Discretization method conservativeness

Unfortunately, discretization methods with large step sizes applied to such problems tend to miss this additional force term [3]. Furthermore, even if the implicit midpoint method is applied to a formulation in local coordinates, similar problems occur [3]. Since the midpoint scheme and its variants (6) and (7) are basically identical in local coordinates, the same problem can be expected for the energy conserving method (6). To demonstrate this, let us consider the following modified model problem [13] ... 

In computational fluid dynamics only the last two methods have been extensively implemented into commercial flow solvers. Especially for CFD problems the FVM has proven robust and stable, and as a conservative discretization scheme it has some built-in mechanism of error avoidance. For this reason, many of the leading commercially available CFD tools, such as CFX4/5, Fluent and Star-CD, are based on the FVM. The oufline on CFD given in this book wiU be based on this method however, certain parts of the discussion also apply to the other two methods. 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

This discretization method obeys a conservation property, and therefore is called conservative. With the exception of the first element and the last element, every element face is a part of two elements. The areas of the coincident faces and the forces on them are computed in exactly the same way (except possibly for sign). Note that the sign conventions for the directions of the positive stresses is important in this regard. The force on the left face of some element is equal and opposite to the force on the right face of its leftward neighbor. Therefore, when the net forces are summed across all the elements, there is exact cancellation except for the first and last elements. For this reason no spurious forces can enter the system through the numerical discretization itself. The net force on the system of elements must be the net force caused by the boundary conditions on the left face of the first element and the right face of the last element. 

Unfortunately, resolution of the conservation problem requires knowledge of species flux, and hence details of the specific problem and discretization method. Therefore it is not possible in the general setting of the present discussion to give a universal solution. Nevertheless, a software author and users of a simulation code must be aware of the difficulty, and consider its resolution when setting up the difference approximations to the particular system of conservation equations. 

The selected mathematical model is represented by a discretization method for approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. Many different approaches are used in reactor engineering , but the most important of them are the simple finite difference methods (FDMs), the flrrx conservative finite volume methods (FVMs), and the accurate high order weighted residual methods (MWRs). 

Equation (12.277) is not necessary conservative due to the finite (i.e., in practice rather coarse) size grid resolution, and some sort of numerical trick must be used to enforce the conservative properties. It is mainly at this point in the formulation of the numerical algorithm that the class method of Hounslow et al [74], the discrete method of Ramkrishna [151] and the multi-group approach used by Carrica et al [24], among others, differs to some extent as discussed earlier. 

In finite-volume methods, the integral formulation of the conservation laws over a small physical control volume is discretized directly. FVM employs a conservative discretization, that is, each species is guaranteed to be conserved, even for coarse meshes. In contrast, many traditional FDMs are not conservative. For example, owing to the nonlinear nature of the constitutive flux equations of ionic species in an a priori unknown electric field, FDM is nonconservative, even when constant physical... 

All symplectic methods preserve the volume, but for a system with no first integrals except the energy, it is known from a theorem of Ge and Marsden [400] that a symplectic method cannot preserve the system energy exactly (unless the symplectic method is itself a time-reparameterization of the exact solution). The options for volume preserving non-symplectic discretization methods that conserve energy exactly are limited (standard form schemes that rely on calculation of the vector field at a few points cannot achieve this [380]). 

J. C. Simo and N. Tarnow. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. ZAMP, 43 757-793, 1992. 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

Although LB therefore nowadays may be considered as a solver for the NS equations, there is definitely more behind it. The method originally stems from the lattice gas automaton (LGA), which is a cellular automaton. In a LGA, a fluid can be considered as a collection of discrete particles having interaction with each other via a set of simple collision rules, thereby taking into account that the number of particles and momentum is conserved. 

For the systems that we have considered so far, the solutions behave smoothly in time and space. Often one can simply inspect the solution and decide if the mesh is sufficiently fine to represent it accurately. Refining mesh sizes and time steps is another simple method to assure oneself that a particular discretization was sufficient. Later we will be much more concerned about numerical accuracy and stability, especially when complex chemistry is considered. For now we take a somewhat cavalier approach, with the objective being mainly to explore the general numerical approaches to solving the conservation equations describing fluid flow. For relatively simple problems we can implement usable solutions with relative ease, for example, in a spreadsheet. 

While ambient pressure studies must rely on discrete changes of crucial parameters, the high pressure method is capable of generating continuous changes of interatomic distances or relative energies of different electronic states. Moreover, at the same time the chemical composition of the rare-earth compound is conserved under pressure, while ambient pressure studies usually have to consider different compounds. In this sense, the application of high pressure can solve physical problems which can not be accessed by any other method. 

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