5.2 Characteristic fields in the 5 Additional Information5 Additional Information

5.1 Riemann problems in locally Minkowskian coordinates 

In [229Jump To The Next Citation Point In The Article], a procedure to integrate the general relativistic hydrodynamic 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 [166].

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, which 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 [229Jump To The Next Citation Point In The Article], we illustrate the procedure for computing the second flux integral in Equation (45Popup Equation), which we call tex2html_wrap_inline5552 . We begin by expressing the integral on a basis tex2html_wrap_inline5554 with tex2html_wrap_inline5556 and tex2html_wrap_inline5558 forming an orthonormal basis in the plane orthogonal to tex2html_wrap_inline4740 with tex2html_wrap_inline5562 normal to the surface tex2html_wrap_inline5564 and tex2html_wrap_inline5566 and tex2html_wrap_inline5568 tangent to that surface. The vectors of this basis verify with the Minkowski metric (in the following, caret subscripts will refer to vector components in this basis).

Denoting by tex2html_wrap_inline5574 the coordinates at the center of the interface at time t, we introduce the following locally Minkowskian coordinate system: tex2html_wrap_inline5578, where the matrix tex2html_wrap_inline5580 is given by tex2html_wrap_inline5582, calculated at tex2html_wrap_inline5574 . In this system of coordinates the equations of general relativistic hydrodynamics transform into the equations of special relativistic hydrodynamics in Cartesian coordinates, but with non-zero sources, and the flux integral reads


(the caret symbol representing the numerical flux in Equation (45Popup Equation) is now removed to avoid confusion) with tex2html_wrap_inline5590, where we have taken into account that, in the coordinates tex2html_wrap_inline5592, tex2html_wrap_inline5564 is described by the equation tex2html_wrap_inline5596 (with tex2html_wrap_inline5598), where the metric elements tex2html_wrap_inline5600 and tex2html_wrap_inline4738 are calculated at tex2html_wrap_inline5574 . Therefore, this surface is not at rest but moves with speed tex2html_wrap_inline5606 .

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 (64Popup Equation) represents the numerical flux across tex2html_wrap_inline5564, which can now be calculated by solving the special relativistic Riemann problem defined with the values at the two sides of tex2html_wrap_inline5564 of two independent thermodynamical variables (namely, the rest mass density tex2html_wrap_inline4640 and the specific internal energy tex2html_wrap_inline4690) and the components of the velocity in the orthonormal spatial basis tex2html_wrap_inline5616 (tex2html_wrap_inline5618).

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 tex2html_wrap_inline5564, simplifying the calculation of the above integral enormously:


where the superscript (*) stands for the value on tex2html_wrap_inline5564 obtained from the solution of the Riemann problem. Notice that the numerical fluxes correspond to the vector fields tex2html_wrap_inline5626, tex2html_wrap_inline5628, tex2html_wrap_inline5630, tex2html_wrap_inline5632, tex2html_wrap_inline5634 and linearized Riemann solvers provide the numerical fluxes as defined in Equation (64Popup Equation). Thus, the additional relation tex2html_wrap_inline5636 has to be used for the momentum equations. The integral in the right hand side of Equation (65Popup Equation) is the area of the surface tex2html_wrap_inline5564 and can be expressed in terms of the original coordinates as


which can be evaluated for a given metric. The interested reader is addressed to [229] for details on the testing and calibration of this procedure.

5.2 Characteristic fields in the 5 Additional Information5 Additional Information

image Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de