High-resolution Godunov Method Research Articles

An HLLC-type Riemann solver and high-resolution Godunov method for a two-phase model of reactive flow with general equations of state

A high-resolution Godunov method is developed for a two-phase model of reactive flow. The model considers general equations of state of Mie-Grüneisen form and different choices for the reaction rate, both relevant for simulations of the dynamical behavior of detonations in PBX-type granular explosives. The numerical approach employs a Riemann solver, and various options ranging from an exact solver to an approximate solver of HLLC type are described. A principal aim of the paper is an assessment of the approximate Riemann solvers in terms of the accuracy and efficiency of numerical solutions of the two-phase model. In particular, this study considers the state-sensitive behavior in different regimes of the solutions in a reverse-impact configuration, including compaction, run to detonation, and the evolution to compaction-led and reaction-led steady detonations. For these regimes, the convergence properties of the numerical approach are examined for the different choices of the Riemann solver, as is the error in the solutions at lower grid resolution. Finally, the computational performance of the time-stepping scheme is assessed for the different solvers.

A well balanced scheme for the shallow water wave equations in open channels with (discontinuous) varying width and bed

Finite volume methods have proven themselves a powerful tool for finding solutions to the shallow water wave equations. They are based on the conservation laws for the mass and momentum, integrated over discrete finite volumes. These methods tend to do well at the difficult problem of capturing solutions involving shocks. However, one area that causes problems is the approximation of steady or near steady states when there is a sloping bed elevation. The problem arises due to a poor balance between the discretisation of the flux terms across the edge of a finite volume and the pressure terms due to the sloping bed. Methods that overcome these difficulties and reproduce the still lake steady state solution, are called well balanced. In this work we are interested in a well balanced scheme for the one dimensional shallow water wave equations but with a modification that allows for varying width in the transverse direction. Here a well balanced method developed by Audusse et al. for the constant width case is extended to the case of varying (possibly discontinuous) width. Numerical validation of this new method is provided. References E. Audusse, F. Bouchut, M. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal of Scientific Computing , 25(6):2050--2065, 2004. doi:10.1137/S1064827503431090 A. Birman and J. Falcovitz. Application of the GRP scheme to open channel flow equations. Journal of Computational Physics , 222:131--154, 2007. doi:10.1016/j.jcp.2006.07.008 Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Rev. , 43(1):89--112 (electronic), 2001. doi:10.1137/S003614450036757X A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton--Jacobi equations. SIAM Journal of Scientific Computing , 23(3):707--740, 2001. doi:10.1137/S1064827500373413 R. LeVeque. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave propagation algorithm. Journal of Computational Physics , 146(1):346--365, 1998. doi:10.1006/jcph.1998.6058 Randall J. LeVeque. Finite volume methods for hyperbolic problems . Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511791253 B. Van Leer. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. Journal of Computational Physics , 32(1):101--136, 1979. doi:10.1016/0021-9991(79)90145-1 J. Zhou, D. Causon, C. Mingham, and D. Ingram. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics , 168:1--25, 2001. doi:10.1006/jcph.2000.6670 Christopher Zoppou and Stephen Roberts. Explicit schemes for dam-break simulations. ASCE, J. Hydraulic Engineering , 129(1):11--34, 2003. doi:10.1061/(ASCE)0733-9429(2003)129:1(11)

An evaluation of the FCT method for high-speed flows on structured overlapping grids

This study considers the development and assessment of a flux-corrected transport (FCT) algorithm for simulating high-speed flows on structured overlapping grids. This class of algorithm shows promise for solving some difficult highly-nonlinear problems where robustness and control of certain features, such as maintaining positive densities, is important. Complex, possibly moving, geometry is treated through the use of structured overlapping grids. Adaptive mesh refinement (AMR) is employed to ensure sharp resolution of discontinuities in an efficient manner. Improvements to the FCT algorithm are proposed for the treatment of strong rarefaction waves as well as rarefaction waves containing a sonic point. Simulation results are obtained for a set of test problems and the convergence characteristics are demonstrated and compared to a high-resolution Godunov method. The problems considered are an isolated shock, an isolated contact, a modified Sod shock tube problem, a two-shock Riemann problem, the Shu–Osher test problem, shock impingement on single cylinder, and irregular Mach reflection of a strong shock striking an inclined plane.

Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations

The last 20 years have seen a wide variety of high-resolution methods that can compute sharp, oscillation-free compressible flows. Here, we combine a complementary set of these methods together in a nonlinear (hybridized) fashion. Our base method is built on a monotone high-resolution Godunov method, the piece-wise parabolic method (PPM). PPM is combined with WENO methods, which reduce the damping of extrema. We find that the relative efficiency of the WENO methods is enhanced by coupling them with the relatively inexpensive Godunov methods. We accomplish our hybridizations through the use of a bounding principle: the approximation used is bounded by two nonlinearly stable approximations. The essential aspect of the method is to have high-order accurate approximations bounded by two non-oscillatory (nonlinearly stable) approximations. The end result is an accuracy-, monotonicity- and extrema-preserving method. These methods are demonstrated on a variety of flows, with quantitative analysis of the solutions with shocks.

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

A numerical method is described for inviscid, compressible, multi-material flow in two space dimensions. The flow is governed by the multi-material Euler equations with a general mixture equation of state. Composite overlapping grids are used to handle complex flow geometry and block-structured adaptive mesh refinement (AMR) is used to locally increase grid resolution near shocks and material interfaces. The discretization of the governing equations is based on a high-resolution Godunov method, but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard, conservative shock-capturing schemes. The energy correction is constructed based on a uniform-pressure–velocity flow and is significant only near the captured interface. A variety of two-material flows are presented to verify the accuracy of the numerical approach and to illustrate its use. These flows assume an equation of state for the mixture based on the Jones–Wilkins–Lee (JWL) forms for the components. This equation of state includes a mixture of ideal gases as a special case. Flow problems considered include unsteady one-dimensional shock–interface collision, steady interaction of a planar interface and an oblique shock, planar shock interaction with a collection of gas-filled cylindrical inhomogeneities, and the impulsive motion of the two-component mixture in a rigid cylindrical vessel.

The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow

This paper considers the Riemann problem and an associated Godunov method for a model of compressible two-phase flow. The model is a reduced form of the well-known Baer–Nunziato model that describes the behavior of granular explosives. In the analysis presented here, we omit source terms representing the exchange of mass, momentum and energy between the phases due to compaction, drag, heat transfer and chemical reaction, but retain the non-conservative nozzling terms that appear naturally in the model. For the Riemann problem the effect of the nozzling terms is confined to the contact discontinuity of the solid phase. Treating the solid contact as a layer of vanishingly small thickness within which the solution is smooth yields jump conditions that connect the states across the contact, as well as a prescription that allows the contribution of the nozzling terms to be computed unambiguously. An iterative method of solution is described for the Riemann problem, that determines the wave structure and the intermediate states of the flow, for given left and right states. A Godunov method based on the solution of the Riemann problem is constructed. It includes non-conservative flux contributions derived from an integral of the nozzling terms over a grid cell. The Godunov method is extended to second-order accuracy using a method of slope limiting, and an adaptive Riemann solver is described and used for computational efficiency. Numerical results are presented, demonstrating the accuracy of the numerical method and in particular, the accurate numerical description of the flow in the vicinity of a solid contact where phases couple and nozzling terms are important. The numerical method is compared with other methods available in the literature and found to give more accurate results for the problems considered.

Application of the high-resolution Godunov method to the multi-fluid flow calculations

In this paper, we have numerically solved the multi-fluid problems using an operator-split two-step high-resolution Godunov PPM (parabolic piecewise method) for the flow in complex geometries. By using the front capturing method, the PPM integrator captures the interface in the solution process. The basic multi-fluid integrator is coupled to a Cartesian grid algorithm where a VOF (volume of fluid) representation of the fluid interface is also used. As an application of this method, we test the 2D interfacial advection example and simulate an experimental hypervelocity launcher model from Sandia National Laboratories. The computational design of the hypervelocity launcher is also given in the paper.

Issues with high-resolution Godunov methods for radiation hydrodynamics

Two issues in the development of high-resolution Godunov methods for radiation hydrodynamics are discussed. First, an analysis of the Riemann problem is given. Second, it is shown that a large class of commonly used high-resolution methods are unable to obtain the equilibrium diffusion limit. The analysis points to several possible fixes.

A Modified Fractional Step Method for the Accurate Approximation of Detonation Waves

The numerical approximation of combustion processes may lead to numerical difficulties, which are caused by different time scales of the transport part and the reactive part of the model equations. Here we consider a modified fractional step method that overcomes this difficulty on standard test problems and allows the use of a mesh width and time step determined by the nonreactive part, without precisely resolving the very small reaction zone. High-resolution Godunov methods are employed and the structure of the Riemann solution is used to determine where burning should occur in each time step. The modification is implemented in the software package CLAWPACK. Numerical results for 1D and 2D detonation waves are shown, including a detonation wave diffracting around a corner.