624 research outputs found
High-order conservative finite difference GLM-MHD schemes for cell-centered MHD
We present and compare third- as well as fifth-order accurate finite
difference schemes for the numerical solution of the compressible ideal MHD
equations in multiple spatial dimensions. The selected methods lean on four
different reconstruction techniques based on recently improved versions of the
weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving
(MP) schemes as well as slope-limited polynomial reconstruction. The proposed
numerical methods are highly accurate in smooth regions of the flow, avoid loss
of accuracy in proximity of smooth extrema and provide sharp non-oscillatory
transitions at discontinuities. We suggest a numerical formulation based on a
cell-centered approach where all of the primary flow variables are discretized
at the zone center. The divergence-free condition is enforced by augmenting the
MHD equations with a generalized Lagrange multiplier yielding a mixed
hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175
(2002) 645-673). The resulting family of schemes is robust, cost-effective and
straightforward to implement. Compared to previous existing approaches, it
completely avoids the CPU intensive workload associated with an elliptic
divergence cleaning step and the additional complexities required by staggered
mesh algorithms. Extensive numerical testing demonstrate the robustness and
reliability of the proposed framework for computations involving both smooth
and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics
(Aug 7 2009
RAM: A Relativistic Adaptive Mesh Refinement Hydrodynamics Code
We have developed a new computer code, RAM, to solve the conservative
equations of special relativistic hydrodynamics (SRHD) using adaptive mesh
refinement (AMR) on parallel computers. We have implemented a
characteristic-wise, finite difference, weighted essentially non-oscillatory
(WENO) scheme using the full characteristic decomposition of the SRHD equations
to achieve fifth-order accuracy in space. For time integration we use the
method of lines with a third-order total variation diminishing (TVD)
Runge-Kutta scheme. We have also implemented fourth and fifth order Runge-Kutta
time integration schemes for comparison. The implementation of AMR and
parallelization is based on the FLASH code. RAM is modular and includes the
capability to easily swap hydrodynamics solvers, reconstruction methods and
physics modules. In addition to WENO we have implemented a finite volume module
with the piecewise parabolic method (PPM) for reconstruction and the modified
Marquina approximate Riemann solver to work with TVD Runge-Kutta time
integration. We examine the difficulty of accurately simulating shear flows in
numerical relativistic hydrodynamics codes. We show that under-resolved
simulations of simple test problems with transverse velocity components produce
incorrect results and demonstrate the ability of RAM to correctly solve these
problems. RAM has been tested in one, two and three dimensions and in
Cartesian, cylindrical and spherical coordinates. We have demonstrated
fifth-order accuracy for WENO in one and two dimensions and performed detailed
comparison with other schemes for which we show significantly lower convergence
rates. Extensive testing is presented demonstrating the ability of RAM to
address challenging open questions in relativistic astrophysics.Comment: ApJS in press, 21 pages including 18 figures (6 color figures
Low-diffusivity scalar transport using a WENO scheme and dual meshing
Interfacial mass transfer of low-diffusive substances in an unsteady flow
environment is marked by a very thin boundary layer at the interface and other
regions with steep concentration gradients. A numerical scheme capable of
resolving accurately most details of this process is presented. In this scheme,
the fourth-order accurate WENO method developed by Liu et al. (1994) was
implemented on a non-uniform staggered mesh to discretize the scalar convection
while for the scalar diffusion a fourth-order accurate central discretization
was employed. The discretization of the scalar convection-diffusion equation
was combined with a fourth-order Navier-Stokes solver which solves the
incompressible flow. A dual meshing strategy was employed, in which the scalar
was solved on a finer mesh than the incompressible flow. The solver was tested
by performing a number of two-dimensional simulations of an unstably stratified
flow with low diffusivity scalar transport. The unstable stratification led to
buoyant convection which was modelled using a Boussinesq approximation with a
linear relationship between flow temperature and density. The order of accuracy
for one-dimensional scalar transport on a stretched and uniform grid was also
tested. The results show that for the method presented above a relatively
coarse mesh is sufficient to accurately describe the fluid flow, while the use
of a refined mesh for the low-diffusive scalars is found to be beneficial in
order to obtain a highly accurate resolution with negligible numerical
diffusion
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