Sudip K. Garain

An efficient class of WENO schemes with adaptive order

Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient.The first advance consists of realizing that WENO schemes require us to carry out stencil operations very efficiently. In this paper we show that the reconstructed polynomials for any one-dimensional stencil can be expressed most efficiently and economically in Legendre polynomials. By using Legendre basis, we show that the reconstruction polynomials and their corresponding smoothness indicators can be written very compactly. The smoothness indicators are written as a sum of perfect squares. Since this is a computationally expensive step, the efficiency of finite difference WENO schemes is enhanced by the innovation which is reported here.The second advance consists of realizing that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order WENO scheme that is nevertheless very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes, which we call WENO-AO (for adaptive order). Thus we arrive at a WENO-AO(5,3) scheme that is at best fifth order accurate by virtue of its centered stencil with five zones and at worst third order accurate by virtue of being non-linearly hybridized with an r=3 CWENO scheme. The process can be extended to arrive at a WENO-AO(7,3) scheme that is at best seventh order accurate by virtue of its centered stencil with seven zones and at worst third order accurate. We then recursively combine the above two schemes to arrive at a WENO-AO(7,5,3) scheme which can achieve seventh order accuracy when that is possible; graciously drop down to fifth order accuracy when that is the best one can do; and also operate stably with an r=3 CWENO scheme when that is the only thing that one can do. Schemes with ninth order of accuracy are also presented.Several accuracy tests and several stringent test problems are presented to demonstrate that the method works very well.

EFFECTS OF COMPTON COOLING ON OUTFLOW IN A TWO-COMPONENT ACCRETION FLOW AROUND A BLACK HOLE: RESULTS OF A COUPLED MONTE CARLO TOTAL VARIATION DIMINISHING SIMULATION

We investigate the effects of cooling of the Compton cloud on the outflow formation rate in an accretion disk around a black hole. We carry out a time-dependent numerical simulation where both the hydrodynamics and the radiative transfer processes are coupled together. We consider a two-component accretion flow in which the Keplerian disk is immersed into an accreting low-angular momentum flow (halo) around a black hole. The soft photons which originate from the Keplerian disk are inverse-Comptonized by the electrons in the halo and the region between the centrifugal pressure supported shocks and the horizon. We run several cases by changing the rate of the Keplerian disk and see the effects on the shock location and properties of the outflow and the spectrum. We show that as a result of Comptonization of the Compton cloud, the cloud becomes cooler with the increase in the Keplerian disk rate. As the resultant thermal pressure is reduced, the post-shock region collapses and the outflow rate is also reduced. Since the hard radiation is produced from the post-shock region, and the spectral slope increases with the reduction of the electron temperature, the cooling produces softer spectrum. We thus find a direct correlation between the spectral states and the outflow rates of an accreting black hole.

Quasi-periodic oscillations in a radiative transonic flow: results of a coupled Monte Carlo–tvd simulation

Low and intermediate frequency quasi-periodic oscillations (QPOs) in black hole candidates are believed to be due to oscillations of the Comptonizing regions in an accretion flow. Assuming that the general structure of an accretion disk is a Two Component Advective Flow (TCAF), we numerically simulate the light curves emitted from an accretion disk for different accretion rates and find how the QPO frequencies vary. We use a standard Keplerian disk residing at the equatorial plane as a source of soft photons. These soft photons, after suffering multiple scattering with the hot electrons of the low angular momentum, sub-Keplerian, flow emerge out as hard radiation. The hydrodynamic and thermal properties of the electron cloud is simulated using a Total Variation Diminishing (TVD) code. The TVD code is then coupled with a radiative transfer code which simulates the energy exchange between the electron and radiation using Monte Carlo technique. The resulting localized heating and cooling are included also. We find that the QPO frequency increases and the spectrum becomes softer as we increase the Keplerian disk rate. However, the spectrum becomes harder if we increase the sub-Keplerian accretion rate. We find that an earlier prediction that QPOs occur when the infall time scale roughly matches with the cooling time scale, originally obtained using a power-law cooling, remains valid even for Compton cooling. Our findings agree with the general observations of low frequency QPOs in black hole candidates.

