#### DMCA

## B.: Heterogeneous multiscale methods for interface tracking of combustion fronts (2006)

Venue: | SIAM Multiscale Model. Simul |

Citations: | 7 - 3 self |

### Citations

1469 |
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...icantly to the physics of the problem, and it is essential to describe them accurately. There are several computational methods for numerical interface tracking. Examples include the level set method =-=[27, 28]-=-, front tracking method [18, 34], phase field method, and segment projection method [33]. The standard situation is the one in which the evolution of the interface can be directly determined ∗Received... |

1082 |
Level Set Methods and Dynamic Implicit Surfaces
- Osher, Fedkiw
- 2003
(Show Context)
Citation Context ...cks, solidification, melting, etching, epitaxial growth of thin films, and multiphase flows. Interfaces are also used as computational tools in high frequency wave propagation and in image processing =-=[27]-=-. The effects of these interfaces often contribute significantly to the physics of the problem, and it is essential to describe them accurately. There are several computational methods for numerical i... |

1007 | Approximate Riemann solvers, parameter vectors and difference schemes
- Roe
- 1981
(Show Context)
Citation Context ...teps. First we assume that no reaction occurs (i.e., (3.8) with K(T ) = 0) and approximate the solution of the nonreactive Euler equations ∂U ∂t + ∂F (U) ∂x = 0.(3.10) We use Roe’s first-order scheme =-=[29]-=- for solving the system (3.10) U n = LΔtRoeU n,(3.11) where LΔtRoe denotes Roe’s solver. Of course, some high-order schemes for conservation laws [21] can be applied to this step. In the second step w... |

700 |
Numerical Methods for Conservation Laws
- LeVeque
- 1992
(Show Context)
Citation Context ...(U) ∂x = 0.(3.10) We use Roe’s first-order scheme [29] for solving the system (3.10) U n = LΔtRoeU n,(3.11) where LΔtRoe denotes Roe’s solver. Of course, some high-order schemes for conservation laws =-=[21]-=- can be applied to this step. In the second step we solve the following system of ODEs for the source terms d dt ⎛ ⎜⎜⎝ ρZ1 ρZ2 . . . ρZN ⎞ ⎟⎟⎠ = ⎛ ⎜⎜⎝ ψ1 ψ2 . . . ψN ⎞ ⎟⎟⎠ with an approximate solution... |

559 |
Adaptive mesh refinement for hyperbolic partial differential equations
- Berger, Oliger
- 1984
(Show Context)
Citation Context ...om step (ii) using the newly evolved front points and their corresponding interface. Our approach is closely related to the general philosophy of the successful adaptive mesh refinement (AMR) methods =-=[5, 4]-=-. As in AMR methods, the end result is to work with a mesh which is locally refined at the interface and coarsened away from the interface. However, there are some important differences on how the mes... |

272 | The heterogeneous multiscale methods
- Engquist, E
(Show Context)
Citation Context ...king, phase field method, Euler equations, combustion AMS subject classifications. 35M10, 65M06, 65M50 DOI. 10.1137/050624844 1. Introduction. The heterogeneous multiscale method (HMM), introduced in =-=[11]-=-, provides a unified framework for designing efficient numerical methods for problems with multiple scales. When a macroscale model is not explicitly given or not valid in localized regions, HMM provi... |

200 | Y.J.: A fronttracking method for the computations of multiphase flow
- Tryggvason, Bunner, et al.
- 2001
(Show Context)
Citation Context ...oblem, and it is essential to describe them accurately. There are several computational methods for numerical interface tracking. Examples include the level set method [27, 28], front tracking method =-=[18, 34]-=-, phase field method, and segment projection method [33]. The standard situation is the one in which the evolution of the interface can be directly determined ∗Received by the editors February 20, 200... |

131 |
Numerical Simulation of Reactive Flow,
- Oran, Boris
- 1987
(Show Context)
Citation Context ... If 552 YI SUN AND BJORN ENGQUIST U represents temperature T , the function K(T ) typically has the Arrhenius form K(T ) = K0e −A/T ,(3.3) where K0 is the rate constant and A is the activation energy =-=[26]-=-. The reaction rate is negligible for low temperature and grows exponentially fast if the temperature is high enough. For computational purposes, a discrete ignition temperature kinetics model may rep... |

