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In this article, we consider the extended Kohn–Sham model for atoms subjected to cylindrically symmetric external potentials. The variational approximation of the model and the construction of appropriate discretization spaces are detailed together with the algorithm to solve the discretized Kohn–Sham equations used in our code. Using this code, we compute the occupied and unoccupied energy levels of all the atoms of the first four rows of the periodic table for the reduced Hartree–Fock (rHF) and the extended Kohn–Sham X models. These results allow us to test numerically the assumptions on the negative spectra of atomic rHF and Kohn–Sham Hamiltonians used in our previous theoretical works on density functional perturbation theory and pseudopotentials. Interestingly, we observe accidental degeneracies between s and d shells or between p and d shells at the Fermi level of some atoms. We also consider the case of an atom subjected to a uniform electric field. For various magnitudes of the electric field, we compute the response of the density of the carbon atom confined in a large ball with Dirichlet boundary conditions, and we check that, in the limit of small electric fields, the results agree with the ones obtained with first-order density functional perturbation theory.
An equation-by-equation (EBE) method is proposed to solve a system of nonlinear equations arising from the moment constrained maximum entropy problem of multidimensional variables. The design of the EBE method combines ideas from homotopy continuation and Newton’s iterative methods. Theoretically, we establish the local convergence under appropriate conditions and show that the proposed method, geometrically, finds the solution by searching along the surface corresponding to one component of the nonlinear problem. We will demonstrate the robustness of the method on various numerical examples, including (1) a six-moment one-dimensional entropy problem with an explicit solution that contains components of order – in magnitude, (2) four-moment multidimensional entropy problems with explicit solutions where the resulting systems to be solved range from – equations, and (3) four- to eight-moment of a two-dimensional entropy problem, whose solutions correspond to the densities of the two leading EOFs of the wind stress-driven large-scale oceanic model. In this case, we find that the EBE method is more accurate compared to the classical Newton’s method, the Matlab generic solver, and the previously developed BFGS-based method, which was also tested on this problem. The fourth example is four-moment constrained of up to five-dimensional entropy problems whose solutions correspond to multidimensional densities of the components of the solutions of the Kuramoto–Sivashinsky equation. For the higher-dimensional cases of this example, the EBE method is superior because it automatically selects a subset of the prescribed moment constraints from which the maximum entropy solution can be estimated within the desired tolerance. This selection feature is particularly important since the moment constrained maximum entropy problems do not necessarily have solutions in general.
We study a class of importance sampling methods for stochastic differential equations (SDEs). A small noise analysis is performed, and the results suggest that a simple symmetrization procedure can significantly improve the performance of our importance sampling schemes when the noise is not too large. We demonstrate that this is indeed the case for a number of linear and nonlinear examples. Potential applications, e.g., data assimilation, are discussed.
We consider the efficient approximation of low Mach number flows by a high-order scheme, coupling a discontinuous Galerkin (DG) discretization in space with an implicit/explicit (IMEX) discretization in time. The splitting into linear implicit and nonlinear explicit parts relies heavily on the incompressible solution. The method has been originally developed for a singularly perturbed ODE and applied to the isentropic Euler equations. Here, we improve, extend, and investigate the so-called RS-IMEX splitting method. The resulting scheme can cope with a broader range of Mach numbers without running into roundoff errors, it is extended to realistic physical boundary conditions, and it is shown to be highly efficient in comparison to more standard solution techniques.
We study the dynamical behavior of compressible fluids evolving on the outer domain of communication of a Schwarzschild background. For both the relativistic Burgers equation and the relativistic Euler system, assuming spherical symmetry we introduce numerical methods that take the Schwarzschild geometry and, specifically, the steady state solutions into account. The schemes we propose preserve the family of steady state solutions and enable us to study the nonlinear stability of fluid equilibria and the behavior of solutions near the black hole horizon. We state and numerically demonstrate several properties about the late-time behavior of perturbed steady states.
A new large time step semi-implicit multiscale method is presented for the solution of low Froude number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite volume discretization is implemented in the framework of second-order Godunov-type methods for conservation laws. The basic properties of the method are validated by numerical tests. This development is a further step in the construction of asymptotically adaptive numerical methods for the computation of large-scale atmospheric flows.
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