ICES REPORT A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws

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1 ICES REPORT 7- August 7 A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws by Todd Arbogast, Chieh-Sen Huang, and Xikai Zhao The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 787 Reference: Todd Arbogast, Chieh-Sen Huang, and Xikai Zhao, "A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws," ICES REPORT 7-, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, August 7.

2 A MULTILEVEL-WENO TECHNIQUE FOR SOLVING NONLINEAR CONSERVATION LAWS TODD ARBOGAST, CHIEH-SEN HUANG, AND XIKAI ZHAO Abstract. We present a multi-level WENO polynomial reconstruction technique. Polynomials of various orders defined over various size and shifted stencils may be combined by using novel weighting strategy. Numerical results show that the technique provides some improvement over standard WENO reconstructions. Weighted essentially non-oscillatory, polynomial reconstruction, hyperbolic con- Key words. servation law AMS subect classifications. 65M8, 65M6, 76M, 76M. Introduction. We consider finite volume and finite difference methods for systems of nonlinear, hyperbolic conservation laws of the form u t + f(x) x =, t >, x R, u R d, d. (.) (For simplicity, we do not treat boundary conditions, although we might also consider a bounded space domain if we impose periodic boundary conditions.) Weighted essentially non-oscillatory (WENO) schemes are a popular choice for solving such systems of equations. They allow one to reconstruct a high order version of the solution merely from approximations of cell averages (in finite volume schemes) or point values (in finite difference schemes). The key is to average approximations defined on various stencils, and to weight them so as to avoid stencils containing a shock discontinuity in the solution. We present a novel multi-level WENO polynomial reconstruction technique. Polynomials of various orders defined over various size and shifted stencils may be combined by using novel weighting strategy. Numerical results in -D for scalar problems and Euler systems show that the technique provides some improvement over standard WENO reconstructions.. Standard WENO framework. To establish the context for our new multilevel WENO technique and to fix notation, we review the standard theory. For simplicity of exposition, we consider the finite volume framework. The finite difference framework is similar. Partition time as = t < t < < t n. In the domain of (.), take arbitrary grid points < x < x < x <, and define the cell I i = [x i, x i+ ], and the length of the cell x i = x i+ x i. The mid point of I i is denoted as x i+ (x i + x i+ ). Let ū n i be the numerically approximated average of u on the cell I i at This work was supported by the U.S. National Science Foundation under grant DMS-875. Department of Mathematics, University of Texas, 55 Speedway, C, Austin, TX 787- and Institute for Computational Engineering and Sciences, University of Texas, East th St., C, Austin, TX (arbogast@ices.utexas.edu) Department of Applied Mathematics and National Center for Theoretical Sciences, National Sun Yat-sen University, Kaohsiung 8, Taiwan, R.O.C. (huangcs@math.nsysu.edu.tw) Department of Mathematics, University of Texas, 55 Speedway, C, Austin, TX 787- and Institute for Computational Engineering and Sciences, University of Texas, East th St., C, Austin, TX (xzhao@math.utexas.edu) =