100 | Analysis of the heterogeneous multiscale method for elliptic homogenization problems
- E, Ming, et al.
(Show Context)
Citation Context ...he missing information from an explicitly given microscale model. Several encouraging results to different classes of problems, including homogenization [1, 12], stiff ordinary differential equations =-=[10, 15]-=-, coupling of molecular dynamics with linear elasticity [13], combining kinetic and hydrodynamic models for complex fluids [24], and coupling kinetic Monte Carlo and continuum models for epitaxial gro... |

96 |
Fast reaction, slow diffusion and curve shortening
- Rubinstein, Sternberg, et al.
- 1989
(Show Context)
Citation Context ... diffusion in (2.26), the front which is a circle should shrink with the radial velocity Vr = κ, where κ is its mean curvature. This phenomenon was also studied by Rubinstein, Sternberg, and Keller =-=[30]-=-. For more information about the motion of fronts in reaction-diffusion equations, the reader is referred to these papers and the references cited therein. However, as we show in Figure 12(a), the fro... |

77 |
A study of numerical methods for hyperbolic conservation laws with stiff source terms,
- LeVeque, Yee
- 1990
(Show Context)
Citation Context ...ding remarks. 2. A reaction-diffusion-convection model problem. A stiff reaction-diffusion-convection equation may contain numerical difficulties due to rapid microscale transition at interior fronts =-=[22, 16]-=-. One example of such a phenomenon is the application of the phase field method. 2.1. Numerical difficulties in the 1-D model. Consider first the simple model problem ∂u ∂t + ∂u ∂x = ψ(u) + ∂2u ∂x2 ... |

75 | Adaptive Mesh Refinement using Wave-Propagation Algorithms for Hyperbolic Systems
- Berger, LeVeque
- 1998
(Show Context)
Citation Context ...om step (ii) using the newly evolved front points and their corresponding interface. Our approach is closely related to the general philosophy of the successful adaptive mesh refinement (AMR) methods =-=[5, 4]-=-. As in AMR methods, the end result is to work with a mesh which is locally refined at the interface and coarsened away from the interface. However, there are some important differences on how the mes... |

60 |
Theoretical and numerical structure for reacting shock waves,
- Colella, Majda, et al.
- 1986
(Show Context)
Citation Context ...lue of u over a large mesh grid, will cause the source terms to be activated over this entire grid in a nonphysical manner. A similar phenomenon has also been observed by Colella, Majda, and Roytburd =-=[8]-=- on a model combustion problem, which will also be studied in section 3 of the present paper. There are a number of existing numerical methods for this problem; see [6, 23, 32, 2, 19] for similar resu... |

52 |
A qualitative model for dynamic combustion
- Majda
- 1981
(Show Context)
Citation Context ... will be discussed in next section. The reason for considering this model is that it is well understood analytically and that it has been used extensively for studying different numerical methods. In =-=[25]-=-, Majda derived a 2 × 2 system, which couples Burgers’ equation to a chemical kinetics equation: Ut + ( 1 2 U2 − q0Z ) x = 0,(3.1) Zx = K(U)Z,(3.2) where U is a variable with some features of pressure... |

50 |
Stable and entropy satisfying approximations for transonic flow calculations
- Engquist, Osher
- 1980
(Show Context)
Citation Context ...xK(U n i ) +K(U n i−1) 2 ) (3.5) and initial condition Zn+1i = 1 for i large enough. In the second step of the scheme we compute Un+1i from {Uni }, {Zn+1i } by applying the Engquist–Osher (EO) scheme =-=[14]-=- to (3.1). This formula for Un+1i is given by Un+1i = U n i − Δt Δx [ (f+(U n i )− f+(Uni−1)) + (f−(Uni+1)− f−(Uni )) ] + q0 Δt Δx (Zn+1i − Zn+1i−1 ), (3.6) where f+(U) = { 0 if U ≤ 0, U2/2 if U ≥ 0, ... |

46 | Heterogeneous multiscale methods for stiff ordinary differential equations
- Engquist, Tsai
(Show Context)
Citation Context ...he missing information from an explicitly given microscale model. Several encouraging results to different classes of problems, including homogenization [1, 12], stiff ordinary differential equations =-=[10, 15]-=-, coupling of molecular dynamics with linear elasticity [13], combining kinetic and hydrodynamic models for complex fluids [24], and coupling kinetic Monte Carlo and continuum models for epitaxial gro... |

39 | A critical analysis of Rayleigh–Taylor growth rates,
- Glimm
- 2001
(Show Context)
Citation Context ...oblem, and it is essential to describe them accurately. There are several computational methods for numerical interface tracking. Examples include the level set method [27, 28], front tracking method =-=[18, 34]-=-, phase field method, and segment projection method [33]. The standard situation is the one in which the evolution of the interface can be directly determined ∗Received by the editors February 20, 200... |

