Cosmos++: Relativistic Magnetohydrodynamics on Unstructured Grids with Local Adaptive Refinement

Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the grey approximation, which reduces Boltzmann seven-dimensional equation to a system of 3+1 equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem. Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit Runge-Kutta commonly employed for the magneto-hydrodynamics and Einstein equations. Finally, we analyze their performance in several numerical tests.

Magnetically polarized matter in astrophysical systems may be relevant in some magnetically dominated regions, for instance, in the funnel that is generated in some highly magnetized disk configurations where relativistic jets are thought to spread, or in pulsars where the fluids are subject to very intense magnetic fields. With the aim of dealing with magnetic media in the astrophysical context, we present for the first time the conservative form of the ideal general relativistic magnetohydrodynamics (GRMHD) equations with a non-zero magnetic polarization vector m μ . Then, we follow the Anile method to compute the eigenvalue structure in the case where the magnetic polarization is parallel to the magnetic field, and it is parameterized by the magnetic susceptibility χ m . This approximation allows us to describe diamagnetic fluids, for which χ m < 0, and paramagnetic fluids, for which χ m > 0. The theoretical results were implemented in the CAFE code to study the role of magnetic polarization in several one-dimensional Riemann problems. We found that independent of the initial condition, the first waves that appear in the numerical solutions are faster in diamagnetic materials than in paramagnetic ones. Moreover, the constant states between the waves change notably for different magnetic susceptibilities. All of these effects are more appreciable if the magnetic pressure is much higher than the fluid pressure. Additionally, with the aim of analyzing magnetic media in a strong gravitational field, we carry out for the first time a test of the magnetized Michel accretion of a magnetically polarized fluid. With this test, we found that the numerical solution is effectively maintained over time (t > 4000), and that the global convergence of the code is ≳2 for χ m ≲ 0.005 for all magnetic field strengths β we considered. Finally, when χ m = 0.008 and β ≥ 10, the global convergence of the code is reduced to a value between the first and second orders.

Incorporating the radiation reaction force into two-fluid plasma in curved space, we get a set of one-fluid general relativistic magnetohydrodynamics (GRMHD) equations with the Landau-Lifshitz radiation reaction force. We analyze the importance of the radiation reaction acting on plasma around an astrophysical compact object.

We investigate the general relativistic magnetohydrodynamic (GRMHD) equations for cold plasma around the Reissner-Nordström black hole. Applying 3+1 spacetime split we linearize the perturbed equations for non-magnetized/magnetized plasma in both rotating and non-rotating background. By Fourier analyze we then derive dispersion relations and investigate the existence of waves with positive angular frequency in the vicinity of the black hole horizon. The analysis finds propagation of negative phase and group velocities for rotating magnetized surroundings.

A directional error estimator for adaptive limit analysis

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

A stencil adaptive algorithm for finite difference solution of incompressible viscous flows

The computational burden associated with multidimensional, multicomponent, numerical solute transport models can be prohibitive. To solve these CPU intensive problems efficiently and accurately, we have implemented and extended a local adaptive grid refinement method of Berger and Oliger (1984) to track error‐prone regions and supply high‐resolution subgrids where they are locally needed while maintaining relatively few nodes elsewhere on a coarse, base grid. Novel features include a unique method for detecting a priori where the numerical error is unacceptable, variable time step control which allows smaller time steps on subgrids than on the base grid, and a modular framework which allows easy exchange of partial differential equation solvers to accommodate different problem formulations. Three examples show how the subgrids track the error‐prone regions, how multiple subgrids are used for geometrically complex front shapes, and how increased resolution is provided for self‐sharpening fronts when multiple reacting species are considered. For all three examples, solutions with accuracies comparable to those achieved with a uniform fine grid are obtained at between 20 and 30% of the computational cost.

