Arbitrary high order finite-volume methods for electromagnetic wave propagation

Problems in electromagnetic wave propagation often require high accuracy approximations with low resolution computational grids. For non-stationary problems such schemes should possess the same approximation order in space and time. In the present article we propose for electromagnetic applications an explicit class of robust finite-volume (FV) schemes for the Maxwell equations. To achieve high accuracy we combine the FV method with the so-called ADER approach resulting in schemes which are arbitrary high order accurate in space and time. Numerical results and convergence investigations are shown for two and three-dimensional test cases on Cartesian grids, where the used FV-ADER schemes are up to 8th order accurate in both space and time.     

A semi-implicit Hall-MHD solver using whistler wave preconditioning

The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form Δt∝²(Δx) for explicit calculations. A new semi-implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It is based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.     

Adaptive Mesh Refinement for conservative systems: multi-dimensional efficiency evaluation

Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multi-dimensional calculations and only 1.5-8% overhead is observed. For AMR calculations of multi-dimensional magnetohydrodynamic problems, several strategies for controlling the constraint are examined. Three source term approaches suitable for cell-centered representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing Lax-Friedrichs method on the coarser levels. This level-dependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.     

OOF3D: An image-based finite element solver for materials science

Recent advances in experimental techniques (micro-CT scans, automated serial sectioning, electron back-scatter diffraction, and synchrotron radiation X-rays) have made it possible to characterize the full, three-dimensional structure of real materials. Such new experimental techniques have created a need for software tools that can model the response of these materials under various kinds of loads. OOF (Object Oriented Finite Elements) is a desktop software application for studying the relationship between the microstructure of a material and its overall mechanical, dielectric, or thermal properties using finite element models based on real or simulated micrographs. OOF provides methods for segmenting images, creating meshes of complex geometries, solving PDE's using finite element models, and visualizing 3D results. We discuss the challenges involved in implementing OOF in 3D and create a finite element mesh of trabecular bone as an illustrative example.     

A highly parallel Black–Scholes solver based on adaptive sparse grids

In this paper, we present a highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided.     

A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids

A Cartesian cut-cell method which allows the solution of two- and three-dimensional viscous, compressible flow problems on arbitrarily refined graded meshes is presented. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative in terms of mass, momentum, and energy. For three-dimensional compressible flows, such a method has not been presented in the literature, yet. Since ghost cells can be arbitrarily positioned in space the proposed method is flexible in terms of shape and size of embedded boundaries. A key issue for Cartesian grid methods is the discretization at mesh interfaces and boundaries and the specification of boundary conditions. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh, which are used to formulate the surface flux. Expressions to impose boundary conditions and to compute the viscous terms on the boundary are derived. The overall discretization is shown to be second-order accurate in L¹. The accuracy of the method and the quality of the solutions are demonstrated in several two- and three-dimensional test cases of steady and unsteady flows.     

Embedded divide-and-conquer algorithm on hierarchical real-space grids: parallel molecular dynamics simulation based on linear-scaling density functional theory

A linear-scaling algorithm has been developed to perform large-scale molecular-dynamics (MD) simulations, in which interatomic forces are computed quantum mechanically in the framework of the density functional theory. A divide-and-conquer algorithm is used to compute the electronic structure, where non-additive contribution to the kinetic energy is included with an embedded cluster scheme. Electronic wave functions are represented on a real-space grid, which is augmented with coarse multigrids to accelerate the convergence of iterative solutions and adaptive fine grids around atoms to accurately calculate ionic pseudopotentials. Spatial decomposition is employed to implement the hierarchical-grid algorithm on massively parallel computers. A converged solution to the electronic-structure problem is obtained for a 32,768-atom amorphous CdSe system on 512 IBM POWER4 processors. The total energy is well conserved during MD simulations of liquid Rb, showing the applicability of this algorithm to first principles MD simulations. The parallel efficiency is 0.985 on 128 Intel Xeon processors for a 65,536-atom CdSe system.     

Staggered grids discretization in three-dimensional Darcy convection

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. It consists of staggered nonuniform grids with five types of nodes, differencing and averaging operators on a two-nodes stencil. The nonlinear terms are approximated using special schemes. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. Branching off of a continuous family of steady states was detected for the problem with zero heat fluxes on two opposite lateral planes.     

A parallel monotone iterative method for the numerical solution of multi-dimensional semiconductor Poisson equation

Various self-consistent semiconductor device simulation approaches require the solution of Poisson equation that describes the potential distribution for a specified doping profile (or charge density). In this paper, we solve the multi-dimensional semiconductor nonlinear Poisson equation numerically with the finite volume method and the monotone iterative method on a Linux-cluster. Based on the nonlinear property of the Poisson equation, the proposed method converges monotonically for arbitrary initial guesses. Compared with the Newton's iterative method, it is easy implementing, relatively robust and fast with much less computation time, and its algorithm is inherently parallel in large-scale computing. The presented method has been successfully implemented; the developed parallel nonlinear Poisson solver tested on a variety of devices shows it has good efficiency and robustness. Benchmarks are also included to demonstrate the excellent parallel performance of the method.     

