First-order Discretization Scheme Research Articles

Error estimates of a sphere-constraint-preserving numerical scheme for Ericksen-Leslie system with variable density

In this paper, a new first-order, sphere-constraint-preserving numerical scheme is constructed for the Ericksen-Leslie equations with variable density. Firstly, by denoting the orientation field vector $ \pmb{d} $ in the polar coordinate, we rewrite the Ericksen-Leslie system into an equivalent system such that the sphere constraint $ \lvert \pmb{d}\rvert = 1 $ can be still preserved at the discrete level. Secondly, we propose a first-order discretization scheme in time for the equivalent new system in which the numerical velocity and modified pressure are determined by a generalized Stokes problem at each time step. Then, the unconditional energy stability and the rigorous error estimates in time are both derived. Finally, some numerical simulations in two dimensions are provided to demonstrate the stability of energy and accuracy of the presented scheme.

Continuous Membrane-Assisted Crystallization To Increase the Attainable Product Quality of Pharmaceuticals and Design Space for Operation

Continuous manufacturing is an important paradigm shift in pharmaceutical industries and has renewed the interest in continuous crystallization. The combination of crystallization and membranes is a promising hybrid technology for separation and purification of pharmaceuticals. The impact of membranes as an extension to conventional continuous crystallization processes on attainable product quality and design space is investigated systematically using model-based optimization. The proposed model is based on a full population balance such that all relevant crystallization phenomena can be included and is solved using a first-order discretization scheme with a hybrid grid. A case study involving continuous crystallization of paracetamol using a series of mixed suspension, mixed product removal (MSMPR) crystallizers is presented to illustrate the approach. The results show that the attainable size and design space can be enlarged significantly by extending conventional crystallization with membranes. In part...

A Comparative Study on Effect of Different Parameters of CFD Modeling for Gas–Solid Fluidized Bed

Simulated results from a commercial computational fluid dynamics (CFD) software package, Fluent 13.0, are compared with the experimental results obtained from fluidization experiments using red mud of size 77 µm. Unsteady behavior of fluidized bed is simulated by using the Eulerian–Eulerian model coupled with kinetic theory of granular flow. Two equation standard k–ϵ model has been used to describe the turbulent quantities. Momentum exchange coefficients are calculated using the Syamlal–O'Brien, Gidaspow, and Wen–Yu drag functions. The solid phase kinetic energy fluctuation is characterized by varying the restitution coefficient values from 0.9 to 0.99. The best model predictions are achieved, using Gidaspow drag model with restitution coefficient of 0.9, convergence criterions as 0.001, free-slip boundary condition of specularity coefficient and first-order discretization scheme. The modeling predictions are found to agree reasonably well with the experimentally observed values for pressure drop, bed expansion ratio, and qualitative gas–solid flow patterns under different operating conditions thereby giving sufficient information hydrodynamic behaviors of fluidized bed. Thus CFD simulation will be able to provide a strong base for the design and operations of bench scale as well as industrial catalytic fluidized bed reactors.

The influence of numerical diffusion on the solution of the radiative transfer equations

We investigate the explicit numerical solution strategies of multi-dimensional radiative transfer equations which are commonly used, e.g., to determine the radiation emerging from astrophysical objects surrounded by absorbing and scattering matter. For explicit grid solvers, we identify numerical diffusion as a severe source of error in first-order discretization schemes, underestimated in former work about radiative transfer. Using the simple example of a beam propagating through vacuum, we illustrate the influence of the diffusion on the solution and discuss various techniques to reduce it. In view of the large required storage for implicit solvers, we propose to use second-order explicit grid techniques to solve 3D radiative transfer problems.

Efficient Monte Carlo Simulation of Security Prices

This paper provides an asymptotically efficient algorithm for the allocation of computing resources to the problem of Monte Carlo integration of continuous-time security prices. The tradeoff between increasing the number of time intervals per unit of time and increasing the number of simulations, given a limited budget of computer time, is resolved for first-order discretization schemes (such as Euler) as well as second- and higher-order schemes (such as those of Milshtein or Talay).