Semi-implicit methods

In the context of our semi-implicit methods, we typically consider the special case Vi = V2 = 0 which leads to the simplified equations of motion... 

Table 1. Maximum error in the energy using the semi-implicit method with the energy conserving method (6) for the strong forces.

EX 52 5.2 Solution of the Qreganator model by semi-implicit method M14,M15,M72... 

The pneumatic drying model was solved numerically for the drying processes of sand particles. The numerical procedure includes discretization of the calculation domain into torus-shaped final volumes, and solving the model equations by implementation of the semi-implicit method for pressure-linked equations (SIMPLE) algorithm [16]. The numerical procedure also implemented the Interphase Slip Algorithm (IPSA) of [17] in order to account the various coupling between the phases. The simulation stopped when the moisture content of a particle falls to a predefined value or when the flow reaches the exit of the pneumatic dryer. 

An alternative, called semi-implicit methods in such texts as [351], avoids the problems, and some of the variants are L-stable (see Chap. 14 for an explanation of this term), a desirable property. This was devised by Rosenbrock in 1962 474]. There are two strong points about this set of formulae. One is that the constants in the implicit set of equations for the k s are chosen such that each can be evaluated explicitly by easy rearrangement of each equation. The other is that the method lends itself ideally to nonlinear functions, not requiring iteration, because it is, in a sense, already built-in. This is explained below. 

The equations are solved numerically using a pressure correction technique, commonly known as SIMPLE and a semi-implicit method to handle the strong coupling between the two phases similar to the IPSA algorithm by Spalding. ... 

One of the popular methods proposed by Patankar and Spalding (1972) is called SIMPLE (semi-implicit method for pressure linked equations). In this method, discretized momentum equations are solved using the guessed pressure field. The discretized form of the momentum equations can be written ... 

SIMPLE Semi-implicit method for pressure linked equations... 

Once the pressure correction has been solved for, the velocities are updated using (12.165) and (12.166). This procedure is known as the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm [140, 141]. 

SIMPLE Semi-Implicit Method for Pressure-Linked Equations... 

What initially seems to be an advantage of the semi-implicit methods is really a negative. In the semi-implicit methods, the Jacobian is directly within the definition of the same method-, in the implicit methods, it is used (if we use a Newton method for the nonlinear system solution) only indirectly, solely for the solution of the nonlinear system. Thus, the semi-implicit methods have the following disadvantages. 

Consequently, the semi-implicit methods can be considered to solve stiff systems only when the Jacobian can be calculated analytically and is not too computationally expensive with respect to the evaluation of the system f. 

In some cases semi-implicit methods can be developed which may only require the solution of a low-dimensional nonlinear system at each step. In Chap. 4, we discuss constrained systems for which implicit methods are needed and, in the case of the SHAKE method, for which the nonlinear system that must be solved at each step is of dimension equal to the number of constraints imposed. This is an example of a semi-implicit method. 

C. Wall, C. Pierce, P. Moin, A semi-implicit method for resolution of acoustic waves in low Mach number flows. Journal of Computational Physics 181 (2) (2002) 545-563. 

Michelsen s third order semi-implicit Runge-Kutta method is a modified version of the method originally proposed by Caillaud and Padmanabhan (1971). This third-order semi-implicit method is an improvement over the original version of semi-implicit methods proposed in 1963 by Rosenbrock. 

So-called false-time-step relaxation is used to achieve stationarity. The semi-implicit method, which considers the pressure-Hnk of the pressure correction equation and the Reynolds equations, is the SIMPLEST algorithm. The sets of algebraic equations for each variable are solved iteratively by means of the ADI technique. An example of the simulated flow field is illustrated in Fig. 3. Good agreement can then be achieved between measured flow details and the simulation results for vessels and impellers of different geometry [1]. 

Several algorithms do exist in the literature for numerical computation of fluid flow problems on the basis of primitive variables, in a finite volume framework. One of the most commonly used algorithms of this kind is the SIMPLE (semi-implicit method for pressure-linked equations) algorithm [2]. With reference to a generic staggered control volume for solution of the momentum equation for u (see Fig. 3) and with similar considerations for Ihe other velocity components, major steps of the SIMPLE algorithm can be summarized as follows ... 

