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25
Three Dimensional Front Tracking
 SIAM J. Sci. Comp
, 1995
"... . We describe a three dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two dimensional computations, we expect three dimensional front tracking to improve significantly computatio ..."
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Cited by 85 (21 self)
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. We describe a three dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two dimensional computations, we expect three dimensional front tracking to improve significantly computational efficiencies for problems dominated by discontinuities. In some cases, for which the interface computations display considerable numerical sensitivity, we expect a greatly enhanced capability. 1. Introduction Front tracking is a numerical method in which surfaces of discontinuity are given explicit computational degrees of freedom; these degrees of freedom are supplemented by degrees of freedom representing continuous solution values at regular grid points. This method is ideal for solutions in which discontinuities are an important feature, and especially where their accurate computation is difficult by other methods. Computational continuum mechanics abounds in such problems, which in...
A simple method for compressible multifluid flows
 SIAM J. Sci. 141
, 1999
"... Abstract. A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunovtype scheme, with approximate ..."
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Cited by 70 (0 self)
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Abstract. A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunovtype scheme, with approximate Riemann solver for the resolution of conservation equations, and a set of nonconservative equations. It is valid for all mesh points and allows the resolution of interfaces. This method works for an arbitrary number of interfaces, for breakup and coalescence. It allows very high density ratios (up to 1000). It is able to compute very strong shock waves (pressure ratio up to 105). Contrary to all existing schemes (which consider the interface as a discontinuity) the method considers the interface as a numerical diffusion zone as contact discontinuities are computed in compressible single phase flows, but the variables describing the mixture zone are computed consistently with the density, momentum and energy. Several test problems are presented in one, two, and three dimensions. This method allows, for example, the computation of the interaction of a shock wave propagating in a liquid with a gas cylinder, as well as Richtmeyer–Meshkov instabilities, or hypervelocity impact, with realistic initial conditions. We illustrate our method with the Rusanov flux. However, the same principle can be applied to a more general class of schemes. Key words. compressible multicomponents flows, compressible multifluid flows, Godunov schemes, nonconservative systems
Conservative multiimplicit spectral deferred correction methods for reacting gas dynamics
 J. Comput. Phys
"... Abstract In most models of reacting gas dynamics, the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales, rendering the reaction part of the model equations stiff. Moreover, nonlinear forcings may introduce into the solutions sharp grad ..."
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Cited by 28 (12 self)
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Abstract In most models of reacting gas dynamics, the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales, rendering the reaction part of the model equations stiff. Moreover, nonlinear forcings may introduce into the solutions sharp gradients or shocks, the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures. This study presents highorder conservative methods for the temporal integration of model equations of reacting flows. By means of a method of lines discretization on the flux difference form of the equations, these methods compute approximations to the cellaveraged or finitevolume solution. The temporal discretization is based on a multiimplicit generalization of spectral deferred correction methods. The advection term is integrated explicitly, and the diffusion and reaction terms are treated implicitly but independently, with the splitting errors reduced via the spectral deferred correction procedure. To reduce computational cost, different time steps may be used to integrate processes with widelydiffering time scales. Numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients; the results also demonstrate the expected convergence rates for the methods of orders three, four, and five for smooth problems.
Highorder multiimplicit spectral deferred correction methods for problems of reactive flow,
 J. Comput. Phys.
, 2003
"... Abstract Models for reacting flow are typically based on advectiondiffusionreaction (ADR) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three subprocesses can be widely different, leading to disparate tim ..."
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Cited by 26 (9 self)
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Abstract Models for reacting flow are typically based on advectiondiffusionreaction (ADR) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three subprocesses can be widely different, leading to disparate timestep requirements for robust and accurate timeintegration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this timescale disparity. The proposed methods are highorder multiimplicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of ADR equations. Spectral deferred correction methods compute a highorder approximation to the solution of a differential equation by using a simple, loworder numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several subprocesses implicitly but independently, while avoiding the splitting errors present in traditional operatorsplitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semiimplicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the subprocess time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific timescales ordering, the generalization to any ordering combination is straightforward.
