Hybrid Riemann Solvers for Large Systems of Conservation Laws

In this paper we present a new family of approximate Riemann solvers for the numerical approximation of solutions of hyperbolic conservation laws. They are approximate, also referred to as incomplete, in the sense that the solvers avoid computing the characteristic decomposition of the flux Jacobian. Instead, they require only an estimate of the globally fastest wave speeds in both directions. Thus, this family of solvers is particularly efficient for large systems of conservation laws, i.e. with many different propagation speeds, and when no explicit expression for the eigensystem is available. Even though only fastest wave speeds are needed as input values, the new family of Riemann solvers reproduces all waves with less dissipation than HLL, which has the same prerequisites, requiring only one additional flux evaluation.Comment: 9 page

Numerical simulation of a highly underexpanded carbon dioxide jet

The underexpanded jets are present in many processes such as rocket propulsion, mass spectrometry, fuel injection, as well as in the process called rapid expansion of supercritical solutions (RESS). In the RESS process a supercritical solution flows through a capillary nozzle until an expansion chamber where the strong changes in the thermodynamic properties of the solvent are used to encapsulate the solute in very fine particles. The research project was focused on the hydrodynamic modeling of an hypersonic carbon dioxide jet produced in the context of the RESS process. The mathematical modeling of the jet was developed using the set of the compressible Navier-Stokes equations along with the generalized Bender equation of state. This set of PDE was solved using an adaptive discontinuous Galerkin discretization for space and the exponential Rosenbrock-Euler method for the time integration. The numerical solver was implemented in C++ using several libraries such as deal.ii and Sacado-Trilinos

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

Investigation of a Jacobian-free Newton-Krylov solution to multiphase flows

The current study is focused on investigating a Jacobian-Free Newton-Krylov (JFNK) method to obtain a fully-implicit solution for two phase flows. In the JFNK formulation, the Jacobian matrix is not directly determined potentially leading to major computational savings compared to a simple Newton's solver. Prior to the implementation of JFNK to solve two-phase flow problem, it is utilized to solve the governing equations corresponding to single phase flow. The objectives of the present study are (i) Application of the JFNK method to two-fluid models, (ii) Investigation of the advantages and disadvantages of the method compared to commonly used explicit methods, and (iii) Comparison of the numerical predictions with those obtained by the current version of the Network thermalhydraulics code, CATHENA. The background information required is presented and the numerical setup for each test case is discussed in detail. Three well-known benchmarks are considered, the 1D dam break problem, the water faucet and the oscillating manometer. For single phase flow simulations, the Shallow Water Wave Equations is selected to model the motion of the fluid and a backward Euler scheme is utilized for the temporal discretization along with a central-upwind Godonuv scheme for the spatial discretization. For the two-phase simulations, an isentropic (four equation) two fluid model is chosen. Time discretization is performed by a backward Euler scheme and the AUSM+ scheme is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. For comparison, one explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented for each set of governing equations. A detailed grid and model parameter sensitivity analysis is performed to identify the advantages and disadvantages of JFNK for each case. For all three benchmarks, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL (Courant–Friedrichs–Lewy ) numbers up to 100 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared to the calculations requiring the determination of the Jacobian matrix. This reduction is in the order of 80%