3 ARBOGAST, HUANG, AND ZHAO time t n, so ū n i u(x, t n ) dx. x i I i Now assume u is smooth. For a kth order approximation of u on the given cell I i, we consider the ordered stencil S i, which contains r cells to the left and s cells to the right of I i with r + s + = k, i.e., S i = {I i r,..., I i,..., I i+s }, r + s + = k, (.) from which we obtain the kth order reconstruction polynomial P i by imposing the interpolation conditions P i (x) dx = ū m, for all I m S i. x m I m.. Background on WENO Reconstruction. Given the cell I i, suppose we are interested in a (k )st general WENO reconstruction. From now and for the remainder of the paper, we fix a value of i and drop it from the notation. First consider all the small stencils S k containing I i and with k cells (k ) S k = {I i k++,..., I i+ }, k. From each S k we can reconstruct P k of degree k. Moreover, we can define the large stencil S k = Sk and reconstruct a higher order polynomial P k of degree (k ). At a fixed point x, the polynomial P k can usually be written as a convex combination of P k, so P k (x) = α P k (x), α =. (.) where we drop the subscript i. We refer to α as a linear weight. When there are discontinuities in the data over stencil S i, we want to make use of the relatively small stencils on which u is smooth in order to achieve the essentially non-oscillatory property. The standard WENO reconstruction is where α = R(x) = ˆα m ˆα, ˆα = m α P k (x), α α =, (.) (ɛ + σ P k, for all, (.) ) τ for some smoothness indicator σ P k and exponent τ (usually τ = and ɛ = 6 > avoids any possibility of division by zero). We refer to α as a nonlinear weight. The smoothness indicator σ defined by Jiang and Shu in [] is normally used to measure the smoothness of the reconstruction polynomials on the cell I i. It is given by k ( xi+ d m σ P k = ( x i ) m P k(x) ) dx m dx, (.5) m= x i

4 A Multilevel-WENO Technique where again k is the degree of the polynomial P k. If the grid is uniform in the cell size x = h, then in the regions where u is smooth, the Taylor expansion of (.5) gives σ P k = (u h) ( + O(h )), which implies α = α + O(h ). If there are discontinuities in u within some of the stencils S k, then σ P = O(). So the weights of the smooth and non-smooth k stencils are O() and O(h τ ), respectively. This leads to an oder of accuracy of the reconstruction R decreasing to k... Levy-Puppo-Russo Reconstructions. In [], Levy, Puppo and Russo describe a third order compact, central WENO scheme. They use a somewhat different WENO reconstruction than the standard one (.), because the linear weights in (.) fail to exist when x = x i+. For the given cell I i, reconstruct the optimal quadratic polynomial P using the large stencil S = {I i, I i, I i+ } and two linear polynomials P and P from the stencils S = {I i, I i } and S = {I i, I i+ }, respectively. The idea is to introduce a centered quadratic polynomial p c such that P = α P + α P + α c p c, α =, α, {, c, }, p c = α c (P α P α P ), where α are the linear weights, subect only to the symmetry α = α. Then the WENO reconstruction is R = α P + α P + α c p c, α =, α, {, c, }, and the nonlinear weights α are computed by (.) with τ = and ɛ =. Here ɛ needs to satisfy: ) ɛ σ in smooth region, ) ɛ σ near discontinuities.. Multilevel-WENO. In this section, we present the multilevel-weno scheme in one space dimension. We proceed as if u were smooth. For the kth order reconstruction (k ), consider the full stencil S i as in (.). We will have hierarchically smaller stencils. Consider the small stencils at level l, where l k. We can construct an (l + )st order accurate reconstruction using the stencils S l+ with (l + ) cells contained in S i. Define S l+ = {I i l+,..., I i+ } S, max(, l r) min(l, s), from which we obtain the (l + )st order reconstruction polynomial P l. Then the kth order multilevel-weno reconstruction is given by the following algorithm, where the nonlinear weights are computed by (.) with corresponding linear weights and smoothness indicators, to be defined after presenting the algorithm. Step. For l =, reconstruct P from the stencil S, and P from the stencil S. Then combine these two polynomials to obtain the second order reconstruction R R = γ P + γ P, γ + γ =, γ, γ, with linear weights γ and smoothness indicators σ P. Perhaps the unbiased choice γ = γ = is most reasonable.

