Go to previous page Go up Go to next page

6.1 Riemann problems in locally-Minkowskian coordinates

In [320Jump To The Next Citation Point], a procedure to integrate the general-relativistic hydrodynamics equations (as formulated in Section 2.1.3), taking advantage of the multitude of Riemann solvers developed in special relativity, was presented. The approach relies on a local change of coordinates in terms of which the spacetime metric is locally Minkowskian. This procedure allows, for 1D problems, the use of the exact solution of the special relativistic Riemann problem [239].

Such a coordinate transformation to locally Minkowskian coordinates at each numerical interface assumes that the solution of the Riemann problem is the one in special relativity and planar symmetry. This last assumption is equivalent to the approach followed in classical fluid dynamics, when using the solution of Riemann problems in slab symmetry for problems in cylindrical or spherical coordinates, as the solution breaks down near the singular points (e.g., the polar axis in cylindrical coordinates). In analogy to classical fluid dynamics, the numerical error depends on the magnitude of the Christoffel symbols, which might be large whenever huge gradients or large temporal variations of the gravitational field are present. Finer grids and improved time-advancing methods will be required in those circumstances.

Following [320Jump To The Next Citation Point], we illustrate the procedure for computing the second flux integral in Equation (76View Equation), which we call ℐ. We begin by expressing the integral on a basis e ˆα with e ≡ n μ ˆ0 and e ˆi forming an orthonormal basis in the plane orthogonal to μ n with eˆ1 normal to the surface Σx1 and eˆ2 and eˆ3 tangent to that surface. The vectors of this basis verify eˆα ⋅ eˆβ = ηˆαˆβ with ηαˆβˆ the Minkowski metric (in the following, caret superscripts will refer to vector components in this basis).

Denoting by α x 0 the coordinates at the center of the interface at time t, we introduce the following locally-Minkowskian coordinate system x ˆα = M ˆα(xα − xα ) α 0, where the matrix M ˆα α is given by ∂ = M ˆαe α α ˆα, calculated at x α 0. In this system of coordinates the equations of general-relativistic hydrodynamics transform into the equations of special relativistic hydrodynamics, in Cartesian coordinates, but with nonzero sources, and the flux integral reads

∫ ∫ ( ˆ1 ) ∘ --- ℐ ≡ √ −-gF 1dx0dx2dx3 = (F ˆ1 − β-F ˆ0) − ˆgdx ˆ0dx ˆ2dxˆ3, (103 ) Σx1 Σx1 α
(the caret symbol representing the numerical flux in Equation (76View Equation) is now removed to avoid confusion) with √ --- − ˆg = 1 + 𝒪 (xˆα), where we have taken into account that, in the coordinates xˆα, Σx1 is described by the equation ˆ1 βˆ1 ˆ0 x − α x = 0 (with ˆi ˆi i β = M iβ), where the metric elements 1 β and α are calculated at α x 0. Therefore, this surface is not at rest but moves with speed βˆ1∕α.

At this point, all the theoretical work developed in recent years on special relativistic Riemann solvers can be exploited. The quantity in parentheses in Equation (103View Equation) represents the numerical flux across Σx1, which can now be calculated by solving the special relativistic Riemann problem defined with the values at the two sides of Σx1 of two independent thermodynamic variables (namely, the rest mass density ρ and the specific internal energy ɛ) and the components of the velocity in the orthonormal spatial basis vˆi (ˆi ˆi i v = M iv).

Once the Riemann problem has been solved, we can take advantage of the self-similar character of the solution of the Riemann problem, which makes it constant on the surface Σx1, simplifying the calculation of the above integral enormously:

( ˆ1 )∗ ∫ ∘ --- ℐ = (F ˆ1 − β-F ˆ0) − ˆgdx ˆ0dxˆ2dxˆ3, (104 ) α Σx1
where the superscript (*) stands for the value on Σx1 obtained from the solution of the Riemann problem. Notice that the numerical fluxes correspond to the vector fields F 1 = {J, T ⋅ n, T ⋅ e ˆ1, T ⋅ eˆ2, T ⋅ eˆ} 3 and linearized Riemann solvers provide the numerical fluxes as defined in Equation (103View Equation). Thus, the additional relation ˆj T ⋅ ∂i = M i (T ⋅ eˆj) has to be used for the momentum equations. The integral in the right-hand side of Equation (104View Equation) is the area of the surface Σx1 and can be expressed in terms of the original coordinates as
∫ ∘--- ∫ ∘ --- − ˆg dxˆ0dx ˆ2dx ˆ3 = γ11√ −-gdx0dx2dx3, (105 ) Σx1 Σx1
which can be evaluated for a given metric. The interested reader is directed to [320] for details on the testing and calibration of this procedure.
  Go to previous page Go up Go to next page