Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Kaps, Peter; Rentrop, P.

doi:10.1007/bf01396495

Cited by 317 publications

(168 citation statements)

References 14 publications

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“…Although the integration of the source term S n+1/2 i, j,k in Eq. (30) is of second order in time, higher order methods, such as Rosenbrock (Kaps & Rentrop 1979), can be chosen.…”

Section: Integration Methodsmentioning

confidence: 99%

Simulation of the growth of the 3D Rayleigh-Taylor instability in supernova remnants using an expanding reference frame

Fraschetti

Teyssier

Ballet³

et al. 2010

A&A

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Context. The Rayleigh-Taylor instabilities that are generated by the deceleration of a supernova remnant during the ejecta-dominated phase are known to produce finger-like structures in the matter distribution that modify the geometry of the remnant. The morphology of supernova remnants is also expected to be modified when efficient particle acceleration occurs at their shocks. Aims. The impact of the Rayleigh-Taylor instabilities from the ejecta-dominated to the Sedov-Taylor phase is investigated over one octant of the supernova remnant. We also study the effect of efficient particle acceleration at the forward shock on the growth of the Rayleigh-Taylor instabilities. Methods. We modified the Adaptive Mesh Refinement code RAMSES to study with hydrodynamic numerical simulations the evolution of supernova remnants in the framework of an expanding reference frame. The adiabatic index of a relativistic gas between the forward shock and the contact discontinuity mimics the presence of accelerated particles. Results. The great advantage of the super-comoving coordinate system adopted here is that it minimizes numerical diffusion at the contact discontinuity, since it is stationary with respect to the grid. We propose an accurate expression for the growth of the RayleighTaylor structures that smoothly connects the early growth to the asymptotic self-similar behaviour. Conclusions. The development of the Rayleigh-Taylor structures is affected, although not drastically, if the blast wave is dominated by cosmic rays. The amount of ejecta that reaches the shocked interstellar medium is smaller in this case. If acceleration were to occur at both shocks, the extent of the Rayleigh-Taylor structures would be similar but the reverse shock would be strongly perturbed.

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“…Although the integration of the source term S n+1/2 i, j,k in Eq. (30) is of second order in time, higher order methods, such as Rosenbrock (Kaps & Rentrop 1979), can be chosen.…”

Section: Integration Methodsmentioning

confidence: 99%

Simulation of the growth of the 3D Rayleigh-Taylor instability in supernova remnants using an expanding reference frame

Fraschetti

Teyssier

Ballet³

et al. 2010

A&A

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“…This means that implicit or semiimplicit methods are necessary to efficiently follow their evolution. To address this problem, we arranged the molar fractions of the 6 species (excluding e − ) into a vector and solved the resulting system of equations using a 4 th order Kaps-Rentrop, or Rosenbrock method (Kaps & Rentrop 1979) as described in more detail in Gray & Scannapieco (2010).…”

Section: Coolingmentioning

confidence: 99%

Identification of a Fundamental Transition in a Turbulently Supported Interstellar Medium

Scannapieco

Gray

Pan

2012

ApJ

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The interstellar medium in star-forming galaxies is a multiphase gas in which turbulent support is at least as important as thermal pressure. Sustaining this configuration requires continuous radiative cooling, such that the overall average cooling rate matches the decay rate of turbulent energy into the medium. Here we carry out a set of numerical simulations of a stratified, turbulently stirred, radiatively cooled medium, which uncover a fundamental transition at a critical one-dimensional turbulent velocity of ≈ 35 km/s. At turbulent velocities below ≈ 35 km/s, corresponding to temperatures below 10 5.5 K, the medium is stable, as the time for gas to cool is roughly constant as a function of temperature. On the other hand, at turbulent velocities above the critical value, the gas is shocked into an unstable regime in which the cooling time increases strongly with temperature, meaning that a substantial fraction of the interstellar medium is unable to cool on a turbulent dissipation timescale. This naturally leads to runaway heating and ejection of gas from any stratified medium with a one-dimensional turbulent velocity above ≈ 35 km/s, a result that has implications for galaxy evolution at all redshifts.

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“…Consequently, explicit source term solvers can be either completely unacceptable or their use may lead to significantly shorter time steps and much more inferior solution accuracy. Thus, we chose to use the 4th-order accurate implicit Rosenbrock method, in particular its implementation by Kaps and Rentrop [25,26]. This method for moderate accuracies (ǫ 10 −4 − 10 −5 in relative error) and modest-sized systems, such as eqs.…”

Section: Methodsmentioning

confidence: 99%

Computation of fluid flows in non-inertial contracting, expanding, and rotating reference frames

Poludnenko

Khokhlov

2007

Journal of Computational Physics

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We present the method for computation of fluid flows that are characterized by the large degree of expansion/contraction and in which the fluid velocity is dominated by the bulk component associated with the expansion/contraction and/or rotation of the flow. We consider the formulation of Euler equations of fluid dynamics in a homologously expanding/contracting and/or rotating reference frame. The frame motion is adjusted to minimize local fluid velocities. Such approach allows to accommodate very efficiently large degrees of change in the flow extent. Moreover, by excluding the contribution of the bulk flow to the total energy the method eliminates the high Mach number problem in the flows of interest. An important practical advantage of the method is that it can be easily implemented with virtually any implicit or explicit Eulerian hydrodynamic scheme and adaptive mesh refinement (AMR) strategy.We also consider in detail equation invariance and existence of conservative formulation of equations for special classes of expanding/contracting reference frames. Special emphasis is placed on extensive numerical testing of the method for a variety of reference frame motions, which are representative of the realistic applications of the method. We study accuracy, conservativity, and convergence properties of the method both in problems which are not its optimal applications as well as in systems in which the use of this method is maximally beneficial. Such detailed investigation of the numerical solution behavior is used to define the requirements that need to be considered in devising problem-specific fluid motion feedback mechanisms.

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Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Cited by 317 publications

References 14 publications

Simulation of the growth of the 3D Rayleigh-Taylor instability in supernova remnants using an expanding reference frame

Simulation of the growth of the 3D Rayleigh-Taylor instability in supernova remnants using an expanding reference frame

Identification of a Fundamental Transition in a Turbulently Supported Interstellar Medium

Computation of fluid flows in non-inertial contracting, expanding, and rotating reference frames

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