We present a comprehensive study of velocity interpolation methods in polygons. These methods are often used as postprocessing procedures for numerical schemes that do not directly calculate the velocity field but only provide cell boundary flux conditions, such as the finite volume schemes. These methods extend the widely used velocity interpolation algorithms, such as the Pollock's algorithm, to more complex geometries such as perpendicular bisection (PEBI) grids, unstructured triangular grids and grids with local refinement. Once the velocity field is interpolated, streamline trajectories and time of flight along the streamlines can be calculated for reservoir simulation, model calibration and waterflood management, for instance. These velocity interpolation methods assume known lower‐order or higher‐order cell boundary fluxes, which satisfy global mass conservation and normal flux continuity. However, they differ in the interpolation of velocities within the interior of the cells. The interpolating velocity may be locally conservative or nonconservative, continuous or discontinuous, lower order or higher order. Results show that the interpolated velocity field has to be locally conservative in order to guarantee the correct volumetric transformation for the calculated streamlines and the time of flight. Velocity continuity is not as important as local conservation for the purpose of streamline applications. Compared to higher‐order interpolation for the streamline trajectories, lower‐order interpolation has the advantage of an analytic solution and an efficient implementation. Based on our analysis, we recommend a lower‐order locally conservative method for the most robust and numerically efficient calculation of streamline trajectories on unstructured grids.

S-splines: A simple surface solution for IGA and CAD

With the fundamental stress mechanism of accretion disks identified—correlated MHD turbulence driven by the magneto-rotational instability—it has become possible to make numerical simulations of accretion disk dynamics based on well-understood physics. A sampling of results from both Newtonian 3-d shearing box and general relativistic global disk MHD simulations is reported. Among other things, these simulations have shown that: contrary to long-held assumptions, stress is continuous through the marginally stable and plunging regions around black holes, so that rotating black holes can give substantial amounts of angular momentum electromagnetically to surrounding matter; the upper layers of accretion disks are primarily supported by magnetic pressure, potentially leading to interesting departures from local black-body emitted spectra; and initially local magnetic fields in accretion flows can, in some cases, spontaneously generate large-scale fields that connect rotating black holes to infinity and mediate strong relativistic jets. 1. Prolog: the classical view of accretion disks It has been understood for decades that accretion through disks can be an extremely powerful source of energy for the generation of both photons and material outflows. When

A Scheme for Generating Unstructured Grids on Spheres with Application to Parallel Computation

Diffusive-Time-of-Flight (DTOF), representing the travel time of pressure front propagation, has found many applications in unconventional reservoir performance analysis. The computation of DTOF typically involves upwind finite difference of the Eikonal equation and solution using the fast marching method (FMM). However, application of the finite difference based FMM to irregular grid systems remains a challenge. In this paper, we present a novel and robust method for solving the Eikonal equation using finite volume discretization and the FMM.The finite volume form of the Eikonal equation is derived by differential manipulation, volume integration and the divergence theorem. Using product rule, the differential term is first converted to the divergence form. Then volume integrals that contain divergence terms are converted to surface integrals using the divergence theorem. Consequently, the spatial coordinates are replaced by cell volumes and transmissibilities which are universal for both structured and unstructured grids in finite volume simulators. When applied with the upstream scheme, the finite volume form evolves into a set of quadratic equations, and fast marching method is implemented to solve these equations.The implementation is first validated with analytical solutions using isotropic and anisotropic models with homogeneous reservoir properties. Consistent DTOF distributions are obtained between the proposed approach and the analytical solutions. Next, the implementation is applied to unconventional reservoirs with hydraulic and natural fractures. Our approach relies on cell volumes and connections (transmissibilities) rather than the grid geometry, and thus can be easily applied to complex grid systems. For illustrative purposes, we present applications of the proposed method to embedded discrete fracture models (EDFM), dual-porosity dual-permeability models, and unstructured PEBI grids with heterogeneous reservoir properties. Visualization of the DTOF provides flow diagnostics such as evolution of the drainage volume of the wells and well interactions.The novelty of the proposed approach is its broad applicability to arbitrary grid systems and ease of implementation in commercial reservoir simulators. This makes the approach well-suited for field applications with complex grid geometry and complex well architecture.

A thin diffusive flame structure in an asymptotic form is resolved by the local polynomial refinement approach using the Gauss-Lobatto formulation. The continuity condition at the grid block interface must be enforced for the diffusion dominant equation. A multiple-point, unequal spaced, differencing approximation to preserve the spectral accurate and to increase computationally efficient is implemented. It is found that the shared first and second derivatives at the gird block interface can be computed from the roots of polynomials of adjacent blocks and effectively retains the numerical resolution in comparison with the L2 norm projection, recovery-based, penalty, and reconstruction methods in applications. The computed flame thickness agrees well with results using a systematic mesh refinement approach.