5 ARBOGAST, HUANG, AND ZHAO Step. For l, first construct the (l + )th order polynomial P l from the stencil S l+ for all. Then following Levy-Puppo-Russo s idea [], the reconstruction R l will be a convex combination of Rl and the centered polynomial p l,c = β l (P l αr l l ), that is, R l = α l R l + β l p l,c, α l + β l =, α l, β l, all, with linear weights α l, βl and using the modified smoothness indicators given below. We generally take α l = 5 and βl = 5. Step. If l = k, then S k is the same as the entire stencil S i, so we stop here. If l < k, then again, using the WENO methodology, we combine all the reconstructions R l to get the final (l + )th reconstruction Rl at level l: R l = γ l R l, γ l =, γ l, all, with linear weights γ and smoothness indicators σ P l... Smoothness indicators. We now define the smoothness indicators. For l k. (a) If = max(, l r), then { σ l,r = σ R l σ P l σ P l + P l, σ l,p = σ P l σ P l + P l. (.) (b) If max(, l r) < < min(l, s), then { σ l,r = σ R l σ P l σp l P σ l P+ l P, l σ,p l = σ P l σp l P σ l P+ l P. (.) l (c) If = min(l, s), then { σ l,r = σ R l σ P l σ P l P l, σ l,p = σ P l σ P l P l. (.). For l = k. { σ k,r = σ R k σ P k, σ,p k = σ P. k (.).. Level. Notice that for the above algorithm, we started from l =. So to add in the level l =, we can first obtain the reconstruction R and then implement Step to Step 5 recursively starting from l =. We will describe the method for reconstructing R below. When l =, we only have one stencil S = {I i } from which we can obtain a first order reconstruction polynomial P = ū i. We will reconstruct R in two steps.

6 A Multilevel-WENO Technique 5. First, reconstruct P from the stencil S and P from the stencil S. Then combine P and P to get R R = α P + β P, =,, where the nonlinear weights are defined by (.) with smoothness indicators σ P and σ P. If we use (.5) to compute the smoothness indicator σ P, then σ P =. This will result in the nonlinear weights being highly biased to α, and therefore we lose accuracy. We want to define σ P such that it has order O(h ) ust like the other smoothness indicators in the smooth region, while maintaining the higher order polynomials when the data fits. So define σ P = σ P +P = (ū i+ ū i ). (.5). Next, combine the reconstructions R and R to get R = γ R + γ R, and the nonlinear weights are computed by (.) with linear weight γ and smoothness indicator σ P. Remark. Suppose instead of using the above method, we let R = P. Then in Step we have R = α R + β p, =,, where we use the smoothness indicator σ R = σ P given by (.5) and σ P to compute nonlinear weights. Near the shock, in theory we expect this modified method to cause oscillations. For instance, when there are shocks on one side of the cell I i, say σ P = O(h ), σ P = O(), then σ R may be much larger than σ P so that the nonlinear weights bias to β and R p. This will cause overshooting or undershooting near the shocks. However, this method seems to give slightly better results than the original method given above for the examples we tested in this paper... Reduction. When implement the algorithm, we can choose which levels to use rather than using all the levels. For example, for the fifth order scheme, we can choose level to level simply by setting α =, for all, or only use level and level by setting α = and β =.. Numerical results in One Dimensional Space. In our one dimensional tests, the L and L errors are computed by the formulas u(x, t n ) dx ū n i x i i x i (L error), (.) I i max i u(x, t n ) dx ū n i x i (L error). (.) I i In all of the tests in this paper, for both third order and fifth order schemes, we use α l = 5, βl = 5 for all l and γ = γ =. For the fifth order scheme, use γ = γ = and γ =. We will not include level in our tests.

7 6 ARBOGAST, HUANG, AND ZHAO.. Linear advection. We first test u t + u x = with periodic boundary conditions on [, ] and initial condition u (x) = sin(πx), at time T = on both uniform and nonuniform grids using the third order ML-WENO scheme with ɛ =, see Tables.., and the fifth order ML-WENO scheme with ɛ = 6, see Tables... We show the results of Shu s linear test in Figures... Finally, we test the initial condition with a sawtooth pattern on a uniform grid on (, ) with cells; that is, ū i = if i is odd and i = if i is even. The results are shown in Figure.. N L error L order L error L order e e e-6..79e e e-7. Table.: Linear advection. ML-WENO error and convergence rate on the uniform grid at T =, with ɛ =, t =.5 x. N L error L order L error L order e e e-6..5e e e Table.: Linear advection. ML-WENO error and convergence rate on the nonuniform grid at T =, with ɛ =. N L error L order L error L order e e e e e e e e e e e e-.896 Table.: Linear advection. ML-WENO5 error and convergence rate on the uniform grid at T =, with ɛ = 6, t =. x.

