Equation ordering algorithm

A Minimum Difficulty Solution Strategy. The rearranged functionality matrix provides information on the simplest strategy available. Hierarchy information from the equation ordering algorithm indicates the presence and size of persistent iteration loops or loops that must be solved simultaneously. 

The results of running the equation ordering algorithm is the functionality matrix of Figure 2.31. There are two hierarchies which indicate that there are two iterative variables needed to solve this problem or that some of the equations must be solved simultaneously. The step equations will be located at rows 5 and 3 of the rearranged functionality matrix. 

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. 

The specification phase of the simulation units makes necessary the analysis of the degrees of freedom (see subchapter 3.2). This is the number of variables that must be set in order to solve the system of equations describing the model. Therefore, the user should have at least an idea about the type of equations and algorithms associated with different modelling units. Sometimes the user has to decide between alternative models, with quite different specification and convergence properties. 

For the sake of simplicity, let us start with a single differential equation. Let us suppose that we are able to estimate the local error of the /(-order algorithm we... 

Equations 12.45, 12.46, 12.47, and 12.48 form a complete set of governing equations which are strongly coupled to each other. Therefore, these equations can be solved by nonlinear iterative procedures [133, 134, 198] and efficient second-order algorithms [1, 71,72,132]. 

There are several levels of analysis. In order of increasing complexity, they involve material balances, material and energy balances, equipment sizing, and profitability analysis. Additional equations are added at each level. New variables are introduced, and the equation-solving algorithms become more complicated. 

In order to find all solutions of the above system of nonlinear equations the algorithm outlined in the previous section was applied. The variables lower and upper bounds are -10 and 1, respectively, which was found by a fast prerun of the algorithm using a larger tolerance of 0.001. The problem was solved utilizing a GAMS [23] implementation of the algorithm. A total of 31 iterations and 1.1 CPU sec were required to identify the solutions within a tolerance of lOE-5. The optimal solution obtained was ... 

Similar expressions to those shown in equations (5)-(7) can be derived for higher-order Runge-Kutta schemes. The fourth-order algorithm is widely used to integrate physical systems. ... 

The simplest of the numerical techniques for the integration of equations of motion is leapfrog-Verlet algorithm (LFV), which is known to be symplectic and of second order. The name leapfrog steams from the fact that coordinates and velocities are calculated at different times. 

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