Sistemas Hiperbólicos no Lineales: Un Nuevo Método para Calcular Flujos Relativistas

Sistemas Hiperbólicos no Lineales: Un Nuevo Método para Calcular Flujos Relativistas Pedro González-Casanova Henríquez Unidad de Investigación en Cómputo Aplicado DGSCA, UNAM Elvira Torondel y Ricard Garrido Departamento de Matemática Aplicada, Universidad de Valencia, España Seminarios de Modelación Matemática y Computacional Instituto de Geofísica, junio 2008

Contents: Nonlinear Hyperbolic Systems of conservation laws. Restricted relativity: Conservative representation. Hybrid Compact-WENO method. Numerical results. Conclusions.

Hyperbolic Systems of Conservation Laws: Basics 1D Let u = (u 1,..., u p ) and x IR u t + f x = 0 Integral representation: udx + t D D The Jacobian A matrix of f is A(u) = f u (= s(u)) fds = 0

Quasi-linear representation -u smooth enough- is given by u t + A(u) u x = 0 The system is hyperbolic if the Jacobian A has d real eigenvalues λ 1 (u)... λ p (u) and p linearly independent eigenvectors If the eigenvalues are all different, the system is called strictly hyperbolic. Initial value problem: u(x, t = 0) = u 0 (x)

Hyperbolic Systems of Conservation Laws: Characteristic Form Consider a complete set of left eigenvectors of A, l j, with i components l i j, then l T j A = λ jl T j T, denotes transpose. Hyperbolic systems can be written in characteristic form as l T j u t + λ jl T j f x = 0 Can we uncouple the system of equations?: nonlinear case the eigenvalues and eigenvectors depend on the solution u.

Hyperbolic Systems of Conservation Laws: Linear Case Suppose A is a constant matrix, i.e.. eigenvectors and eigenvalues constant. Then l T j u t + λ j l T j f x = 0 this is an uncoupled system of equations: 1D hyperbolic scalar conservation laws. Initial conditions: lj T u(x, t = 0) = lt j u 0(x) for j = 1,..., d. We can now solve d scalar problems in the characteristic space, and then, using the right eigenvectors go back to the physical space.

In the non linear case the Jacobian and thus the eigenvalues and eigenvectors depends on u. Solution: frozen the Jacobian at some u. In the numerical algorithm, this implies that we can locally solve the problem in the characteristic space.

Hyperbolic Systems of Conservation Laws: Back to reality, IR d Let u = (u 1,..., u p ) and x = (x 1,..., x d ) IR d and f j (u) = (f 1j, f 2j,,..., f pj ) is the vector of fluxes u t + d j=1 f j x j = 0 (= s(u)) Integral representation: t D d udx + j=1 D f j (u) nds = 0 For all j = 1,..., d the Jacobian A j matrix of f j is A j (u) = f j u

Summary: We have one Jacobian for each spatial direcction. For each Jacobian apply the characteristic algorithm: Use left eigenvalues to uncouple the system. Solve the 1D problem in each characteristic direction. Use the right eigenvalues to return to the physical space. Joint the solutions corresponding to each spatial dimension.

Special Relativistic Fluid Hydrodynamics Relativistic simulations are more difficult than Newtonian simulations (Norman and Winkler 1986) The characteristic wave speeds in the relativistic case not only depends on the fluid velocity components in the wave propagation direction, but also depends on the normal velocity components. This coupling complicates the solution of the Riemann problem severely, (Donat et al. 1998).

Relativistic Fluid Dynamics: Local conservation laws Restricted relativity: Relativistic flux local conservation laws: Local conservation of rest mass, µ (ρu µ ) = 0 Local conservation of stress-enery of a fluid, µ T µν = 0 ρ is the rest mass density measured in the fluid frame, U µ = W (c, u) the fluid 4-velocity vector, W is the Lorentz factor, c the speed of light, u is the classical three-velocity.

Also, T µν is the energy-momentum tensor, which for a perfect fluid can be written as T µν = ρhu µ u ν + pg µν where p is the pressure and h = 1 + ɛ + p ρ is the specific enthalpy, with ɛ being the specific internal energy. The tensor g µν defines the metric of the space-time where the fluid evolves.

Relativistic Fluid Dynamics: System of conservation Laws The first step is to rewrite the local conservation laws in conservative form, (see Martí and Müller, 1999): Consider the following variables: D = ρw S j = ρhw 2 v j τ = ρhw 2 p ρw which are, respectively, the rest mass, momentum and total energy densities, measured in the laboratory frame. Where v i are the components of the threevelocity of the fluid and W is the Lorentz factor.