Effects of Compton cooling on the hydrodynamic and the spectral properties of a two-component accretion flow around a black hole

We carry out a time-dependent numerical simulation where both the hydrodynamics and the radiative transfer are coupled together. We consider a two-component accretion flow in which the Keplerian disc is immersed inside an accreting low angular momentum flow (halo) around a black hole. The injected soft photons from the Keplerian disc are reprocessed by the electrons in the halo. We show that in presence of an axisymmetric soft-photon source the spherically symmetric Bondi flow loses its symmetry and becomes axisymmetric. The low angular momentum flow was observed to slow down close to the axis and formed a centrifugal barrier which added new features into the spectrum. Using the Monte Carlo method, we generated the radiated spectra as functions of the accretion rates. We find that the transitions from a hard state to a soft state is determined by the mass accretion rates of the disc and the halo. We separate out the signature of the bulk motion Comptonization and discuss its significance. We study how the net spectrum is contributed by photons suffering different number of scatterings and spending different amounts of time inside the Compton cloud. We study the directional dependence of the emitted spectrum as well.

Segregation of a Keplerian disc and sub-Keplerian halo from a transonic flow around a black hole by viscosity and cooling processes

A black hole accretion is necessarily transonic. In presence of sufficiently high viscosity and cooling effects, a low-angular momentum transonic flow can become a standard Keplerian disc except close to the where hole where it must pass through the inner sonic point. However, if the viscosity is not high everywhere and cooling is not efficient everywhere, the flow cannot completely become a Keplerian disc. In this paper, we show results of rigorous numerical simulations of a transonic flow having vertically varying viscosity parameter (being highest on the equatorial plane) and optical depth dependent cooling processes to show that the flow indeed segregates into two distinct components as it approaches a black hole. The component on the equatorial plane has properties of a standard Keplerian disc, though the flow is not truncated at the innermost stable circular orbit. This component extends till the horizon as a sub-Keplerian flow. This standard disc is found to be surrounded by a hot, low angular momentum component forming a centrifugal barrier dominated oscillating shock wave, consistent with the Chakrabarti-Titarchuk two component advective flow configuration.

A two-dimensional Riemann solver with self-similar sub-structure - Alternative formulation based on least squares projection

Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The self-similar formulation of Balsara 16 proves especially useful for this purpose. While that work is based on a Galerkin projection, in this paper we present an analogous self-similar formulation that is based on a different interpretation. In the present formulation, we interpret the shock jumps at the boundary of the strongly-interacting state quite literally. The enforcement of the shock jump conditions is done with a least squares projection (Vides, Nkonga and Audit 67). With that interpretation, we again show that the multidimensional Riemann solver can be endowed with sub-structure. However, we find that the most efficient implementation arises when we use a flux vector splitting and a least squares projection. An alternative formulation that is based on the full characteristic matrices is also presented. The multidimensional Riemann solvers that are demonstrated here use one-dimensional HLLC Riemann solvers as building blocks.Several stringent test problems drawn from hydrodynamics and MHD are presented to show that the method works. Results from structured and unstructured meshes demonstrate the versatility of our method. The reader is also invited to watch a video introduction to multidimensional Riemann solvers on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.

Comparing Coarray Fortran (CAF) with MPI for several structured mesh PDE applications

Language-based approaches to parallelism have been incorporated into the Fortran standard. These Fortran extensions go under the name of Coarray Fortran (CAF) and full-featured compilers that support CAF have become available from Cray and Intel; the GNU implementation is expected in 2015. CAF combines elegance of expression with simplicity of implementation to yield an efficient parallel programming language. Elegance of expression results in very compact parallel code. The existence of a standard helps with portability and maintainability. CAF was designed to excel at one-sided communication and similar functions that support one-sided communication are also available in the recent MPI-3 standard. One-sided communication is expected to be very valuable for structured mesh applications involving partial differential equations, amongst other possible applications. This paper focuses on a comparison of CAF and MPI for a few very useful applications areas that are routinely used for solving partial differential equations on structured meshes. The three specific areas are Fast Fourier Techniques, Computational Fluid Dynamics, and Multigrid Methods.For each of those applications areas, we have developed optimized CAF code and optimized MPI code that is based on the one-sided messaging capabilities of MPI-3. Weak scalability studies that compare CAF and MPI-3 are presented on up to 65,536 processors. Both paradigms scale well, showing that they are well-suited for Petascale-class applications. Some of the applications shown (like Fast Fourier Techniques and Computational Fluid Dynamics) require large, coarse-grained messaging. Such applications emphasize high bandwidth. Our other application (Multigrid Methods) uses pointwise smoothers which require a large amount of fine-grained messaging. In such applications, a premium is placed on low latency. Our studies show that both CAF and MPI-3 offer the twin advantages of high bandwidth and low latency for messages of all sizes. Even for large numbers of processors, CAF either draws level with MPI-3 or shows a slight advantage over MPI-3. Both CAF and MPI-3 are shown to provide substantial advantages over MPI-2.In addition to the weak scalability studies, we also catalogue some of the best-usage strategies that we have found for our successful implementations of one-sided messaging in CAF and MPI-3. We show that CAF code is of course much easier to write and maintain, and the simpler syntax makes the parallelism easier to understand.