33 |
Heterogeneous Multiscale Method: A General Methodology for Multiscale Modeling
- Weinan, Engquist, et al.
- 2003
(Show Context)
Citation Context .... Several encouraging results to different classes of problems, including homogenization [1, 12], stiff ordinary differential equations [10, 15], coupling of molecular dynamics with linear elasticity =-=[13]-=-, combining kinetic and hydrodynamic models for complex fluids [24], and coupling kinetic Monte Carlo and continuum models for epitaxial growth [31] have demonstrated the potential of HMM. Interfaces ... |

27 |
A modified fractional step method for the accurate approximation of detonation waves,
- Helzel, LeVeque, et al.
- 2000
(Show Context)
Citation Context ...+ p u(E + p) ρuZ ⎞ ⎟⎟⎠, S(U) = ⎛ ⎜⎜⎝ 0 0 0 −K(T )ρZ ⎞ ⎟⎟⎠.(3.14) Figure 16 shows numerical results at time t = 0.4 for the reaction rate constant K0 = 1000 with the initial data (same as Example 1 in =-=[19]-=-){ ρl = 1.4, ul = 0, pl = 1, Zl = 0 if x ≤ 0, ρr = 0.887565, ur = −0.577350, pr = 0.191709, Zr = 1 if x > 0. (3.15) The other parameters are set to γ = 1.4, q0 = 1, andR = 1. In this case, the fractio... |

25 |
W.: Finite difference heterogeneous multi-scale method for homogenization problems
- Abdulle, E
- 2003
(Show Context)
Citation Context ...icient, and stable strategy for supplementing the missing information from an explicitly given microscale model. Several encouraging results to different classes of problems, including homogenization =-=[1, 12]-=-, stiff ordinary differential equations [10, 15], coupling of molecular dynamics with linear elasticity [13], combining kinetic and hydrodynamic models for complex fluids [24], and coupling kinetic Mo... |

25 | Two-dimensional front tracking based on high resolution wave propagation methods
- LeVeque, Shyue
- 1996
(Show Context)
Citation Context ...2. We take Δx = 0.01 and CFL number 0.5. There are a number of existing numerical methods for this problem. One could use AMR or front tracking schemes; see, e.g., Bourlioux [6] and LeVeque and Shyue =-=[23]-=-. Sjögreen and Engquist [32] introduced a projection step that eliminates intermediate states which are not in equilibrium. A random projection method was developed by Bao and Jin [2]. The ignition t... |

23 |
The piecewise-parabolic method (PPM) for gas-dynamical simulations
- Colella, Woodward
- 1984
(Show Context)
Citation Context ...n rate K(T ) = { K0 if T ≥ Tign, 0 if T < Tign (3.4) with Tign, the ignition temperature. Colella, Majda, and Roytburd [8] apply the Godunov method and a high resolution extension of Godunov’s method =-=[9]-=- to this problem. The techniques of splitting and solving the resulting ODEs exactly handle the source term well, and the solutions are stable. However, the numerical results are qualitatively wrong. ... |

15 |
The heterogeneous multi-scale method for homogenization problems
- Engquist, E
- 2005
(Show Context)
Citation Context ...icient, and stable strategy for supplementing the missing information from an explicitly given microscale model. Several encouraging results to different classes of problems, including homogenization =-=[1, 12]-=-, stiff ordinary differential equations [10, 15], coupling of molecular dynamics with linear elasticity [13], combining kinetic and hydrodynamic models for complex fluids [24], and coupling kinetic Mo... |

13 | The Random Projection Method for Hyperbolic Conservation Laws with Stiff Reaction Terms
- Bao, Jin
- 2000
(Show Context)
Citation Context ...by Colella, Majda, and Roytburd [8] on a model combustion problem, which will also be studied in section 3 of the present paper. There are a number of existing numerical methods for this problem; see =-=[6, 23, 32, 2, 19]-=- for similar results and recent methods. We now use numerical results to show the potential of the incorrect propagation velocities of discontinuities. Consider the following initial data for (2.1): u... |

13 | Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth
- Schulze, Smereka, et al.
(Show Context)
Citation Context ...upling of molecular dynamics with linear elasticity [13], combining kinetic and hydrodynamic models for complex fluids [24], and coupling kinetic Monte Carlo and continuum models for epitaxial growth =-=[31]-=- have demonstrated the potential of HMM. Interfaces or internal boundaries are present in many different applications, such as evolution of shocks, solidification, melting, etching, epitaxial growth o... |