Definining now the conserved quantities u = (D, S j, τ ) T and the fluxes f i (u) = (Dv i, S j v i + pδ ij, S i Dv i) T where, δ ij is the Kroneker symbol and v i is the velocity. We obtain the desired conservative representation u t + f i = 0 x i i This system, which is closed by an equation of state: p = p(ρ, ɛ), is hyperbolic.

Thus using the spectral decomposition of the three 5 5 Jacobian matices. (Donat, et al.1998) B i = f i (u) u numerical methods can be applied.

Compact-WENO schemes Compact schemes very accurate in smooth regions with spectral-like resolution (see Lele 1992), however produce non-physical oscillations (Gibbs phenomena) when applied to flow with discontinuities. WENO schemes are less accurate in smooth regions but are good to approximate discontinuous flows. Hybrid schemes: combines the advantages of compact schemes in smooth regions with the sharp WENO technique near the discontinuities.

The finite difference equation: 1D Problem Scalar hyperbolic conservation law: u t + f x = 0 where f = f(u) Semi-discrete conservative finite difference scheme u j t + 1 ( ) ˆf h j+1/2 ˆf j 1/2 = 0 ˆf j+1/2 is a numerical flux function 1 ( ˆfj+1/2 ˆf ) ( ) f h j 1/2 = x j + O(h k )

The time integration: three-stage, TVD Runge- Kutta scheme. Let L j (u) = 1 h ( ˆfj+1/2 ˆf j 1/2 ) the TVD Runge-Kutta scheme is given by u (1) j = u n j + tl j(u n ) (1) u (2) j = 3 4 un j + 1 4 u(1) j + 1 4 tl j(u (1) ) (2) u n+1 j = 1 3 un j + 2 3 u(2) j + 2 3 tl j(u (2) ) (3)

Compact-WENO scheme in 1D: Numerical fluxes Step 1.- Compute the numerical flux function using fifth-order conservative upwind compact scheme proposed by Pirozzoli, 2002. φ j+1/2 ˆfj 1/2 + ˆf j+1/2 + ψ j+1/2 ˆfj+3/2 = ˆb j+1/2 Step 2.- Compute the numerical flux function of fifth-order WENO scheme ˆf W ENO j+1/2 = 2 γ=0 ω γ f γ j+1/2 Step 3.- Combine both fluxes: a wight σ j+1/2, depending on a threshold value r c, measure the smoothness of the numerical solution. If

the solution is smooth the compact scheme is used, otherwise the WENO method is utilized. Step 4.- The scalar Compact-WENO scheme, is given by σ j+1/2 φ j+1/2 ˆf j 1/2 + ˆf j+1/2 +σ j+1/2 ψ j+1/2 ˆf j+3/2 = ĉ j+1/2 The weight smoother indicator is σ j+1/2 = min(1, r j+1/2 r c ) the threshold value r c is chosen in an appropriated form.

Compact-WENO scheme for RFD 1. Upwind method. At each fixed x j+1/2, the average state u j+1/2 is computed by the simple mean: u j+1/2 = 1 2 (u j + u j+1 ) 2. Freeze the Jacobian. Compute right λ (i) j+1/2 (i = 1, 2, 3) and left eigenvectors l (i) at u j+1/2. j+1/2 (i = 1, 2, 3) 3. Characteristic projection. Compute local characteristic projected fluxes. w (i) m = l (i) j+1/2 f m, i = 1, 2, 3; m = j 1,..., j+2.

4. 1D compact-weno method For i = 1, 2, and 3, hybrid compact-weno scalar scheme applied to the local characteristic variables w (i) (i = 1, 2, 3) σ j+1/2 φ (i) j+1/2ŵ(i) j 1/2 + ŵ(i) j+1/2 + where σ j+1/2 ψ (i) j+1/2ŵ(i) j+3/2 = ĉ(i) j+1/2 5. For i = 1, 2, and 3, the characteristic compact- WENO is Φ j+1/2 ˆfj 1/2 + L j+1/2 ˆfj+1/2 + Ψ j+1/2 ˆfj+3/2 = ĉ j+1/2

where Φ j+1/2 = Ψ j+1/2 = L j+1/2 = σ (1) j+1/2 φ(1) j+1/2 l(1) j+1/2 σ (2) j+1/2 φ(2) j+1/2 l(2) j+1/2 σ (3) j+1/2 φ(3) j+1/2 l(3) j+1/2 l (1) j+1/2 l (2) j+1/2 l (3) j+1/2 σ (1) j+1/2 ψ(1) j+1/2 l(1) j+1/2 σ (2) j+1/2 ψ(2) j+1/2 l(2) j+1/2 σ (3) j+1/2 ψ(3) j+1/2 l(3) j+1/2

and ĉ j+1/2 = ĉ (1) j+1/2 ĉ (2) j+1/2 ĉ (3) j+1/2

Numerical test: Relativistic shock tube problem, 1D 1. Hot dense fluid on one side is separated by a membrane from a cool rarefied gas on the other. 2. The membrane is removed and the high density fluid pushes into the low density fluid.