Faster ray tracing using adaptive grids

A new hybrid approach is presented which outperforms the regular grid technique in scenes with highly irregular object distributions by a factor of hundreds, and combined with an area interpolator, by a factor of thousands. Much has been said about scene independence of different acceleration techniques and the alleged superiority of one approach over another. Several theoretical and practical studies conducted in the past have led to the same conclusion: a space partitioning method that allows the fastest rendering of one scene often fails with another. Specialization may be the answer. This has always been pursued, consciously or not, in developing various ray-tracing systems. Despite our new algorithm's impressive efficiency, we don't interpret the new method as the fastest ray-tracing scene decomposition possible. This is because our recent groundwork experiments with a derivative method produced in some of the test scenes presented in this article produced timings that were better by approximately 50%     

2-dimensional implicit hydrodynamics on adaptive grids

We present a numerical scheme for two-dimensional hydrodynamics computations using a 2D adaptive grid together with an implicit discretization. The combination of these techniques has offered favorable numerical properties applicable to a variety of one-dimensional astrophysical problems which motivated us to generalize this approach for two-dimensional applications. Due to the different topological nature of 2D grids compared to 1D problems, grid adaptivity has to avoid severe grid distortions which necessitates additional smoothing parameters to be included into the formulation of a 2D adaptive grid. The concept of adaptivity is described in detail and several test computations demonstrate the effectivity of smoothing. The coupled solution of this grid equation together with the equations of hydrodynamics is illustrated by computation of a 2D shock tube problem.     

A parallel viscous flow solver on multi-block overset grids

A multi-block overset grid method is presented to accurately simulate viscous flows around complex configurations. A combination of multi-block and overlapping grids is used to discretize the flow domain. A hierarchical grid system with layers of grids of varying resolution is developed to ensure inter-grid connectivity within a framework suitable for multi-grid and parallel computations. At each stage of the numerical computation, information is exchanged between neighboring blocks across either or both matched block boundaries and overlapping boundaries. Coarse-grain parallel processing is facilitated by the multi-block system. Numerical results of flows over multi-element airfoils and three-dimensional turbulent flows around wing–body aerodynamic configurations demonstrate the utility and efficiency of the method.     

Finite-difference time-domain simulation of electromagnetic propagation in magnetized plasma

The electromagnetic propagation through a homogeneous magnetized plasma slab is studied using the finite-difference time-domain method based on Z transforms. The reflection and transmission coefficients of the magnetized plasma layer for the right-hand circularly polarized wave are computed. The comparison of the numerical results of the Z transform and recursive convolution algorithms with analytic values indicates that the Z transform algorithm is more accurate than the recursive convolution algorithm. The dependence of the absorption coefficient on frequency is presented.     

Numerical simulation of pyroclastic density currents using locally refined Cartesian grids

Pyroclastic density currents are ground hugging, hot, gas–particle flows representing the most hazardous events of explosive volcanism. Their impact on structures is a function of dynamic pressure, which expresses the lateral load that such currents exert over buildings. Several critical issues arise in the numerical simulation of such flows, which involve a rheologically complex fluid that evolves over a wide range of turbulence scales, and moves over a complex topography. In this paper we consider a numerical technique that aims to cope with the difficulties encountered in the domain discretization when an adequate resolution in the regions of interest is required. Without resorting to time-consuming body-fitted grid generation approaches, we use Cartesian grids locally refined near the ground surface and the volcanic vent in order to reconstruct the steep velocity and particle concentration gradients. The grid generation process is carried out by an efficient and automatic tool, regardless of the geometric complexity. We show how analog experiments can be matched with numerical simulations for capturing the essential physics of the multiphase flow, obtaining calculated values of dynamic pressure in reasonable agreement with the experimental measurements. These outcomes encourage future application of the method for the assessment of the impact of pyroclastic density currents at the natural scale.     

A mixed Lagrangian-Eulerian method for non-linear free surface flows using multigrid on hierarchical Cartesian grids

This article describes a two-dimensional finite difference method for the numerical simulation of fully non-linear irrotational water waves. The computational domain is discretised in an Eulerian fashion using hierarchical Cartesian meshes, whilst the free surface location is explicitly tracked using a Lagrangian approach. The accuracy of the method is strongly dependent on the quality of the approximation to the free surface velocities and a novel method to compute these is proposed. A multigrid strategy is implemented to take advantage of the hierarchical nature of the grids, incorporating an efficient technique to generate the coarser grids. The method is verified through simulations of quasi-linear low-amplitude waves and through a comparative study using an asymmetric sloshing wave. The non-linear behaviour of waves of moderate amplitude is also simulated.     

Towards a more consistent discretization scheme for adaptive implicit RHD computations

We present a new implicit numerical discretization for the equations of radiation hydrodynamics (RHD) which is based on a more geometrical representation of a finite volume scheme suitable for spherical systems. In particular, the motion of the grid points is directly included by appropriate volume changes. Several examples illustrate the accuracy gained by this improved difference scheme.     

Computing zeros of analytic functions in the complex plane without using derivatives

We present a package in Fortran 90 which solves f(z)=0, where z∈W⊂C without requiring the evaluation of derivatives, f^′(z). W is bounded by a simple closed curve and f(z) must be holomorphic within W.We have developed and tested the package to support our work in the modeling of high frequency and optical wave guiding and resonant structures. The respective eigenvalue problems are particularly challenging because they require the high precision computation of all multiple complex roots of f(z) confined to the specified finite domain. Generally f(z), despite being holomorphic, does not have explicit analytical form thereby inhibiting evaluation of its derivatives.

Program summary

Title of program:EZEROCatalogue identifier:ADXY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXY_v1_0Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputer:IBM compatible desktop PCOperating system:Fedora Core 2 Linux (with 2.6.5 kernel)Programming languages used:Fortran 90No. of bits in a word:32No. of processors used:oneHas the code been vectorized:noNo. of lines in distributed program, including test data, etc.:21045Number of bytes in distributed program including test data, etc.:223 756Distribution format:tar.gzPeripherals used:noneMethod of solution:Our package uses the principle of the argument to count the number of zeros encompassed by a contour and then computes estimates for the zeros. Refined results for each zero are obtained by application of the derivative-free Halley method with or without Aitken acceleration, as the user wishes.