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

Helzel, Christiane; LeVeque, Randall J.; Warnecke, Gerald

doi:10.1137/s1064827599357814

Cited by 59 publications

(63 citation statements)

References 20 publications

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“…It is also related to the modified split-scheme proposed in [25] in the context of reactive Euler equations. At last, the split-schemes investigated in [1] are obtained just by inserting a projection stage onto piecewise constant functions similar to (19) between the free transport step and the relaxation process, that is, right between (16) and (18).…”

Section: Lemmamentioning

confidence: 99%

Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

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Section: Lemmamentioning

confidence: 99%

Localization effects and measure source terms in numerical schemes for balance laws

Gosse

2001

Math. Comp.

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“…To further understand equation (17) let us consider a conservation law in which q is comprised of a scalar quantity ρ( x, t) (e.g., density) and a vector quantity µ( x, t) (e.g., momentum):…”

Section: Conservation Laws On Curved Manifoldsmentioning

confidence: 99%

“…The method is numerically conservative, second order accurate in smooth regions, non-dispersive in regions of large gradients, and shockcapturing. This method has been successfully applied in the past to several applications areas, including gas dynamics [21], acoustics [9,10], elasticity and plasticity [8,22,26,27], combustion and detonation waves [16,17], relativistic hydrodynamics [1], and numerical relativity [3,18].…”

Section: Introductionmentioning

confidence: 99%

A wave propagation algorithm for hyperbolic systems on curved manifolds

Rossmanith

Bale

LeVeque

2004

Journal of Computational Physics

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An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327-353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and high-resolution shockcapturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary one-dimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package clawpack and is freely available on the web.

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“…In some cases this approach works well (see Langseth, Tveito, and Winther [50] for a favorable error estimate for operator splitting), but may also give unacceptably large errors for reasonable choices of ∆t. This is especially evident if k is large, so that the ordinary differential equation is stiff (see LeVeque and Yee [52]), an exception being stiff source terms of dissipative nature as in Tang [68], see also Helzel, LeVeque, and Warnecke [28] for a modified splitting method for approximating detonation waves.…”

Section: Introductionmentioning

confidence: 99%

Front Tracking for Scalar Balance Equations

Karlsen

Risebro

Towers

2004

J. Hyper. Differential Equations

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Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL-condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady-state solutions (or achieving them in the long time limit) with good accuracy.

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

Cited by 59 publications

References 20 publications

Localization effects and measure source terms in numerical schemes for balance laws

Localization effects and measure source terms in numerical schemes for balance laws

A wave propagation algorithm for hyperbolic systems on curved manifolds

Front Tracking for Scalar Balance Equations

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