Fractional step methods

The fractional step, or time splitting, concept is more a generic operator splitting approach than a particular solution method [35, 56, 106, 144, 260]. 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. 

The fractional step concept is frequently used to solve scalar transport equations on the generic form [12, 99, 159]  

This balance equation is written on the conservative flux form, the non-conservative form of the equation is easily obtained by use of the continuity equation. 

To illustrate the fractional step method, the explicit Euler advancement of the property ijj can be written in symbolic form  

The fractional step, or time splitting, concept can be adapted to the discretized equation by splitting the discrete scalar equation into a three step method  

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The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

The pressure-based method was introduced by Harlow and Welch [67] and Chorin [30] for the calculation of unsteady incompressible viscous flows (parabolic equations). In Chorines fractional step method, an incomplete form of the momentum equations is integrated at each time step to 3ueld an approximate velocity field, which will in general not be divergence free, then a correction is applied to that velocity field to produce a divergence free velocity field. The correction to the velocity field is an orthogonal projection in the sense that it projects the initial velocity field into the divergence free... 

The fractional-step concept can be used to solve the governing equations for the fluid motion as well. To illustrate the overall method with emphasis on the pressure-velocity coupling, a FVM variant of the fractional-step method used by Chorin [30], Fortin et al [56] and Andersson and Kristoffersen [3] for solving the unsteady Navier-Stokes and continuity equations for incompressible viscous flows is outlined. We consider the equations of motion of an incompressible viscous fluid ... 

In application of the fractional step method to the incompressible Navier-Stokes equations, the pressure may be interpreted as a projection operator which projects an arbitrary vector field into a divergence-free vector field. 

Fractional step methods have become quite popular. There are many variations of them, due to a vast choice of approaches to time and space discretizations, but they are all based on the principles described above. To... 

Implicit Fractional Step Method for Solving the Two-Fluid Granular Flow Model Equations Applied to Fluidized Bed Flow... 

Kim J, Moin P (1985) Apphcation of a Fractional-Step Method to Incompressible Navier-Stokes Equations. J Comput Phys 59 308-323... 

The fractional step method is used to advance each RK substep, and a Poisson equation needs to be solved for pressure update. The current geometry leads to a nine-diagonal matrix for the Poisson solver. An LU-decomposition solver was made to solve this nine-diagonal matrix. The singularity of the matrix is removed by prescribing the reference value of zero at the last cell in... 

Equation (25.85), the basis of every atmospheric model, is a set of time-dependent, nonlinear, coupled partial differential equations. Several methods have been proposed for their solution including global finite differences, operator splitting, finite element methods, spectral methods, and the method of lines (Oran and Boris 1987). Operator splitting, also called the fractional step method or timestep splitting, allows significant flexibility and is used in most atmospheric chemical transport models. 

Solve Navier-Stokes equations using the fractional steps method. 

The fractional step methods have become quite popular. To predict an accurate time history of the flow, higher order discretizations must be employed. Kim and Moin [106], for example, used a second order explicit Adams-Bashforth scheme for the convective terms and a second order implicit Crank-Nicholson scheme for the viscous terms. Boundary conditions for the intermediate velocity fields in timesplitting methods are generally a complex issue [3, 106]. There are many variations of the fractional step methods, due to a vast choice of approaches to time and space discretizations, but they are generally based on the principles described above. 

Auxiliary factor in multiphase fractional step method... 

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