In the macroscale, the particles can also be considered as moving mesh nodes, as it is so conceived in SPH [86,96] and moving particles semi-implicit method (MPS) [96]. Both of the algorithms belong to the wider class of approximation methods defined in Ref. [33]. The particle system is defined by mass m, distribution in space, where fluid density in r is given, by the approximation formula... 

Koshizuka, S. and Ikeda, H., MPS, moving particles semi-implicit method, http //www.tokai.t.u. tokyo.ac.jp/usr/rohonbu/ikeda/mps/mps.html, 1999. 

The solution strategy is somewhat varied by the last step since the approach used to linearize and solve the discretized equations varies with the solver type. The two commonly employed solvers in the FVM 2se pressure-based and density-based solvers [ 12,16]. In both methods the velocity field is obtained from the momentum equations. In the density-based approach, the continuity equation is used to obtain the density field, while the pressure field is determined from an equation of state. On the other hand, the pressure-based solver extracts the pressure field by solving the pressure or pressure correction equation, which is obtained by manipulation of the momentum and continuity equations [16]. Implementation of the pressure-based solver via the so-called Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm [12] is explained later. Details of the density-based solver are extensively covered elsewhere [16] and will not be discussed here. 

The semi-implicit method for pressure-finked equations (SIMPLE) was initially developed for the simulation of incompressible flows [9]. The strategy is to write an iteractive scheme whose final result is a pressure correction that is applied to update the velocities at time level based on time level f. This method was extended to solve compressible flows. The idea was to introduce a thermodynamic coupling between the pressure and the density using the mass conservation equation to obtain the pressure correction [11]. 

The computational cost of the LU-SSOR scheme is comparable to that of the two-step explicit scheme. The damping properties of the error of the LU-SSOR method tend to be a bit worse when compared to explicit multistep methods, such as the simplified Runge-Kutta method. However, implicit or semi-implicit methods are preferred to solve stiff systems of equations. 

The semi-implicit method for pressure-linked equations (SIMPLE) algorithm [73] is used for the computation of pressure and velocity vectors of the fluid in each cell. 

By this means, the partial differential equation is transferred into an ordinary differential equation in the discrete cosine space. A semi-implicit method is used to trade-off the stability, computing time, and accuracy [39,40]. In order to remove the shortcomings with the small time-step size associated with the exphcit Euler scheme to achieve convergence, the linear fourth-order operators can be treated implicitly while the nonlinear terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is ... 

SIMPLE Algorithm. For 3D simulations, the three equations of motion [eq. (5-6)] and the equation of continuity [eq. (5-5)] combine to form four equations for four unknowns the pressure and the three velocity components. Because there is no explicit equation for the pressure, special techniques have been devised to extract it in an alternative manner. The best known of these techniques is the SIMPLE algorithm, semi-implicit method for pressure-linked equations (Patankar, 1980). Indeed, a family of algorithms has been derived from this basic one, each of which has a small modification that makes it well suited to one application or another. 

Koshizuka, S., Nobe, A., Oka, Y. (1998). Numerical analysis of breaking waves using the moving particle semi-implicit method. International Journal for Numerical Methods in Fluids, 26, 751-769. 

Simple High-Accuracy Resolution Program Semi-Implicit Method for Pressure-Linked Equations SIMPLE Consistent Simple Line Interface Calculation Spectral Method... 

It is necessary to note that application of a semi-implicit method, at conservation of simplicity of scalings, secures with much wider limits of stability of computing process, rather than obvious methods. So the overhead basil of a resistant to integration step on a time increases by two order in case of application of a method of Euler. 

Another way of classifying the integration techniques depends on whether or not the method is explicit, semi-implicit, or implicit. The implicit and semi-implicit methods play an important role in the numerical solution of stiff differential equations. To maintain the continuity of the section, we will first describe the explicit integration techniques in the context of one-step and multistep methods. The concept of stiffness and implicit methods are considered in a separate subsection, which also marks the end of this section. 

---