High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates

High-order reconstruction schemes for the solution of hyperbolic conservation laws in orthogonal curvilinear coordinates are revised in the finite volume approach. The formulation employs a piecewise polynomial approximation to the zone-average values to reconstruct left and right interface states from within a computational zone to arbitrary order of accuracy by inverting a Vandermonde-like linear system of equations with spatially varying coefficients. The approach is general and can be used on uniform and non-uniform meshes although explicit expressions are derived for polynomials from second to fifth degree in cylindrical and spherical geometries with uniform grid spacing. It is shown that, in regions of large curvature, the resulting expressions differ considerably from their Cartesian counterparts and that the lack of such corrections can severely degrade the accuracy of the solution close to the coordinate origin. Limiting techniques and monotonicity constraints are revised for conventional reconstruction schemes, namely, the piecewise linear method (PLM), third-order weighted essentially non-oscillatory (WENO) scheme and the piecewise parabolic method (PPM). The performance of the improved reconstruction schemes is investigated in a number of selected numerical benchmarks involving the solution of both scalar and systems of nonlinear equations (such as the equations of gas dynamics and magnetohydrodynamics) in cylindrical and spherical geometries in one and two dimensions. Results confirm that the proposed approach yields considerably smaller errors, higher convergence rates and it avoid spurious numerical effects at a symmetry axis.Comment: 37 pages, 12 Figures. Accepted for publication in Journal of Compuational Physic

Numerical simulations of radiative magnetized Herbig-Haro jets: the influence of pre-ionization from X-rays on emission lines

We investigate supersonic, axisymmetric magnetohydrodynamic (MHD) jets with a time-dependent injection velocity by numerical simulations with the PLUTO code. Using a comprehensive set of parameters, we explore different jet configurations in the attempt to construct models that can be directly compared to observational data of microjets. In particular, we focus our attention on the emitting properties of traveling knots and construct, at the same time, accurate line intensity ratios and surface brightness maps. Direct comparison of the resulting brightness and line intensity ratios distributions with observational data of microjets shows that a closer match can be obtained only when the jet material is pre-ionized to some degree. A very likely source for a pre-ionized medium is photoionization by X-ray flux coming from the central object.Comment: Accepted for publication in Ap

A Second-Order Unsplit Godunov Scheme for Cell-Centered MHD: the CTU-GLM scheme

We assess the validity of a single step Godunov scheme for the solution of the magneto-hydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The proposed scheme employs a cell-centered representation of the primary fluid variables (including magnetic field) and conserves mass, momentum, magnetic induction and energy. A variant of the scheme, which breaks momentum and energy conservation, is also considered. Divergence errors are transported out of the domain and damped using the mixed hyperbolic/parabolic divergence cleaning technique by Dedner et al. (J. Comput. Phys., 175, 2002). The strength and accuracy of the scheme are verified by a direct comparison with the eight-wave formulation (also employing a cell-centered representation) and with the popular constrained transport method, where magnetic field components retain a staggered collocation inside the computational cell. Results obtained from two- and three-dimensional test problems indicate that the newly proposed scheme is robust, accurate and competitive with recent implementations of the constrained transport method while being considerably easier to implement in existing hydro codes.Comment: 31 Pages, 16 Figures Accepted for publication in Journal of Computational Physic

An HLLC Riemann Solver for Relativistic Flows: I. Hydrodynamics

We present an extension of the HLLC approximate Riemann solver by Toro, Spruce and Speares to the relativistic equations of fluid dynamics. The solver retains the simplicity of the original two-wave formulation proposed by Harten, Lax and van Leer (HLL) but it restores the missing contact wave in the solution of the Riemann problem. The resulting numerical scheme is computationally efficient, robust and positively conservative. The performance of the new solver is evaluated through numerical testing in one and two dimensions.Comment: 12 pages, 12 figure

The National Security Dividend of Global Carbon Mitigation

Energy and environmental security objectives are often conflated in political circles and in the popular press. Results from a well-established integrated assessment model suggest that policies designed to stabilize atmospheric carbon dioxide concentrations at levels above ~500 ppm generally do not align with policies to curb global oil dependence, because these atmospheric objectives can be achieved largely through reductions in global coal consumption. Policies designed to stabilize atmospheric carbon dioxide at levels below ~500 ppm, on the other hand, directly facilitate the alignment of environmental and security objectives because atmospheric targets in this range demand significant reductions in both coal and oil use. Greater recognition that investment in carbon mitigation can yield significant security dividends may alter the political cost-benefit calculus of energy-importing nations and could increase the willingness of some key global actors to seek binding cooperative targets under any post-Kyoto climate treaty regime.

The Dynamics of Radiative Shock Waves: Linear and Nonlinear Evolution

33 pages, 12 figures, accepted for publication on the Astrophysical Journa

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

A five-wave HLL Riemann solver for relativistic MHD

We present a five-wave Riemann solver for the equations of ideal relativistic magnetohydrodynamics. Our solver can be regarded as a relativistic extension of the five-wave HLLD Riemann solver initially developed by Miyoshi and Kusano for the equations of ideal MHD. The solution to the Riemann problem is approximated by a five wave pattern, comprised of two outermost fast shocks, two rotational discontinuities and a contact surface in the middle. The proposed scheme is considerably more elaborate than in the classical case since the normal velocity is no longer constant across the rotational modes. Still, proper closure to the Rankine-Hugoniot jump conditions can be attained by solving a nonlinear scalar equation in the total pressure variable which, for the chosen configuration, has to be constant over the whole Riemann fan. The accuracy of the new Riemann solver is validated against one dimensional tests and multidimensional applications. It is shown that our new solver considerably improves over the popular HLL solver or the recently proposed HLLC schemes.Comment: 15 pages, 19 figures. Accepted for Publication in MNRA