CWENO Schemes for HJ Equations approximation exact.5 approximation exact x x com- Fig.. left: Convex Hamiltonian right: non-convex Hamiltonian at T =.5 π pared with the exact solution, N = 00. References. Barles G., Solution de viscosité des équations de Hamilton-Jacobi, Springer- Verlag, Berlin, Bryson S., Levy D., Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations, NASA Technical Report NAS 0 0, 00, submitted. 3. Crandall M.G., Evans L.C., Lions P.-L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 8, 984), pp Crandall M.G., Ishii H., Lions P.-L., User s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 7, 99), pp Crandall M.G., Lions P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., ), pp Gottlieb S., Shu C.-W., Tadmor E., Strong stability-preserving high order time discretization methods, SIAMReview, 43, 00), pp Harten A., Engquist B., Osher S., Chakravarthy S., Uniformly High Order Accurate Essentially Non-oscillatory Schemes III, JCP, 7, 987), pp Kruzkov S.N., The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first order with several variables, Soviet Math. Dokl., 5, 964), pp Jiang G.-S., Peng D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comp.,, 000), pp Jiang G.-S., Shu C.-W., Efficient Implementation of Weighted ENO Schemes, JCP, 6, 996), pp Kurganov A., Noelle S., Petrova G., Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAMJ.Sci. Comp., to appear.. Kurganov A., Tadmor E., New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations, JCP, 60, 000), pp Levy D., Puppo G., Russo G., Central WENO schemes for hyperbolic systems of conservation laws, Math. Model. and Numer. Anal., 33, no ), pp Levy D., Puppo G., Russo G., Compact central WENO schemes for multidimensional conservation laws, SIAMJ. Sci. Comp.,, 000), pp Lions P.L., Generalized solutions of Hamilton-Jacobi equations, Pitman, London, 98.

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