8 A Multilevel-WENO Technique 7 N L error L order L error L order e e e e e e e e e e e e-.977 Table.: Linear advection. ML-WENO5 error and convergence rate on the nonuniform grid at T =, with ɛ = (a) time step. N = (b) time steps. N =. Fig..: Shu s linear test. t = x/. The plots are WENO5-JS (black line) and ML-WENO5 (blue line) (a) time step. N = (b) time steps. N =. Fig..: Shu s linear test. t = x/. The plots are WENO5-JS (black line) and ML-WENO5 with levels - (red line) (a) time step (b) time steps. Fig..: Sawtooth. t = x/. The plots are WENO5-JS (black line) and ML- WENO5 (blue line).

9 8 ARBOGAST, HUANG, AND ZHAO.. Burger s equation. We solve the Burger s equation u t + (u /) x = with the initial condition u (x) = sin(πx) on [, ] at time T =. The results are shown in Tables N L error L order L error L order e e e e e Table.5: Burger s equation. ML-WENO error and convergence rate on the uniform grid at T =, with ɛ =, t =. x. N L error L order L error L order e e e e e-6.99 Table.6: Burger s equation. ML-WENO error and convergence rate on the nonuniform grid at T =, with ɛ =. N L error L order L error L order e e e e e e e-8.97 Table.7: Burger s equation. ML-WENO5 error and convergence rate on the uniform grid at T =, with ɛ = 6, t =. x... Buckley-Leverett equation. In this example we test the Buckley-Leverett equation, which has the nonconvex flux f(u) = u u + ( u).

10 A Multilevel-WENO Technique 9 N L error L order L error L order e e e e e e e Table.8: Burger s equation. ML-WENO5 error and convergence rate on the nonuniform grid at T =, with ɛ = 6. We use the initial condition x, for x.5, u (x) =.5, for.5 x.,, otherwise. The results are shown in Figures. and.5 on uniform grids using N = cells at t =., t =., t =.6 and dt =. x. We take ɛ = 6 for both of the tests (a) t =.. (b) t =.. (c) t =.6. Fig..: Buckley-Leverett equation. N =, t =. x. The plots are WENO5- JS (black line) and ML-WENO5 (blue line) (a) t =.. (b) t =.. (c) t =.6. Fig..5: Buckley-Leverett equation. N =, t =. x. The plots are WENO5- JS (black line) and ML-WENO5 with levels - (red line).

11 ARBOGAST, HUANG, AND ZHAO.. D Euler system. The one dimensional Euler system of gas dynamics is given by ρ m + t x E m ρu + p =, (.) u(e + p) where m = ρu, E = p/(γ ) + ρu / and ρ, u, m, p, E are the density, velocity, momentum, pressure and energy, respectively, and γ =.. For all the tests below, we implemented ML-WENO with ɛ = 6, ML-WENO5 with ɛ = and WENO- JS with ɛ = 6, WENO5-JS with ɛ = Accuracy test. We first test the order of convergence on the Euler equations on a smooth problem. The initial condition is given by ρ(x, ) = +. sin(πx), u(x, ) =, p(x, ) =, with periodic boundary conditions. The exact solution is See Tables.9.. ρ(x, t) = +.sin(πx πt), u =, p =. N L error L order L error L order e-5..75e e e-6..8e e e e e-9..5e Table.9: Euler s equations. ML-WENO error and convergence rate on the nonuniform grid at T =.5 and ɛ =. N L error L order L error L order e e e-7.6.8e e e e-9.7.7e-9.97 Table.: Euler s equations. ML-WENO error and convergence rate on the nonuniform grid at T =.5 and ɛ = 6.

