First-Order System Least Squares and the Energetic Variational Approach for Two-Phase Flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.Comment: 22 pages, 8 figures submitted to Journal of Computational Physic

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

In this paper we present a full-fledged scheme for the second order accurate, divergence-free evolution of vector fields on an adaptive mesh refinement (AMR) hierarchy. We focus here on adaptive mesh MHD. The scheme is based on making a significant advance in the divergence-free reconstruction of vector fields. In that sense, it complements the earlier work of Balsara and Spicer (1999) where we discussed the divergence-free time-update of vector fields which satisfy Stoke's law type evolution equations. Our advance in divergence-free reconstruction of vector fields is such that it reduces to the total variation diminishing (TVD) property for one-dimensional evolution and yet goes beyond it in multiple dimensions. Divergence-free restriction is also discussed. An electric field correction strategy is presented for use on AMR meshes. The electric field correction strategy helps preserve the divergence-free evolution of the magnetic field even when the time steps are sub-cycled on refined meshes. The above-mentioned innovations have been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics which supports both non-relativistic and relativistic MHD. Several rigorous, three dimensional AMR-MHD test problems with strong discontinuities have been run with the RIEMANN framework showing that the strategy works very well.Comment: J.C.P., figures of reduced qualit

Monolithic Multigrid for Magnetohydrodynamics

The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems

ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement

We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a standard tree-type algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space--time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.Comment: With updated bibliography informatio

Relativistic resistive magnetohydrodynamic reconnection and plasmoid formation in merging flux tubes

We apply the general relativistic resistive magnetohydrodynamics code {\tt BHAC} to perform a 2D study of the formation and evolution of a reconnection layer in between two merging magnetic flux tubes in Minkowski spacetime. Small-scale effects in the regime of low resistivity most relevant for dilute astrophysical plasmas are resolved with very high accuracy due to the extreme resolutions obtained with adaptive mesh refinement. Numerical convergence in the highly nonlinear plasmoid-dominated regime is confirmed for a sweep of resolutions. We employ both uniform resistivity and non-uniform resistivity based on the local, instantaneous current density. For uniform resistivity we find Sweet-Parker reconnection, from $\eta = 10^{-2}$ down to $\eta = 10^{-4}$, for a reference case of magnetisation $\sigma = 3.33$ and plasma-$\beta = 0.1$. {For uniform resistivity $\eta=5\times10^{-5}$ the tearing mode is recovered, resulting in the formation of secondary plasmoids. The plasmoid instability enhances the reconnection rate to $v_{\rm rec} \sim 0.03c$ compared to $v_{\rm rec} \sim 0.01c$ for $\eta=10^{-4}$.} For non-uniform resistivity with a base level $\eta_0 = 10^{-4}$ and an enhanced current-dependent resistivity in the current sheet, we find an increased reconnection rate of $v_{\rm rec} \sim 0.1c$. The influence of the magnetisation $\sigma$ and the plasma-$\beta$ is analysed for cases with uniform resistivity $\eta=5\times10^{-5}$ and $\eta=10^{-4}$ in a range $0.5 \leq \sigma \leq 10$ and $0.01 \leq \beta \leq 1$ in regimes that are applicable for black hole accretion disks and jets. The plasmoid instability is triggered for Lundquist numbers larger than a critical value of $S_{\rm c} \approx 8000$.Comment: Matching accepted version in MNRA

Development of low dissipative high order filter schemes for multiscale Navier–Stokes/MHD systems

Recent progress in the development of a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier–Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described. The four main features of this class of schemes are: (a) multiresolution wavelet decomposition of the computed flow data as sensors for adaptive numerical dissipative control, (b) multistep filter to accommodate efficient application of different numerical dissipation models and different spatial high order base schemes, (c) a unique idea in solving the ideal conservative MHD system (a non-strictly hyperbolic conservation law) without having to deal with an incomplete eigensystem set while at the same time ensuring that correct shock speeds and locations are computed, and (d) minimization of the divergence of the magnetic field numerical error. By design, the flow sensors, different choice of high order base schemes and numerical dissipation models are stand-alone modules. A whole class of low dissipative high order schemes can be derived at ease, making the resulting computer software very flexible with widely applicable. Performance of multiscale and multiphysics test cases are illustrated with many levels of grid refinement and comparison with commonly used schemes in the literature