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The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. The general equation for steady diffusion can be easily derived from the general transport equation for property Φ by deleting transient and convective terms [1]
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. [1] In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then ...
The method is based on the idea of a so-called fraction function . It is a scalar function, defined as the integral of a fluid's characteristic function in the control volume, namely the volume of a computational grid cell. The volume fraction of each fluid is tracked through every cell in the computational grid, while all fluids share a single ...
This is the only physical step of the whole procedure. The discontinuities at the interfaces are resolved in a superposition of waves satisfying locally the conservation equations. The original Godunov method is based upon the exact solution of the Riemann problems. However, approximate solutions can be applied as an alternative.
Banker's algorithm is a resource allocation and deadlock avoidance algorithm developed by Edsger Dijkstra that tests for safety by simulating the allocation of predetermined maximum possible amounts of all resources, and then makes an "s-state" check to test for possible deadlock conditions for all other pending activities, before deciding whether allocation should be allowed to continue.
Since this is only the time derivative of volume, a scalar quantity, the volumetric flow rate is also a scalar quantity. The change in volume is the amount that flows after crossing the boundary for some time duration, not simply the initial amount of volume at the boundary minus the final amount at the boundary, since the change in volume ...
Allocation is an ongoing process based on flow or volume measurements, and gives the distribution of contributing sources, often with a final calculation per day, which in turn provides the basis for a daily production report in the case of a field that produces hydrocarbons.
Optimization-based methods aim to attain, for each instance, an allocation that is "as fair as possible" for this instance. An allocation is "fair" if = for all agents i; in this case, we say that the "unfairness" of the allocation is 0. If this equality is violated, one can define a measure of "total unfairness", and try to minimize it.