Deterministic proton transport solving a one dimensional Fokker-Planck equation

The transport of protons through matter is characterized by many interactions which cause small deflections and slight energy losses. The few which are catastrophic or cause large angle scattering can be viewed as extinction for many applications. The transport of protons at this level of approximation can be described by a Fokker Planck Equation. This equation is solved using a deterministic multigroup differencing scheme with a highly resolved set of discrete ordinates centered around the beam direction which is adequate to properly account for deflections and energy losses due to multiple Coulomb scattering. Comparisons with LAHET for a large variety of problems ranging from 800 MeV protons on a copper step wedge to 10 GeV protons on a sandwich of material are presented. The good agreement with the Monte Carlo code shows that the solution method is robust and useful for approximate solutions of selected proton transport problems

Real-Space Mesh Techniques in Density Functional Theory

This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.Comment: 70 pages, 11 figures. To be published in Reviews of Modern Physic

An adaptive weighted diamond differencing method for three-dimensional, XYZ geometry

About sixteen years ago, Bengt Carlson introduced a method for discretizing the neutral particle transport equation to achieve a positive solution while at the same time retaining much of the accuracy of the diamond differencing method. About six years later Russian researchers applied this work to their problems and extended it somewhat to enhance the flexibility of the method to incorporate monotonic properties of the solution. This latter work came to the attention of US researchers in late 1991 where it verified much of Carlson`s conclusions in theory and in test problems. This method, called the adaptive weighted diamond (AWDD) method, is based upon a weighted diamond discretization of the transport equation with the weights chosen from a diamond difference prediction of the solution so as to correct it for positively and monotonicity. This work re-examines the method and extends it to three-dimensional XYZ geometry and demonstrates its potential for solving such problems accurately while achieving a much smoother solution than diamond with set-to-zero fixup and is as effective as the theta-weighted fixup method 3 while theoretically and operationally more satisfying

Boundary Projection Acceleration: A New Approach to Synthetic Acceleration of Transport Calculations

We present a new class of synthetic acceleration methods which can be applied to transport calculations regardless of geometry, discretization scheme, or mesh shape. Unlike other synthetic acceleration methods which base their acceleration on P1 equations, these methods use acceleration equations obtained by projecting the transport solution onto a coarse angular mesh only on cell boundaries. We demonstrate, via Fourier analysis of a simple model problem as well as numerical calculations of various problems, that the simplest of these methods are unconditionally stable with spectral radius less than or equal toc/3 (c being the scattering ratio), for several different discretization schemes in slab geometry. 28 refs., 4 figs., 3 tabs