Additive Semi-Implicit Runge–Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows

Zhong, Xiaolin

doi:10.1006/jcph.1996.0193

Cited by 188 publications

(176 citation statements)

References 17 publications

(21 reference statements)

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“…when H(·) is non zero). This will be improved in the future by the use of a real third order semi-implicit Runge-Kutta scheme [35]. Our experience, however, indicates that the influence of this loss of accuracy does not affect significantly the overall resolution of the flow structure (see section 5).…”

Section: The Time Discretizationmentioning

confidence: 87%

A high-resolution penalization method for large Mach number flows in the presence of obstacles

2009

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A penalization method is applied to model the interaction of large Mach number compressible flows with obstacles. A supplementary term is added to the compressible Navier-Stokes system, seeking to simulate the effect of the Brinkmanpenalization technique used in incompressible flow simulations including obstacles. We present a computational study comparing numerical results obtained with this method to theoretical results and to simulations with Fluent software. Our work indicates that this technique can be very promising in applications to complex flows.

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Section: The Time Discretizationmentioning

confidence: 87%

A high-resolution penalization method for large Mach number flows in the presence of obstacles

2009

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“…When studying the linear stability of semi-implicit methods, one must specify how the standard model problem Equation (11) is decomposed into explicit and implicit parts. Numerous choices of the splitting have appeared in the literature [Frank et al 1997;Ascher et al 1995;Pareschi and Russo 2001;Zhong 1996;Pareschi and Russo 2005]. The most general approach is to decompose the problem into explicit and implicit terms by…”

Section: Linear Stability Analysismentioning

confidence: 99%

“…where λ E and λ I are complex constants [Frank et al 1997;Pareschi and Russo 2001;Liotta et al 2000;Pareschi and Russo 2005;Zhong 1996]. Then additional constraints are made to define a stability region which depends only on a single complex number.…”

Section: Linear Stability Analysismentioning

confidence: 99%

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Minion

2007

CAMCoS

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High-order semi-implicit Picard integral deferred correction (SIPIDC) methods have previously been proposed for the time-integration of partial differential equations with two or more disparate time scales. The SIPIDC methods studied to date compute a high-order approximation by first computing a provisional solution with a first-order semi-implicit method and then using a similar semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. This study assesses the efficiency of SIPIDC methods that instead use standard semi-implicit methods with orders two through four to compute the provisional solution. Numerical results indicate that using a method with more than first-order accuracy in the computation of the provisional solution increases the efficiency of SIPIDC methods in some cases. First-order PIDC corrections can improve the efficiency of semi-implicit integration methods based on backward difference formulae (BDF) or Runge-Kutta methods while maintaining desirable stability properties. Finally, the phenomenon of order reduction, which may be encountered in the integration of stiff problems, can be partially alleviated by the use of BDF methods in the computation of the provisional solution.

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“…Besides, the viscous stress and heat flux terms in the boundary layers can cause the stiffness too. The source terms are stiff because the thermal-chemical non-equilibrium reactive processes possess a wide range of time scales and some of them are much smaller than that of hydrodynamic flow [1]. The simulation will be inefficient when the explicit methods rather than the implicit methods are used, because the time-step sizes dictated by the stability restraint in explicit methods are much smaller than those required by the CFL condition.…”

mentioning

confidence: 99%

“…The additive semi-implicit meth-ods resolve ODEs into the stiff part and non-stiff part, in which the stiff part is computed implicitly while the nonstiff part explicitly. Zhong [1] conducted a detailed study on additive semi-implicit methods and proposed a stiff accurate semi-implicit Runge-Kutta method up to the third order. The proposed methods had been applied in reactive flow computation.…”

mentioning

confidence: 99%

Additive Runge-Kutta methods for H2/O2/Ar detonation with a detailed elementary chemical reaction model

Ren

Ning

2013

Chin. Sci. Bull.

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We report here the additive Runge-Kutta methods for computing reactive Euler equations with a stiff source term, and in particular, their applications in gaseous detonation simulations. The source term in gaseous detonation is stiff due to the presence of wide range of time scales during thermal-chemical non-equilibrium reactive processes and some of these time scales are much smaller than that of hydrodynamic flow. The high order, L-stable, additive Runge-Kutta methods proposed in this paper resolved the stiff source term into the stiff part and non-stiff part, in which the stiff part was solved implicitly while the non-stiff part was handled explicitly. The proposed method was successfully applied to simulating the gaseous detonation in a stoichiometric H 2 /O 2 /Ar mixture based on a detailed elementary chemical reaction model comprised of 9 species and 19 elementary reactions. The results showed that the stiffly accurate additive Runge-Kutta methods can capture the discontinuity well, and describe the detonation complex wave configurations accurately such as the triple wave structure and cellular pattern. Reacting flows, specifically in gaseous combustion, have been a significant topic of active research for more than one hundred years. The strong coupling between hydrodynamic flow and chemical kinetics is complex and even today many phenomena are not very well understood yet. Gaseous detonation is a process of supersonic combustion in which a shock wave is propagated and supported by the energy release in a reaction zone behind it. It is the more powerful and destructive of the two general classes of combustion, the other one being deflagration. The primary difficulty in computing reacting flows is the source term stiffness inherent in the reactive Euler equations in temporal integrations. Besides, the viscous stress and heat flux terms in the boundary layers can cause the stiffness too. The source terms are stiff because the thermal-chemical non-equilibrium reactive processes possess a wide range of time scales and some of them are much smaller than that of hydrodynamic flow [1]. The simulation will be inefficient when the explicit methods rather than the implicit methods are used, because the time-step sizes dictated by the stability restraint in explicit methods are much smaller than those required by the CFL condition. Due to these limitations in explicit methods, the implicit methods are normally required to simulate gaseous detonation. The practical implicit methods for gaseous detonation simulation can be categorized into two classes, i.e. the time-splitting method and the additive semi-implicit method [2,3].The time-splitting methods [4][5][6][7][8], resolve the source term of reactive Euler equations into ( ) ( ),   t

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Additive Semi-Implicit Runge–Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows

Cited by 188 publications

References 17 publications

A high-resolution penalization method for large Mach number flows in the presence of obstacles

A high-resolution penalization method for large Mach number flows in the presence of obstacles

Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Additive Runge-Kutta methods for H2/O2/Ar detonation with a detailed elementary chemical reaction model

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