A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws I: Scalar Equations
 in Hyperbolic Conservation Laws II: Systems of Equations, (in preparation
, 1998
"... A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite di#erence approach, analogous to the method of lines scheme presented in [20]. The zero of a level set function is used to specify the location of the discontinuity. Si ..."
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Cited by 14 (3 self)
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A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite di#erence approach, analogous to the method of lines scheme presented in [20]. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the "real" state, and one corresponding to a "ghost node" state, analogous to the "Ghost Fluid Method" of [6]. High order pointwise convergence is demonstrated for linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions. The solutions are compared to standard high order shock capturing schemes. This paper focuses on scalar conservation laws. Level set tracking for systems of conservation laws in multidimensions will be pres...
Numerical Simulation of Three Dimensional Free Surface Flows with Bubbles
 INT. J. NUM. METH. FLUIDS
, 2003
"... A numerical model is presented for the simulation of free surface flows. The unknowns are the volume fraction of liquid, the velocity and pressure in the liquid and the pressure in the bubbles of gas which can appear in the liquid flow. The volume friction ..."
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Cited by 14 (4 self)
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A numerical model is presented for the simulation of free surface flows. The unknowns are the volume fraction of liquid, the velocity and pressure in the liquid and the pressure in the bubbles of gas which can appear in the liquid flow. The volume friction
Unconditionally Stable Splitting Methods For The Shallow . . .
, 1998
"... The fronttracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The fronttracking method is unconditionally stabl ..."
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Cited by 10 (2 self)
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The fronttracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The fronttracking method is unconditionally stable, but for practical computations feasible cfl numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical
B.: Heterogeneous multiscale methods for interface tracking of combustion fronts
 SIAM Multiscale Model. Simul
, 2006
"... Abstract. In this paper we investigate the heterogeneous multiscale methods (HMM) for interface tracking and apply the technique to the simulation of combustion fronts. Our goal is to overcome the numerical difficulties, which are caused by different time scales between the transport part and the r ..."
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Cited by 7 (3 self)
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Abstract. In this paper we investigate the heterogeneous multiscale methods (HMM) for interface tracking and apply the technique to the simulation of combustion fronts. Our goal is to overcome the numerical difficulties, which are caused by different time scales between the transport part and the reactive part in the model equations of some interface tracking problems, such as combustion processes. HMM relies on an efficient coupling between the macroscale and microscale models. When the macroscale model is not fully known explicitly or not valid in localized regions, HMM provides a procedure for supplementing the missing data from a microscale model. Here we design and analyze a multiscale scheme in which a localized microscale model resolves the details in the model and a phase field or a front tracking method defines the interface on the macroscale. This multiscale technique overcomes the difficulty of stiffness of common problems in combustion processes. Numerical results for Majda’s model and reactive Euler equations in one and two dimensions show substantially improved efficiency over traditional methods.
Developments in Cartesian cut cell methods
"... This paper describes the Cartesian cut cell method, which provides a flexible and efficient alternative to traditional boundary fitted grid methods. The Cartesian cut cell approach uses a background Cartesian grid for the majority of the flow domain with special treatments being applied to cells whi ..."
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Cited by 7 (0 self)
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This paper describes the Cartesian cut cell method, which provides a flexible and efficient alternative to traditional boundary fitted grid methods. The Cartesian cut cell approach uses a background Cartesian grid for the majority of the flow domain with special treatments being applied to cells which are cut by solid bodies, thus retaining a boundary conforming grid. The development of the method is described with applications to problems involving both moving bodies and moving material interfaces.
High order finite difference methods with subcell resolution for advection equations with stiff source terms
 J. Comput. Phys
, 2012
"... doi:10.4208/cicp.250214.130814a ..."
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