MONTE-CARLO SIMULATIONS OF COMPTONIZATION PROCESS IN A TWO COMPONENT ACCRETION FLOW AROUND A BLACK HOLE IN PRESENCE OF AN OUTFLOW

A black hole accretion may have both the Keplerian and the sub-Keplerian component. In the so-called Chakrabarti–Titarchuk scenario, the Keplerian component supplies low-energy (soft) photons while the sub-Keplerian component supplies hot electrons which exchange their energy with the soft photons through Comptonization or inverse Comptonization processes. In the sub-Keplerian component, a shock is generally produced due to the centrifugal force. The postshock region is known as the CENtrifugal pressure–supported BOundary Layer (CENBOL). In this paper, we compute the effects of the thermal and the bulk motion Comptonization on the soft photons emitted from a Keplerian disk by the CENBOL, the preshock sub-Keplerian disk and the outflowing jet. We study the emerging spectrum when the converging inflow and the diverging outflow (generated from the CENBOL) are simultaneously present. From the strength of the shock, we calculate the percentage of matter being carried away by the outflow and determine how the emerging spectrum depends on the outflow rate. The preshock sub-Keplerian flow is also found to Comptonize the soft photons significantly. The interplay between the up-scattering and down-scattering effects determines the effective shape of the emerging spectrum. By simulating several cases with various inflow parameters, we conclude that whether the preshock flow, or the postshock CENBOL or the emerging jet is dominant in shaping the emerging spectrum depends strongly on the geometry of the flow and the strength of the shock in the sub-Keplerian flow.

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part I, second-order FVTD schemes

Abstract While classic finite-difference time-domain (FDTD) solutions of Maxwells equations have served the computational electrodynamics (CED) community very well, formulations based on Godunov methodology have begun to show advantages. We argue that the formulations presented so far are such that FDTD schemes and Godunov-based schemes each have their own unique advantages. However, there is currently not a single formulation that systematically integrates the strengths of both these major strains of development. While an early glimpse of such a formulation was offered in Balsara et al. [16] , that paper focused on electrodynamics in plasma. Here, we present a synthesis that integrates the strengths of both FDTD and Godunov-based schemes into a robust single formulation for CED in material media. Three advances make this synthesis possible. First, from the FDTD method, we retain (but somewhat modify) a spatial staggering strategy for the primal variables. This provides a beneficial constraint preservation for the electric displacement and magnetic induction vector fields via reconstruction methods that were initially developed in some of the first authors papers for numerical magnetohydrodynamics (MHD). Second, from the Godunov method, we retain the idea of upwinding, except that this idea, too, has to be significantly modified to use the multi-dimensionally upwinded Riemann solvers developed by the first author. Third, we draw upon recent advances in arbitrary derivatives in space and time (ADER) time-stepping by the first author and his colleagues. We use the ADER predictor step to endow our method with sub-cell resolving capabilities so that the method can be stiffly stable and resolve significant sub-cell variation in the material properties within a zone. Overall, in this paper, we report a new scheme for numerically solving Maxwells equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time–step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.

Images and spectra of time-dependent two-component advective flow in presence of outflows

Two Component Advective Flow (TCAF) successfully explains the spectral and tem- poral properties of outbursting or persistent sources. Images of static TCAF with Compton cloud or CENtrifugal pressure supported Boundary Layer (CENBOL) due to gravitational bending of photons have been studied before. In this paper, we study time dependent images of advective flows around a Schwarzschild black hole which include cooling effects due to Comptonization of soft photons from a Keplerian disks well as the self-consistently produced jets and outflows. We show the overall image of the disk-jet system after convolving with a typical beamwidth. A long exposure image with time dependent system need not show the black hole horizon conspicuously, un- less one is looking at a soft state with no jet or the system along the jet axis. Assuming these disk-jet configurations are relevant to radio emitting systems also, our results would be useful to look for event horizons in high accretion rate Supermassive Black Holes in Seyfert galaxies, RL Quasars.