Scalar Conservation Laws

LeVeque, Randall J.

doi:10.1007/978-3-0348-8629-1_3

Cited by 44 publications

(31 citation statements)

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“…A crucial check that the CSS solutions of the ODEs are indeed the unique critical solutions involves a comparison with the solutions of the full Einstein/fluid equations. Although relativistic fluid equations are known to be difficult to solve (particularly relative to 'fundamental-field' equations, such as the EMKG system), modern high-resolution shockcapturing schemes [32][33][34][35] allow one to calculate flows with shocks of almost arbitrary strength. However, perhaps the greatest challenge in finding the perfect-fluid critical solutions (especially as → 2) is the accurate treatment of flows with very large Lorentz factors.…”

Section: Simulation Of Critical Solutionsmentioning

confidence: 99%

Critical phenomena in perfect fluids

Neilsen

Choptuik

2000

Class. Quantum Grav.

117

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We investigate the gravitational collapse of a spherically symmetric, perfect fluid with equation of state P = (Γ − 1)ρ. We restrict attention to the ultrarelativistic ("kinetic-energy-dominated", "scale-free") limit where black hole formation is anticipated to turn on at infinitesimal black hole mass (Type II behavior). Critical solutions (those which sit at the threshold of black hole formation in parametrized families of collapse) are found by solving the system of ODEs which result from a self-similar ansatz, and by solving the full Einstein/fluid PDEs in spherical symmetry. These latter PDE solutions ("simulations") extend the pioneering work of Evans and Coleman (Γ = 4/3) and verify that the continuously self-similar solutions previously found by Maison and Hara et al for 1.05 ≤ Γ 1.89 are (locally) unique critical solutions. In addition, we find strong evidence that globally regular critical solutions do exist for 1.89 Γ ≤ 2, that the sonic point for Γ dn ≃ 1.8896244 is a degenerate node, and that the sonic points for Γ > Γ dn are nodal points, rather than focal points as previously reported. We also find a critical solution for Γ = 2, and present evidence that it is continuously self-similar and Type II. Mass-scaling exponents for all of the critical solutions are calculated by evolving near-critical initial data, with results which confirm and extend previous calculations based on linear perturbation theory. Finally, we comment on critical solutions generated with an ideal-gas equation of state.

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Section: Simulation Of Critical Solutionsmentioning

confidence: 99%

Critical phenomena in perfect fluids

Neilsen

Choptuik

2000

Class. Quantum Grav.

117

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“…A semidiscrete implementation of the high-resolution finite volume method (HR-FVM) , reduced the partial differential equations to a system of ordinary differential equations (ODEs). The ODE was solved numerically with a medium-order Runge–Kutta method in Matlab R2022b (with the application of the ode23 function).…”

Section: Methodsmentioning

confidence: 99%

Dynamic Modeling and Optimal Design Space Determination of Pharmaceutical Crystallization Processes: Realizing the Synergy between Off-the-Shelf Laboratory and Industrial Scale Data

Orosz,

Szilágyi,

Spaits

et al. 2024

Ind. Eng. Chem. Res.

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Well-designed and operated pharmaceutical crystallization can enhance the key features of the active pharmaceutical ingredients, which can be taken to the next level by a model-based design. Model development not only requires computer implementation, kinetic identification, and advanced programming skills but also a suitable number of information-rich experiments. Numerous products have been crystallized for a long time, with many related laboratory-scale experiments and plant-scale manufacturing data accumulated from past developments and productions. The question arises: can these historical data be utilized to build process models, which can be used subsequently to optimize processes? We aim to demonstrate in this study that this approach can be feasible. To illustrate this, we used the data of two commercial crystallization production campaigns, totaling 16 industrial crystallizations, and six laboratory experiments, four of which were formerly performed for different purposes. Our tailored PBM involves primary and secondary nucleation, crystal growth, and dissolution and can simultaneously reproduce laboratory-and plant-scale dynamics. Despite relying on nontargeted experiments and measuring/sampling strategy, the estimated parameters were accurate, with an average deviation between the nominal values and 95% confidence interval bonds of 16.1%. The model was employed to construct an optimal design space (DS) for a temperature cycling operation involving, as a constraint, 2, 3, and 4 cycles: the goal was to define a temperature domain with minimal batch time and heating/cooling energy demand that respects the constraints on the product PSD and heating/cooling rates. The problem was solved as a constrained robust optimization, where discrete temperature stamps and corresponding time stamps were optimized. The optimal operation halved the current batch time. The optimized temperature profile was validated on a laboratory scale for two particle size specifications. More expansive DS (1 vs 0.5 h random temperature variation allowed around the nominal temperature profile) was observed to translate to longer batch times (30 vs 25 h) and deeper temperature cycles of 40 vs 20 °C, reflecting a tradeoff between nominal performance and robustness. The optimal laboratory-scale operation was validated successfully by repeated experiments.

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“…The PBE was solved in discretized form using the high-resolution FVM (HRFVM). This is a generic partial differential equation solver that has been successfully applied for solving PBE-s for the past decades. , The resulting ordinary differential equation system was solved in MATLAB (MathWorks Inc., version R2019a). To improve the computational performance, the ode23 code was applied for the solution of the differential equations, which is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. , …”

Section: Model Equations and Kinetic Parameter Estimationmentioning

confidence: 99%

Population Balance Modeling of Diastereomeric Salt Resolution

Bosits

Orosz

Szalay

et al. 2023

Crystal Growth & Design

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Diastereomeric salt crystallization is a classical, widely applicable chiral resolution technique, which enables the separation of the enantiomers of both racemate and conglomerate-forming compounds. A resolution method for racemic pregabalin with l-tartaric acid was developed to obtain pure (S)-pregabalin l-tartrate monohydrate crystals with the yield ranging from 43 to 50%. A series of designed resolution experiments were executed at different cooling rates and temperature end points to estimate the crystallization kinetics using population balance modeling. Inline ATR-FTIR measurements of these experiments were used to calculate concentrations in the crystallization phase and to collect solid–liquid equilibrium data by the solubility trace method after product sampling. Since diastereomeric salts are dissociable ionic compounds, solubility product expressions were used in the model to express the relative supersaturation as the driving force of crystallization. As a result, a population balance model with secondary nucleation, growth, and agglomeration mechanisms was identified, and the simulated supersaturation profiles and product size distributions well reproduced the measured data.

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Scalar Conservation Laws

Cited by 44 publications

References 0 publications

Critical phenomena in perfect fluids

Critical phenomena in perfect fluids

Dynamic Modeling and Optimal Design Space Determination of Pharmaceutical Crystallization Processes: Realizing the Synergy between Off-the-Shelf Laboratory and Industrial Scale Data

Population Balance Modeling of Diastereomeric Salt Resolution

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