12 |
The segment projection method for interface tracking
- Tornberg, Engquist
(Show Context)
Citation Context ...e are several computational methods for numerical interface tracking. Examples include the level set method [27, 28], front tracking method [18, 34], phase field method, and segment projection method =-=[33]-=-. The standard situation is the one in which the evolution of the interface can be directly determined ∗Received by the editors February 20, 2005; accepted for publication (in revised form) March 15, ... |

10 |
Numerical Approximation to Hyperbolic Conservation Laws with Stiff Terms, UCLA
- Engquist, Sjogren
- 1989
(Show Context)
Citation Context ...by Colella, Majda, and Roytburd [8] on a model combustion problem, which will also be studied in section 3 of the present paper. There are a number of existing numerical methods for this problem; see =-=[6, 23, 32, 2, 19]-=- for similar results and recent methods. We now use numerical results to show the potential of the incorrect propagation velocities of discontinuities. Consider the following initial data for (2.1): u... |

6 |
The random projection method for stiff multispecies detonation capturing,
- Bao, Jin
- 2002
(Show Context)
Citation Context ...grid size Δx = 0.01, microscale grid size δx = 0.01/50 = 2× 10−4, and a fixed CFL number 0.5. 3.2. 1-D multispecies reactive Euler equations. Next we discuss 1-D multispecies reactive Euler equations =-=[17, 3]-=- ∂U ∂t + ∂F (U) ∂x = S(U),(3.8) where U = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ ρ ρu E ρZ1 ρZ2 . . . ρZN ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠ , F (U) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ ρu ρu2 + p u(E + p) ρuZ1 ρuZ2 . . . ρuZN ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠ , S(U) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0 0 0 ψ1 ψ2 .... |

6 |
Numerical Study of Unstable Detonations
- Bourlioux
- 1991
(Show Context)
Citation Context ...by Colella, Majda, and Roytburd [8] on a model combustion problem, which will also be studied in section 3 of the present paper. There are a number of existing numerical methods for this problem; see =-=[6, 23, 32, 2, 19]-=- for similar results and recent methods. We now use numerical results to show the potential of the incorrect propagation velocities of discontinuities. Consider the following initial data for (2.1): u... |

6 |
The heterogeneous multiscale method for interface dynamics
- Cheng, E
- 2003
(Show Context)
Citation Context ...s that we can choose other conventional methods for interface dynamics, such as the level set method [28] or the segment projection method [33] instead of the front tracking method here. In the paper =-=[7]-=-, Cheng and E use the level set method as the macroscopic solver. There the interface is described by the level set function Φ in a globally defined velocity field v, and a PDE for evolution of the in... |

4 |
Stochastic models of polymeric fluids at small Deborah number
- Li, Vanden-Eijnden, et al.
- 2004
(Show Context)
Citation Context ...luding homogenization [1, 12], stiff ordinary differential equations [10, 15], coupling of molecular dynamics with linear elasticity [13], combining kinetic and hydrodynamic models for complex fluids =-=[24]-=-, and coupling kinetic Monte Carlo and continuum models for epitaxial growth [31] have demonstrated the potential of HMM. Interfaces or internal boundaries are present in many different applications, ... |

1 | Front motion in multi-dimensional viscous conservation laws with stiff source terms driven by mean curvature and front thickness, Quart
- Fan, Jin
(Show Context)
Citation Context ...ding remarks. 2. A reaction-diffusion-convection model problem. A stiff reaction-diffusion-convection equation may contain numerical difficulties due to rapid microscale transition at interior fronts =-=[22, 16]-=-. One example of such a phenomenon is the application of the phase field method. 2.1. Numerical difficulties in the 1-D model. Consider first the simple model problem ∂u ∂t + ∂u ∂x = ψ(u) + ∂2u ∂x2 ... |

1 |
Hyperbolic Conservation Laws with Stiff Source Terms
- Klingenstein
- 1995
(Show Context)
Citation Context ...which all the chemical energy is released, followed by a shock wave traveling more slowly. The reaction wave always travels at the velocity of one grid per time step, which is totally nonphysical. In =-=[20]-=-, Klingenstein showed an error analysis of the shock locations and constructed an adaptation of the step size so that the error of the shock location remains sufficiently small. Let us describe the fr... |