Niel K. Madsen

Bio: Niel K. Madsen is an academic researcher from Lawrence Livermore National Laboratory. The author has contributed to research in topics: Differential equation & Numerical partial differential equations. The author has an hindex of 10, co-authored 14 publications receiving 1017 citations.

Algorithm 540: PDECOL, General Collocation Software for Partial Differential Equations [D3]

Abstract: PDECOL, new computer software package for numerically solving coupled systems of nonlinear partial differential equations (PDE's) in one space and one time dimension, is discussed. The package implements finite element collocation methods based on piecewise polynomials for the spatial discretization techniques. The time integration process is then accomplished by widely acceptable procedures that are generalizations of the usual methods for treating time-dependent partial differental equations. PDECOL is unique because of its flexibiility both in the class of problems it addresses and in the variety of methods it provides for use in the solution process. High-order methods (as well as low-order ones) are readily available for use in both the spatial and time discretization procedures. The time integration methods used feature automatic time step size and integration formula order selection so as to solve efficiently the problem at hand and yet achieve a user-specific time integration error level. PDECOL consists of a collection of 19 subroutines written in reasonably standard Fortran, and therefore is quite portable. No special hardware features are required. PDECOL is designed to solve broad classes of difficult systems of partial differential equations that descrobe physical processes. 4 figures, 1 table. (RWR)

Software for Nonlinear Partial Differential Equations

Abstract: The numerical solution of physically realistic nonlinear partial differential equations (PDEs) is a complicated and highly problem-dependent process which usually requires the scientist to undertake the difficult and time-consuming task of developing his own computer program to solve his problem. This paper presents a software interface which can eliminate much of the expensive and time-consuming effort involved in the solution of nonlinear PDEs. The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators. Besides being portable, efficient, and easy to use, the software interface along with an ODE integrator will discretize the problem, select the time step and order, solve the nonlinear equations (checking for convergence, etc.), and maintain a user-specified time integration accuracy, all automatically and reliably. Physically realistic examples are given to illustrate the use and capability of the software.

A Three-Dimensional Modified Finite Volume Technique for Maxwell's Equations

Abstract: A modified finite volume method for solving Maxwell's equations in the time-domain is presented. This method, which allows the use of general nonorthogonal mixed-polyhedral grids, is a direct generalisation of the canonical staggered-grid finite difference method. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) this method allows more accurate modeling of non-rectangular structures. The traditional “stair-stepped” boundary approximations associated with the orthogonal grid based finite difference methods ate avoided. Numerical results demonstrating the accuracy of this new method are presented.

Iterative Methods for Sparse Linear Systems

Abstract: Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

9 – Computational Electromagnetics: The Finite-Difference Time-Domain Method

Abstract: Prior to abour 1990, the modeling of electromagnetic engineering systems was primarily implemented using solution techniques for the sinusoidal steady-state Maxwell's equations. Before about 1960, the principal approaches in this area involved closed-form and infinite-series analytical solutions, with numerical results from these analyses obtained using mechanical calculators. After 1960, the increasing availability of programmable electronic digital computers permitted such frequency-domain approaches to rise markedly in sophistication. Researchers were able to take advantage of the capabilities afforded by powerful new high-level programming languages such as Fortran, rapid random-access storage of large arrags of numbers, and computational speeds that were orders of magnitude faster than possible with mechanical calculators. In this period, the principal computational approaches for Maxwell's equations included the high-frequency asymptotic methods of Keller (1962) as well as Kouyoumjian and Pathak (1974) and the integral equation techniques of Harrington (1968) .

Data assimilation in meteorology and oceanography

Abstract: Publisher Summary This chapter provides a review of current operational practice and of advanced data assimilation techniques in meteorology. Numerical models can be used to assimilate meteorological and oceanographic data, creating a dynamically consistent, complete and accurate “movie” of the two geofluids, atmosphere, and ocean in motion. The ocean's strong stratification helps determine the most energetic scales and processes for the global ocean circulation. Active research on data assimilation is burgeoning rapidly in both meteorology and oceanography. Operational NWP requirements have produced a mature data-assimilation technology in meteorology, from which climatic research has benefitted as well. Ocean is characterized by transient, energetic motions with a broad spectrum in frequency and wave number. A steady component of the circulation may not even exist, and be only a model resulting from the analysis of data sets sparse in space and time, like hydrographic data sets, for which steadiness is assumed a priori. Thus, in oceanic data-assimilation problems, the choice of a model and related data assimilation scheme and the definition of success of the assimilation process itself depend crucially on the scientific issue of interest as the starting point.