Hyperbolic equation conservation

Streamline Methods For some years, originating with [69] there has been interest in applying the method of characteristics to the solution of the hyperbolic equation. Such methods do not possess a local mass conservation... 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

In 1959, Godunov [64] introduced a novel finite volume approach to compute approximate solutions to the Euler equations of gas d3mamics that applies quite generally to compute shock wave solutions to non-linear systems of hyperbolic conservation laws. In the method of Godunov, the numerical approximation is viewed as a piecewise constant function, with a constant value on each finite volume grid cell at each time step and the time evolution... 

Since v is known, this closed hyperbolic advection equation for the disperse-phase volume fraction is in conservative form. 

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

Kinetic theory is introduced and developed as the initial step toward understanding microscopic transport phenomena. It is used to develop relations for the thermal conductivity which are compared to experimental measurements for a variety of solids. Next, it is shown that if the time- or length scale of the phenomena are on the order of those for scattering, kinetic theory cannot be used but instead Boltzmann transport theory should be used. It was shown that the Boltzmann transport equation (BTE) is fundamental since it forms the basis for a vast variety of transport laws such as the Fourier law of heat conduction, Ohm s law of electrical conduction, and hyperbolic heat conduction equation. In addition, for an ensemble of particles for which the particle number is conserved, such as in molecules, electrons, holes, and so forth, the BTE forms the basis for mass, momentum, and energy conservation equa-... 

This modification removes the degeneration for = 1. Furthermore, it reveals more of the mathematical nature of the problem. If we omit the capillary pressure in the second equation, which would describe the case of vanishing capillary forces, we see that the resulting equation is nonlinear hyperbolic. This observation gives rise to numerous numerical schemes using techniques from hyperbolic conservation laws. 

By solving (17.13), the distribution of C is obtained and thus free surface location can be identified. In a physical sense, the equation implies mass conservation of one phase in the mixture. Numerically, this equation is characterized as a hyperbolic or pure convection equation. 

The axial dispersion term, the first term in equation (6.19), can be omitted if pure plug flow can be assumed. The conservation equation then reduces to first-order hyperbolic form. If axial dispersion is significant, then the first term in equation (6.19) must be retained, and the flow is known as axially dispersed plug flow. It is generally undesirable to have radially dispersed flow in an adsorption bed, and thus this aspect is generally not incorporated into flow models for adsorption. 

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