Operator Splitting Scheme Research Articles

Projection functions for particle‐grid methods

AbstractRandom walk particle methods (RWPM) can be used in operator splitting schemes to simulate reactive solute transport in porous media. Projection functions are used to transfer particle location and mass information to concentrations at selected spatial points. Because of the stochastic nature of RWPM, concentration estimates made from particle distributions include a “noisy” error component. In some cases of reactive or density‐dependent flows, this type of error may be propagated forward in time. It can be reduced by using larger numbers of particles or by using different projection functions. The effects of using different projection functions or numbers of particles in different flow regimes or dimensions are explored using concentration solutions for a set of one‐, two‐, and three‐dimensional nonreactive test problems. Resulting solutions are compared with analytic results and classical random walk error estimates. A piecewise linear projection function provides a reasonable improvement in accuracy over the more convenient box methods at a modest increase in cost. The support of the projection functions should be O(Δx) to avoid excessive smearing. Multidimensional projection functions may be advantageously formed by products of different one‐dimensional projection functions.

On the interaction between equilibration processes and wet or dry deposition

Atmospheric equilibration processes between two phases with different deposition velocities have the potential to affect significantly the amount of total material deposited on the ground. The magnitude of the effects of the equilibration processes depends primarily on the ratio of the deposition velocities of the two phases, on the production/emission rate of the gas phase species, and on the initial distribution of species between the two phases. The deposition of a condensible species equilibrating between gas and aerosol phases can increase by as much as 20 times over that when equilibration processes are not present under appropriate conditions (very large aerosol particles, most of the material initially in the gas phase and high gas-phase production rate) or to decrease by as much as 15 times (very small aerosol particles, most of the material initially in the gas phase and high gas-phase production rate). In fog episodes, the deposition of a gaseous species with a Henry's Law constant between 10 3 and 10 6 M atm −1 (e.g. SO 2 for pH between 4.5 and 7, H 2O 2, HCHO etc) can be enhanced by as much as a factor of 3 because of its transfer to the aqueous phase. For the NH 3HNO 3NH 4NO 3 system the total deposition can be reduced by as much as a factor of 3 for typical conditions in a polluted atmosphere and small initial concentration of aerosol NH 4NO 3 with NH 3 initially dominating HNO 3 in the gas phase. If an operator splitting scheme is used in a mathematical both equilibration and removal processes should be included in the same operator or very small operator time steps (typically less than 1 min) will be necessary.

Time Dependent Operator-split and Unsplit Schemes for One Dimensional Premixed Flames

This paper is concerned with a study of an operator split scheme and unsplit scheme for the computation of adiabatic freely propagating one-dimensional premixed flames. The study uses unsteady method for both split and unsplit schemes employing implicit chemistry and explicit diffusion, a combination which is stable and convergent. Solution scheme is not sensitive to the initial starting estimate and provides steady state even with straight line profiles (far from steady state) in small number of time steps. Two systems H2-Air and H2-NO (involving complex nitrogen chemistry) are considered in presentinvestigation. Careful comparison shows that the operator split approach is slightly superior than the unsplit when chemistry becomes complex. Comparison of computational times with those of existing steady and unsteady methods seems to suggest that the method employing implicit-explicit algorithm is very efficient and robust.

NUMERICAL STUDY OF CONVECTION-DIFFUSION-REACTION EQUATIONS FOR LARGE DAMKOHLER AND CELL REYNOLDS NUMBERS

The issues of the large cell Reynolds number and large Damkohler number are examined. Thirteen finite-difference schemes for solving a system of highly nonlinear convection-diffusion-reaction integrodifferential equations are studied. The system models the propagation of a one-dimensional unsteady laminar flame in closed or open combustors. The numerical schemes include an explicit scheme with upwind, central, and hybrid differencing, a predictor-corrector explicit scheme, a sequential implicit scheme, a quasi-linear implicit scheme, a block implicit scheme, the standard Crank-Nicolson implicit scheme, two methods of lines, a projection implicit scheme, an operator-splitting scheme, and the QUICK scheme. The efficiency, accuracy, and stability of these schemes art discussed. In particular, it is shown that the treatment of convection terms and of reaction terms, as well as the flow conditions, has a strong impact on the selection of an optimum numerical technique. Generally, for a large Damkohler number t...

A MULTILEVEL-MULTIGRID TECHNIQUE FOR RECIRCULATING FLOWS

A solution algorithm has been developed for the prediction of recirculating flows. Brandt's multilevel acceleration technique is used with Leonard's QUICK differencing scheme and a modified pressure implicit operator splitting scheme. Intermediate calculations enable a tau error distribution to be used for the identification of regions for local grid refinement, i.e., multigrid. The algorithm was tested for the prediction of laminar flow in a shear-driven and a buoyancy-driven cavity.

A Multilevel-Multigrid Technique for Recirculating Flows

A solution algorithm has been developed for the prediction of recirculating flows. Brandt's multilevel acceleration technique is used with Leonard's QUICK differencing scheme and a modified pressure implicit operator splitting scheme. Intermediate calculations enable a tau error distribution to be used for the identification of regions for local grid refinement, i.e., multigrid. The algorithm was tested for the prediction of laminar flow in a shear-driven and a buoyancy-driven cavity.

On the Construction and Comparison of Difference Schemes

On the Construction and Comparison of Difference Schemes