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Concept# Finite difference method

Summary

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis.
Today, FDM are one of the most common approaches to the numerical solution of PDE, alon

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Let d(n) denote Dirichlet's divisor function for positive integer numbers. This work is primarily concerned with the study of We are interested, in the error term where Ρ3 is a polynomial of degree 3 ; more precisely xΡ3(log x) is the residue of in s = 1. A. Ivić showed that E(x) = Ο(x1/2+ε) for all ε > 0 (cf. [9], p.394). We will prove that for all x > 0, we have With this intention, we apply Perron's formula to the generating function ζ4(s)/ ζ (2s) and Landau's finite difference method. It was conjectured that E(x) = Ο(x1/4+ε) for ε > 0. The existence of non-trivial zeros of the Riemann ζ function implies that we cannot do better, that is The study of the Riesz means for ρ sufficiently large shows that their error term, is an infinite series , on the zeros of the Riemann Zeta function added with a development , into a series of Hardy-Voronoï's type, both being convergent. To find the "meaning" of , one could consider the difference But the series (probably) doesn't converge.We will thus substract only a finite part of , weighted by a smooth function ω, the number of terms of the finite part depending on x. If we consider this new error term , we obtain, using a classical method due to Hardy, that for x ≥ 1.

This thesis presents a systematic study on the merits and limitations on using pin-by-pin resolution and transport theory based approaches for nuclear core design calculations. Starting from the lattice codes and an optimal cross section generation scheme, it compares different methods, transport approximations, and spatial discretizations used in pin-by-pin homogenized codes.
It is in the interest of nuclear power plant operators to employ more heterogeneous core loadings in order to improve the fuel utilization and decrease the amount of spent fuel. This necessarily increases the requirements on the accuracy of the computation tools used for the core design and safety analysis. One possibility is employing 3D core solvers with higher spatial resolution, e.g. pin-cell wise.
The comparison of several lattice codes indicates that already the proper generation of diffusion coefficients and higher-order scattering moments for pin-cell geometry is not straightforward. Out of three available lattice codes, two generated unphysical diffusion coefficient when using the inscatter approximation, while the last code was not able to provide the higher scattering moments.
Several few-assembly test cases with either high neutron leakage effects or MOX/uranium interfaces showed that the quality of the results of diffusion and SP3 solvers depends critically on the choice of the diffusion coefficient: in the outscatter approximation, it can cause major deviations, while the inscatter seems overall more adequate. Regarding the transport solvers, results obtained with the SN solver DORT showed very good performance for MOX/uranium interfaces, but major deviations occurred for problems with large power gradients. On the other hand, the MOC solver nTRACER showed in all cases small average error, but 2 - 3 % error around the interface of different assemblies.
A study on spatial discretization indicated that the finite difference method applied on pin-cells does not properly capture the big flux changes between MOX and uranium fuel, while the nodal expansion method is more accurate but too slow. It was suggested to use the finite difference method with finer mesh in the outer assembly pin-cells, which increases the required computation time by only 50 % and decreases the pin power errors below 1 % with respect to lattice code results.
Due to some problems which were observed with the available diffusion/SP3 solvers, a new SP3 solver was implemented in the DORT-TD platform. Several core tests showed that the SP3 pin-by-pin solver can significantly outperform the state-of-the-art nodal solver SIMULATE-5, in particular for reactor cores with inserted control rods.
Finally, the pin-by-pin solvers were coupled to a depletion solver. For that, an accurate and fast interpolation routine had to be implemented. The obtained results of full-cycle depletion with complex core loading showed very good performance in comparison to a heterogeneous transport-based fine-group calculations.

Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.

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