A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.

doi:10.1016/j.jcp.2013.08.021

Cited by 115 publications

(107 citation statements)

References 30 publications

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“…The anisotropic diffusion equation has non-positive eigenvalues; i.e., none of the Fourier modes grow with time. Following Meyer et al 2014 (see their section 2), we can write the diffusion equation in a semi-discrete ODE form…”

Section: Super-time-stepping: Rkc Rkl Aagmentioning

confidence: 99%

“…Following Meyer et al (2014), in Appendix A we illustrate the second order temporal accuracy of the RKL-2 scheme along with explicit formulae for an s = 3 stage scheme. The key advantage of RKL schemes over RKC is that they are more robust because LPs, unlike CPs, are always smaller than unity in magnitude if their argument lies in (-1,1).…”

Section: Super-time-stepping: Rkc Rkl Aagmentioning

confidence: 99%

“…This Appendix is closely based on Meyer et al (2012Meyer et al ( , 2014 and is included here for completeness. Second order accuracy for the RKL-2 scheme can be achieved by matching the first three terms in Eqs.…”

Section: Appendix A: Illustration Of Rkl-2 Schemementioning

confidence: 99%

“…A variant of this method was popularized by Alexiades et al 1996 (AAG-STS). More recently, super-time-stepping based on Legendre polynomials, known as Runge-Kutta Legendre STS (RKL-STS), was proposed by Meyer et al (2012Meyer et al ( , 2014.…”

Section: Introductionmentioning

confidence: 99%

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Scalable explicit implementation of anisotropic diffusion with Runge–Kutta–Legendre super-time stepping

Vaidya

Prasad

Mignone

et al. 2017

Monthly Notices of the Royal Astronomical Society

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An important ingredient in numerical modelling of high temperature magnetised astrophysical plasmas is the anisotropic transport of heat along magnetic field lines from higher to lower temperatures.Magnetohydrodynamics (MHD) typically involves solving the hyperbolic set of conservation equations along with the induction equation. Incorporating anisotropic thermal conduction requires to also treat parabolic terms arising from the diffusion operator. An explicit treatment of parabolic terms will considerably reduce the simulation time step due to its dependence on the square of the grid resolution (∆x) for stability. Although an implicit scheme relaxes the constraint on stability, it is difficult to distribute efficiently on a parallel architecture. Treating parabolic terms with accelerated super-time stepping (STS) methods has been discussed in literature but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work we highlight a second order (in time) Runge Kutta Legendre (RKL) scheme (first described by Meyer et al. 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first order super time stepping schemes with standard tests and astrophysical applications. We also show that explicit conduction is particularly robust in handling saturated thermal conduction. Parallel scaling of explicit conduction using RKL scheme is demonstrated up to more than 10 4 processors.

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Section: Super-time-stepping: Rkc Rkl Aagmentioning

confidence: 99%

Section: Super-time-stepping: Rkc Rkl Aagmentioning

confidence: 99%

Section: Appendix A: Illustration Of Rkl-2 Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scalable explicit implementation of anisotropic diffusion with Runge–Kutta–Legendre super-time stepping

Vaidya

Prasad

Mignone

et al. 2017

Monthly Notices of the Royal Astronomical Society

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“…FastRad3D has a variety of thermal conduction solvers available, including conjugate-gradient, linerelaxation, and alternating-direction-implicit schemes, 67 as well as a selection of explicit solvers, including RKL2. 68 For the targets simulated in this study, an implicit fractional-step algorithm 69 was used to solve Eq. (7).…”

Section: The Numerical Modelmentioning

confidence: 99%

Numerical simulations of the ablative Rayleigh-Taylor instability in planar inertial-confinement-fusion targets using the FastRad3D code

et al. 2016

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The ablative Rayleigh-Taylor (RT) instability is a central issue in the performance of laser-accelerated inertial-confinement-fusion targets. Historically, the accurate numerical simulation of this instability has been a challenging task for many radiation hydrodynamics codes, particularly when it comes to capturing the ablatively stabilized region of the linear dispersion spectrum and modeling ab initio perturbations. Here, we present recent results from two-dimensional numerical simulations of the ablative RT instability in planar laser-ablated foils that were performed using the Eulerian code FastRad3D. Our study considers polystyrene, (cryogenic) deuterium-tritium, and beryllium target materials, quarter- and third-micron laser light, and low and high laser intensities. An initial single-mode surface perturbation is modeled in our simulations as a small modulation to the target mass density and the ablative RT growth-rate is calculated from the time history of areal-mass variations once the target reaches a steady-state acceleration. By performing a sequence of such simulations with different perturbation wavelengths, we generate a discrete dispersion spectrum for each of our examples and find that in all cases the linear RT growth-rate γ is well described by an expression of the form γ=α [kg/(1+ϵ kLm)]1/2−βkVa, where k is the perturbation wavenumber, g is the acceleration of the target, Lm is the minimum density scale-length, Va is the ablation velocity, and ϵ is either one or zero. The dimensionless coefficients α and β in the above formula depend on the particular target and laser parameters and are determined from two-dimensional simulation results through the use of a nonlinear curve-fitting procedure. While our findings are generally consistent with those of Betti et al. (Phys. Plasmas 5, 1446 (1998)), the ablative RT growth-rates predicted in this investigation are somewhat smaller than the values previously reported for the same target and laser parameters. It is speculated that differences in the equation-of-state and opacity models are largely responsible for the discrepancy. Resolution of this issue awaits the development of better experimental diagnostics capable of measuring small-wavelength (5–20 μm) perturbation growth due to the ablative RT instability in the linear regime.

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Average reduced model to simulate solutions for heat and mass transfer through porous material

Berger

Abdykarim

2021

Heat Trans

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The design of numerical tools to model the behavior of building materials is a challenging task. The crucial point is to save computational costs and maintain the high accuracy of predictions. There are two main limitations on the time scale choice, which places an obstacle to solving the above issues. The first one is the numerical restriction. A number of research studies are dedicated to overcome this limitation and it is shown that it can be relaxed with innovative numerical schemes. The second one is the physical restriction. It is imposed by the properties of a material, the phenomena itself, and the corresponding boundary conditions. This study is focused on the study of a methodology that enables to overcome the physical restriction on the time grid; a so-called average reduced model is suggested. It is based on smoothing the timedependent boundary conditions. Besides this, the approximate solution is decomposed into average and fluctuating components. The primer is obtained by integrating the equations over time, whereas the latter is a user-defined EM. The methodology is investigated for both heat diffusion and coupled heat and mass transfer. It is demonstrated that the signal core of the boundary conditions is preserved and the physical restriction can be relaxed. The model proved to be reliable, accurate, and efficient also in comparison with

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A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

Cited by 115 publications

References 30 publications

Scalable explicit implementation of anisotropic diffusion with Runge–Kutta–Legendre super-time stepping

Scalable explicit implementation of anisotropic diffusion with Runge–Kutta–Legendre super-time stepping

Numerical simulations of the ablative Rayleigh-Taylor instability in planar inertial-confinement-fusion targets using the FastRad3D code

Average reduced model to simulate solutions for heat and mass transfer through porous material

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