Numerical test: Relativistic shock tube problem, 1D Sketch of the shock tube solution: 1. (1) the undisturbed high density fluid; 2. (2) the rarefraction wave; 3. (3) a region of constant velocity and pressure which features a contact discontinuity separating regions of different density; 4. (4) the shock itself; 5. (5) the undisturbed low density fluid.

Numerical test: 1D, problem, No. 1 Riemann Problem: The initial condition for the first problem is: {p L = 13.3, ρ L = 10, v L = 0} x [0, 0.5[ {p R = 0, ρ R = 1, v R = 0} x [0.5, 1] The results are showed in the following tables.

Numerical test: 1D, Problem 1 Schemes: Hybrid and WENO for problem 1 in 1D with N = 625 points, T = 0.4, t x = 0.8 HYB r c CPU 2 Loc. err. % WENO 0.1 0.15 38.187 0.0891 1.4349 0.2668 4.43 0.2 38.046 0.0938 1.6113 0.2617 5.05 0.3 38.046 0.1003 1.7615 0.2775 5.52 0.4 38.030 0.1042 1.8361 0.2896 5.70 0.5 38.483 0.1054 1.8568 0.2943 6.01 0.6 38.436 0.1065 1.8777 0.2978 6.39 0.7 38.577 0.1075 1.8931 0.3016 7.05 WEN 43.7 0.1156 2.0125 0.3300 100

Numerical test: 1D, Problem 2 Riemann Problem: The second problem initial condition is: {D(x) = 10, S(x) = 0, τ (x) = 19.95} L x [0, 0.5[ {D(x) = 1+0.2 sin(20πx), S(x) = 0, τ (x) = 9.9 10 7 } R x [0.5, 1] Turbulent right side conditions.

Numerical test: 1D, Problem 2 Schemes: Hybrid and WENO for problem 2 in 1D with N = 625 points, T = 0.4, t x = 0.8 HYB r c CPU 2 Loc. err. % WENO 0.15 44 0.1179 1.456 0.4407 4.63 0.2 44.3 0.1190 1.4079 0.4447 5.22 0.3 44.62 0.1219 1.4874 0.4557 5.83 0.7 45 0.1290 1.6329 0.4761 8.09 WEN 53.7 0.1465 1.7936 0.5496 100

Numerical test: 2D Problem. A square [0, 1] 2, initially there is a uniformly distributed gas whose density is everywhere. At t = 0, an inflow flux is injected into the square from the region 0.4 < x < 0.6, y = 0, while other boundary conditions are set to be absorbent.

Schemes for problem in 2D: Hybrid ( x t = 0.5), WENO ( x t = 0.5), N N = 125 125 points, T = 0.8, v y 0 = 0.9c HYBRID r c CPU ERROR % WENO 0.2 0.3 1408 10.2821 15.75 0.5 1444 11.0893 19.86 0.8 1477 11.4797 25.73 WENO 1853 12.6638 100

Comparative density ρ between hybrid compact- WENO (λ = 0.5, r c = 0.3) and WENO (λ = 0.5) for the 2D problem with 125 125 points

Schemes for problem 1 in 2D: Hybrid ( x t = 0.5), WENO ( x t = 0.5), N N = 250 250 points, T = 0.8, v y 0 = 0.9c HYBRID r c CPU-Time (s) ERROR % WENO 0.2 0.3 8508 5.7804 9.56 0.5 11088 6.4827 12.71 0.8 11135 7.0383 18 WENO 15279 8.1697 100

Comparative density ρ between hybrid compact- WENO (λ = 0.5, r c = 0.3) and WENO (λ = 0.5) for the 2D problem with 250 250 points.

Conclusions Hybrid compact-weno method has been formulated which accurately solve relativistic flow problems characterized by having strong shocks and turbulence.

Conclusions Hybrid compact-weno method has been formulated which accurately solve relativistic flow problems characterized by having strong shocks and turbulence. Numerical experiments in 1D and 2D for two methods applied to relativistic problems were performed. We show that the hybrid method gives, a better resolution in 1D and 2D, owing to its spectral like resolution property.

Further work is in progress to calculate ultrarelativistic problems for v y 0 = 0.999c.