12 A Multilevel-WENO Technique N L error L order L error L order e e e e e e e e e e- 5.5 Table.: Euler s equations. ML-WENO5 error and convergence rate on the nonuniform grid at T = Shock tube test with Sod s and Lax s initial conditions. Sod s initial condition is { ρ l =, m l =, E l =.5, for x <.5, (ρ, m, E) = ρ r =.5, m r =, E r =.5, for x >.5, and Lax s initial condition is { ρ l =.5, m l =., E l = 8.98, for x <.5, (ρ, m, E) = ρ r =.5, m r =, E r =.75. for x >.5. The results are shown in Figures.6.8 using cells and t =. x. The ML- WENO5 result is somewhat better than WENO5-JS (a) Sod. N =. Third order scheme (b) Sod. N =. Zoom (c) Lax. N =. Third order scheme (d) Lax. N =. Zoom. Fig..6: Sod and Lax density at T =.6 using N = cells. The plots are the reference solution (green), WENO-JS (black) and ML-WENO (blue).

13 ARBOGAST, HUANG, AND ZHAO (a) Sod. N =. Fifth order scheme (b) Sod. N =. Zoom (c) Lax. N =. Fifth order scheme (d) Lax. N =. Zoom. Fig..7: Sod and Lax density at T =.6 using N = cells. The plots are the reference solution (green), WENO5-JS (black) and ML-WENO5 (blue) (a) Sod. N =. Fifth order scheme (b) Sod. N =. Zoom (c) Lax. N =. Fifth order scheme (d) Lax. N =. Zoom. Fig..8: Sod and Lax density at T =.6 using N = cells. The plots are the reference solution (green), WENO5-JS (black) and ML-WENO5 with levels - (red).

14 A Multilevel-WENO Technique... Shu and Osher s shock interaction with entropy waves. Next we solve the shock interaction with entropy waves problem given in [], which has a moving Mach shock interacting with sine waves in density. The initial condition is { ρ l =.857, u l =.6996, p l =., for < x <., (ρ, u, p) = ρ r = +.sin(5(x 5)), u r =, p r =, for. < x <. We compute the density at T =.6 using t =. x and N = cells. The results are shown in Figures (a) N =. Third order scheme (b) N =. Zoom. Fig..9: Shu and Osher s shock interaction with entropy waves density at T =.6 using N = cells. The plots are WENO-JS (black) and ML-WENO (blue) (a) N =. Fifth order scheme (b) N =. Zoom. Fig..: Shu and Osher s shock interaction with entropy waves density at T =.6 using N = cells. The plots are WENO5-JS (black) and ML-WENO5 (blue) (a) N =. Fifth order scheme (b) N =. Zoom. Fig..: Shu and Osher s shock interaction with entropy waves density at T =.6 using N = cells. The plots are WENO5-JS (black) and ML-WENO5 with levels - (red).

15 ARBOGAST, HUANG, AND ZHAO... Woodward and Colella s double blast test. This test uses the initial condition ρ l =, m l =, E l = /(γ ), for < x <., (ρ, m, E) = ρ m =, m m =, E m =./(γ ), for. < x <.9, ρ r =, m r =, E r = /(γ ), for.9 < x <. See Figures. and (a) N = 99. Third order scheme (b) N = 99. Zoom Fig..: Woodward and Colellas double blast test density at T =.8 using N = 99 cells. The plots are the reference solution (green), WENO-JS (black) and ML-WENO (blue) (a) N = 99. Fifth order scheme (b) N = 99. Zoom Fig..: Woodward and Colellas double blast test density at T =.8 using N = 99 cells. The plots are the reference solution (green), WENO5-JS (black), ML-WENO5 (blue) and ML-WENO5 with levels - (red). REFERENCES [] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 6 (996), pp. 8. [] R. J. Leveque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., (996), pp [] D. Levy, G. Puppo, and G. Russo, Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., (), pp [] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes, ii, Journal of Computational Physics, 8 (989), pp. 78.