2,043 research outputs found
ECHO: an Eulerian Conservative High Order scheme for general relativistic magnetohydrodynamics and magnetodynamics
We present a new numerical code, ECHO, based on an Eulerian Conservative High
Order scheme for time dependent three-dimensional general relativistic
magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at
providing a shock-capturing conservative method able to work at an arbitrary
level of formal accuracy (for smooth flows), where the other existing GRMHD and
GRMD schemes yield an overall second order at most. Moreover, our goal is to
present a general framework, based on the 3+1 Eulerian formalism, allowing for
different sets of equations, different algorithms, and working in a generic
space-time metric, so that ECHO may be easily coupled to any solver for
Einstein's equations. Various high order reconstruction methods are implemented
and a two-wave approximate Riemann solver is used. The induction equation is
treated by adopting the Upwind Constrained Transport (UCT) procedures,
appropriate to preserve the divergence-free condition of the magnetic field in
shock-capturing methods. The limiting case of magnetodynamics (also known as
force-free degenerate electrodynamics) is implemented by simply replacing the
fluid velocity with the electromagnetic drift velocity and by neglecting the
matter contribution to the stress tensor. ECHO is particularly accurate,
efficient, versatile, and robust. It has been tested against several
astrophysical applications, including a novel test on the propagation of large
amplitude circularly polarized Alfven waves. In particular, we show that
reconstruction based on a Monotonicity Preserving filter applied to a fixed
5-point stencil gives highly accurate results for smooth solutions, both in
flat and curved metric (up to the nominal fifth order), while at the same time
providing sharp profiles in tests involving discontinuities.Comment: 20 pages, revised version submitted to A&
The Cauchy-Lagrangian method for numerical analysis of Euler flow
A novel semi-Lagrangian method is introduced to solve numerically the Euler
equation for ideal incompressible flow in arbitrary space dimension. It
exploits the time-analyticity of fluid particle trajectories and requires, in
principle, only limited spatial smoothness of the initial data. Efficient
generation of high-order time-Taylor coefficients is made possible by a
recurrence relation that follows from the Cauchy invariants formulation of the
Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430).
Truncated time-Taylor series of very high order allow the use of time steps
vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the
accuracy of the solution. Tests performed on the two-dimensional Euler equation
indicate that the Cauchy-Lagrangian method is more - and occasionally much more
- efficient and less prone to instability than Eulerian Runge-Kutta methods,
and less prone to rapid growth of rounding errors than the high-order Eulerian
time-Taylor algorithm. We also develop tools of analysis adapted to the
Cauchy-Lagrangian method, such as the monitoring of the radius of convergence
of the time-Taylor series. Certain other fluid equations can be handled
similarly.Comment: 30 pp., 13 figures, 45 references. Minor revision. In press in
Journal of Scientific Computin
Blended numerical schemes for the advection equation and conservation laws
In this paper we propose a method to couple two or more explicit numerical
schemes approximating the same time-dependent PDE, aiming at creating new
schemes which inherit advantages of the original ones. We consider both
advection equations and nonlinear conservation laws. By coupling a macroscopic
(Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of
multiscale numerical method
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