Operator splitting

NVT, and in die course of the simulation the volume V of the simulation box is allowed to vary, according to the new equations of motion. A usefid variant allows the simulation box to change shape as well as size [89, 90], It is also possible to extend the Liouville operator-splitting approach to generate algoritlnns for MD in these ensembles examples of explicit, reversible, integrators are given by Martyna et al [91],... 

A different long-time-step method was previously proposed by Garci a-Archilla, Sanz-Serna, and Skeel [8]. Their mollified impulse method, which is based on the concept of operator splitting and also reduces to the Verlet scheme for A = 0 and admits second-order error estimates independently of the frequencies of A, reads as follows when applied to (1) ... 

A quite successful integrator for (16), which is based on the idea of operator splitting, is the PICKABACK scheme of Nettesheim et al. [19]. With the shorthand notation... 

Wu, J. K. and Chen, H. X., Operator Splitting Method to Calculate Pressure of Ultra-Thin Gas Film of Magnetic Head/ Disk," Tribology, Vol. 23, No. 5, 2003, pp. 402-405 (in Chinese). 

To implement the vector potential in the Jacobi coordinate system (R,r,j), one proceeds as follows. The Jacobi kinetic energy operator splits into three parts [61] ... 

A reactive transport model in a more general sense treats a multicomponent system in which a number of equilibrium and perhaps kinetic reactions occur at the same time. This problem requires more specialized solution techniques, a variety of which have been proposed and implemented (e.g., Yeh and Tripathi, 1989 Steefel and MacQuarrie, 1996). Of the techniques, the operator splitting method is best known and most commonly used. 

Fig. 21.1. Operator splitting method for tracing a reactive transport simulation. To step forward from t = 0, the initial condition, to t = At, evaluate transport of the chemical components into and out of each nodal block, using the current distribution of mass. The net transport is the amount of component mass accumulating in a block over the time step. Once the updated component masses are known, evaluate the chemical equations to give a revised distribution of mass at each block. Repeat procedure, stepping to t = 2At, t = 3 At, and so on, until the simulation endpoint is reached.

To march forward in time according to the operator splitting method, we divide the procedure for advancing the time step into four substeps ... 

Once the advective and dispersive mass fluxes across each face are known, the rate of mass accumulation of component i, required in substep (iii) of the operator splitting procedure, is given by,... 

Vol. 9. Augmented Lagrangian and Operator-Splitting Method in Nonlinear Mechanics... 

FID detector installed in parallel with an olfactometer capable of delivering the effluent from the column to the human sniffer. The column is installed in either the FID port or the olfactometer port, depending upon the mode of operation. Split effluents are not recommended. 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

The exchange of concentration tendencies, rather than concentrations, is a special and perhaps unique feature of the GRG coupled system. The formulation of the tendency terms has to reflect the operator splitting and time stepping in the both the CTMs and the IPS as well as the relation between the tendency and the respective concentration value, and the cost (memory, time) of the exchange. 

The CTM use an operator-splitting approach in which chemistry, emission injection, diffusion and deposition are called in sequence and the update of the concentration follows directly within each subroutine. 

This operator split neglects relaxation effects and assumes that the tt rf pulse is short enough that all other terms in the Hamiltonian may be neglected during its application. In the following discussion we will consider the details of this basic sequence, but we must bear in mind that it may be just the joint in the chain of a more sophisticated sequence. The effect of the nonuniform magnetic field is given by... 

Issa, R. I. (1986) Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting, Journal of Computational Physics, Vol. 62(1), pp. 40-65. 

Issa, R.I. (1986), Solution of the implicitly discretized fluid flow equations by operator splitting, J. Computat. Phys., 62, 40-65. 

The explicit operator splitting techniques based on sequential one-dimensional updates may contain large splitting errors. To reduce the splitting errors... 

The fractional step, or time splitting, concept is more a generic operator splitting approach than a particular solution method [30, 211, 124, 92, 49]. It is essentially an approximate factorization of the methods applied to the different operators in an equation or a set of equations. The overall set of operators can be solved explicitly, implicitly or by a combination of both implicit and explicit discretization schemes. 

COSMIC Conservative Operator Splitting for Multidimensions with Inherent Consistancy... 

Strictly speaking, the conservative form refers to conservation equations without a source term. When one is solving moment-transport equations, operator splitting wherein the transport step is solved with S = 0 is often employed. 

Because the collision time t is arbitrary and can lead to stiff equations, it is best to solve Eq. (8.96) using operator splitting, i.e. solve first... 

The numerical solution of the moment-transport equation is again done by using operator splitting to update first for the spatial fluxes and then for the drag. The realizable scheme... 

The reader should note that in Eqs. (B.2)-(B.5) the spatial derivative appears on the right-hand side, and therefore it will be necessary to define a realizable high-order FVM for each case. In contrast, the source term S in Eq. (B.l) contains no spatial derivatives and hence is local in each finite-volume grid cell. In other words, with operator splitting the source term leads to a (stiff) ordinary differential equation (ODE) for each grid cell. 

For clarity, we will describe the advection term in Eq. (B.2) using operator splitting in each direction. Formally, this procedure can be represented as... 

For multidimensional diffusion with diffusion coefficient T t, x, ), we will again use operator splitting in each spatial direction (Vikas et al, 2012). In the x direction, the diffusion equation is... 

Celnik, M., Patterson, R., Kraft, M. Wagner, W. 2007 Coupling a stochastic soot population balance to gas-phase chemistry using operator splitting. Combustion and Flame 148,158-176. 

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