WENO Scheme with Eigensystem for Euler Equations The Annual Report in Advanced Research in Applied Mathematics and Scientific Computing 2018, OUC, China Kangbo Tian, School of Mathematical Sciences, Ocean University of China I NTRODUCTION Consider the Euler equation ∂ Q ∂t + ∂ F ∂x =0, (1) where the conservative variables Q = (ρ, ρu, E ) T , flux F =(ρu, ρu 2 + P, (E + P )u) T , ρ is the density, u is the velocity, E is the total energy and P is the pressure. A fifth- order WENO scheme is used to numerically approximate the solution of equation (1). The performance will be better if the fluxes are projected on the characteristic fields before WENO reconstruction procedure is carried out. P REPARE OF WENO S CHEME x i x i+1 x i+2 x i-1 x i-2 x i+1/2 The semi-discretized form of Eq. (1), by the method of lines, yields a system of ordinary differential equations dQ i (t) dt = - ∂f ∂x x=x i , i =0,...,N. (2) By implicitly defining the numerical flux function h(x) as f (x)= 1 Δx Z x+ Δx 2 x- Δx 2 h(ξ )dξ, (3) Eq. (2) is exactly obtained by a conservative finite differ- ence formula at the cell boundaries, dQ i (t) dt = - 1 Δx h i+ 1 2 - h i- 1 2 . (4) The reconstruction of h i± 1 2 and many of the details regard- ing WENO schemes, as well as a three-stage third order TVD Runge-Kutta scheme for solving the ordinary equa- tion (2), can be found in the lecture note [2]. F LUX S PLITTING One would need to perform a flux splitting, for example, the global Lax-Friedrichs flux splitting, to split the flux F into the positive F + and negative F - fluxes, that is, F ± = 1 2 (F ± αQ) ,α = max x∈Ω max Q ∂ F ∂ Q , (5) where α is a vector of maximum absolute eigenvalues of the Jacobian. R OE - AVERAGES E IGENSYSTEM The system (1) can be written as ∂ Q ∂t + ∂ F ∂x = ∂ Q ∂t + A ∂ Q ∂x =0, (6) where R OE - AVERAGES E IGENSYSTEM Q = ρ ρU E = Q 1 Q 2 Q 3 , (7) F = Q 2 1 2Q 1 ((3 - γ )Q 2 2 + 2(γ - 1)Q 1 Q 3 ) 1 2Q 2 1 ((1 - γ )Q 3 2 +2γQ 1 Q 2 Q 3 ) , (8) and A = ∂ F ∂ Q is the Jacobian of the system A = 0 1 0 (γ - 3)U 2 (3 - γ )U γ - 1 ((γ - 1)U 2 - h)U h - (γ - 1)U 2 γU , where c is the sound speed, h is the specific total enthalpy and U 2 = 1 2 U 2 is specific kinetic energy. Here are some of the relationships between the variables, h = E + P ρ ,c = r γP ρ . (9) The eigensystem of the Jacobian can be re-written as A = RΛL, where Λ = U - c 0 0 0 U 0 0 0 U + c , (10) R = 1 1 1 U - c U U + c h - cU U 2 h + cU , (11) L = 1 2c 2 cU + bU 2 -c - bU b 2c 2 - 2bU 2 2bU -2b -cU + bU 2 c - bU b , (12) where b =(γ - 1), L = R -1 . For the content of the Roe- averaged states, we refer to [3] for the details. C HARACTERISTIC P ROJECTION In the WENO reconstruction procedure, at each cell center x i ,i =0,...N , we need to • perform a characteristic projection of the flux F ± i+j ,j = -r, . . . , r onto the characteristic space as G ± i+j = L i+ 1 2 F ± i+j , j = -r, . . . , r. (13) • perform WENO reconstruction procedure with data G ± i+j ,j = -r, . . . , r - 1 to obtain ˆ G ± i+ 1 2 . • perform a characteristic projection of the flux ˆ G ± i+ 1 2 back onto the physical space as h ± i+ 1 2 = R i+ 1 2 ˆ G ± i+ 1 2 . (14) A CCURACY OF THE METHOD Table 1: The initial condition is set as (ρ, u, P ) = (1 + 0.1 sin(πx), 1, 1) for the one-dimensional Euler equation, x ∈ [0, 2] with periodical boundary condition, and Δt is bounded by Δx 5 3 for the tests of accuracy at final time t =2. WENO-Z5 WENO-Z5-E WENO-Z5 WENO-Z5-E WENO-Z5 WENO-Z5-E N L ∞ err. Order L ∞ err. Order L 2 err. Order L 2 err. Order L 1 err. Order L 1 err. Order 10 2.2e-3 — 2.2e-3 — 2.9e-3 — 2.9e-3 — 4.8e-3 — 4.8e-3 — 20 7.1e-5 4.93 7.1e-5 4.93 7.6e-5 5.26 7.6e-5 5.26 1.1e-4 5.47 1.1e-4 5.47 40 2.2e-6 4.98 2.2e-6 4.98 2.3e-6 5.07 2.3e-6 5.07 3.0e-6 5.18 3.0e-6 5.18 80 7.0e-8 4.99 7.0e-8 4.99 7.0e-8 5.01 7.0e-8 5.01 9.1e-8 5.05 9.1e-8 5.05 160 2.2e-9 5.00 2.2e-9 4.95 2.2e-9 5.00 2.2e-9 5.00 2.8e-9 5.01 2.8e-9 5.01 320 7.1e-11 4.95 7.1e-11 5.00 6.9e-11 5.00 6.9e-11 5.00 8.8e-11 5.00 8.8e-11 5.00 1D E ULER E QUATIONS Sod (N = 200) Lax (N = 200) Shu-Osher (N = 800) x Rho -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 Reference WENO-Z5 WENO-Z5-E -2.4 -2.2 -2 -1.8 -1.6 -1.4 0.25 0.3 0.35 0.4 x Rho -4 -2 0 2 4 0.2 0.4 0.6 0.8 1 1.2 1.4 Reference WENO-Z5 WENO-Z5-E 2 2.5 3 1 1.1 1.2 1.3 x Rho -5 0 5 10 15 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Reference WENO-Z5 WENO-Z5-E 6.5 7 7.5 8 3.6 3.8 4 4.2 2D DMR P ROBLEM x y 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 22.4 21.4 20.4 19.4 18.4 17.4 16.4 15.4 14.4 13.4 12.4 11.4 10.4 9.4 8.4 7.4 6.4 5.4 4.4 3.4 2.4 1.4 x y 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 0 0.1 0.2 0.3 0.4 0.5 F UTURE W ORK Do more numerical experiments of the characteristic-wise WENO scheme. A CKNOWLEDGEMENT Thanks to the support by Prof. Don Wai Sun, Prof. Gao Zhen, Prof. Xie Shusen, Mr. Wang Baoshan, I got a chance to study here this summer. R EFERENCES [1] R. Borges, M. Carmona, B. Costa, W.S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conser- vation laws, J. Comput. Phys. (2008) 227(6) 3191-3211. [2] W.S. Don, R. Borges, Introduction of WENO Finite Differ- ence Scheme for Hyperbolic Conservation Laws, https://www. researchgate.net/publication/309194721. [3] P.L. Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes, J. Comput. Phys. (1981) 43 357-372.