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All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

On the Dynamics of Some Discretizations of Convection-Diffusion Equations

Numerical discretizations of differential equations which model physical processes can possess dynamics quite different from that of the equations themselves. Recently the emphasis has been on the the dynamics of numerical discretizations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is more complex. First, the spatial discretisation may introduce dynamics not present in the original equations; second, the solution of the resulting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this talk we present some results of our recent work on the dynamics of discretizations of convection-diffusion equations, including those produced using Total Variation Diminishing (TVD) schemes and adaptive grid techniques. A more general overview of the area may be found on our accompanying poster presentation.

Sweby, Peter K.↗
NASA NTRS1984

Multigrid techniques for the numerical solution of the diffusion equation

An accurate numerical solution of diffusion problems containing large local gradients can be obtained with a significant reduction in computational time by using a multigrid computational scheme. The spatial domain is covered with sets of uniform square grids of different sizes. The finer grid patterns overlap the coarse grid patterns. The finite-difference expressions for each grid pattern are solved independently by iterative techniques. Two interpolation methods were used to establish the values of the potential function on the fine grid boundaries with information obtained from the coarse grid solution. The accuracy and computational requirements for solving a test problem by a simple multigrid and a multilevel-multigrid method were compared. The multilevel-multigrid method combined with a Taylor series interpolation scheme was found to be best.

Phillips, R. E.↗
NASA NTRS1976

Periodic solutions to systems of reaction-diffusion equations

Necessary and sufficient conditions are derived for the existence of temporally periodic 'dissipative structure' solutions in weak diffusion with the reaction rate terms dominant in a generic system of reaction-diffusion differential equations. The enumerator index i of the equations denotes the density or concentration of the ith participating molecular or biological species, and D sub i is the diffusivity constant for the ith species while Q sub i (c), an algebraic function of the n-tuple c, expresses the local rate of production of the ith species due to chemical reactions or biological interactions.

Rosen, G.↗
NASA NTRS1990

Application of implicit numerical techniques to the solution of the three-dimensional diffusion equation

Implicit techniques for calculating three-dimensional, time-dependent heat diffusion in a cube are tested with emphasis on storage efficiency, accuracy, and speed of calculation. For this purpose, a tensor product technique with both Chebyshev collocation and finite differences and a generalized conjugate gradient technique with finite differences are used in conjunction with Crank-Nicolson discretization. An Euler explicit finite difference calculation is performed for use as a benchmark. The implicit techniques are found to be competitive with the Euler explicit method in terms of storage efficiency and speed of calculation and offer advantages both in accuracy and stability. Mesh stretching in the finite difference calculations is shown to markedly improve the accuracy of the solution.

Peltier, Leonard Joel↗
NASA NTRS1959

Computer Program for Solving Nine-Group Diffusion Equations for Cylindrical Reactors

A method is presented for determining the critical size of a cylindrical reactor by a one-dimensional group-diffusion method extended to a two-dimensional solution by prescribing values of axial leakage based on assumed flux levels. The neutron energy spectrum is divided into nine groups, and the reactor into four concentric cylinders. A computing machine program for an IBM 650 computer with attachments and a method for using the program with nine, or less, groups and four, or less, regions are given.

Miser, James W.↗
NASA NTRS1975

Steady-state solution of the semi-empirical diffusion equation for area sources

The problem investigated can be solved exactly in a simple manner if the equations are written in terms of a similarity variable. The exact solution is used to explore two questions of interest in the modelling of urban air pollution, taking into account the distribution of surface concentration downwind of an area source and the distribution of concentration with height.

Lebedeff, S. A.↗
NASA NTRS1984

A numerical study of the steady scalar convective diffusion equation for small viscosity

A time-independent convection diffusin equation is studied by means of a compact finite difference scheme and numerical solutions are compared to the analytic inviscid solutions. The correct internal and external boundary layer behavior is observed, due to an inherent feature of the scheme which automatically produces upwind differencing in inviscid regions and the correct viscous behavior in viscous regions.

Giles, M. B.↗
NASA NTRS2005

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of di usion. Gauss-Seidel relaxation converges 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel.

Atkins, H. L.↗
NASA NTRS2007

The SMM Model as a Boundary Value Problem Using the Discrete Diffusion Equation

A generalized single step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.

Campbell, Joel↗
NASA NTRS2011

Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.

Cacio, Emanuela↗
NASA NTRS2014

A Radiation Chemistry Code Based on the Green's Function of the Diffusion Equation

Stochastic radiation track structure codes are of great interest for space radiation studies and hadron therapy in medicine. These codes are used for a many purposes, notably for microdosimetry and DNA damage studies. In the last two decades, they were also used with the Independent Reaction Times (IRT) method in the simulation of chemical reactions, to calculate the yield of various radiolytic species produced during the radiolysis of water and in chemical dosimeters. Recently, we have developed a Green's function based code to simulate reversible chemical reactions with an intermediate state, which yielded results in excellent agreement with those obtained by using the IRT method. This code was also used to simulate and the interaction of particles with membrane receptors. We are in the process of including this program for use with the Monte‐Carlo track structure code Relativistic Ion Tracks (RITRACKS). This recent addition should greatly expand the capabilities of RITRACKS, notably to simulate DNA damage by both the direct and indirect effect.

Plante, Ianik↗
NASA NTRS1988

An asymptotic induced numerical method for the convection-diffusion-reaction equation

A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run time results demonstrate the viability of the method.

Scroggs, Jeffrey S.↗
NASA NTRS1972

Applications of NASTRAN to nuclear problems

The extent to which suitable solutions may be obtained for one physics problem and two engineering type problems is traced. NASTRAN appears to be a practical tool to solve one-group steady-state neutron diffusion equations. Transient diffusion analysis may be performed after new levels that allow time-dependent temperature calculations are developed. NASTRAN piecewise linear anlaysis may be applied to solve those plasticity problems for which a smooth stress-strain curve can be used to describe the nonlinear material behavior. The accuracy decreases when sharp transitions in the stress-strain relations are involved. Improved NASTRAN usefulness will be obtained when nonlinear material capabilities are extended to axisymmetric elements and to include provisions for time-dependent material properties and creep analysis. Rigid formats 3 and 5 proved to be very convenient for the buckling and normal-mode analysis of a nuclear fuel element.

Spreeuw, E.↗
NASA NTRS2001

A Hydrodynamic Theory for Spatially Inhomogeneous Semiconductor Lasers: Microscopic Approach

Starting from the microscopic semiconductor Bloch equations (SBEs) including the Boltzmann transport terms in the distribution function equations for electrons and holes, we derived a closed set of diffusion equations for carrier densities and temperatures with self-consistent coupling to Maxwell's equation and to an effective optical polarization equation. The coherent many-body effects are included within the screened Hartree-Fock approximation, while scatterings are treated within the second Born approximation including both the in- and out-scatterings. Microscopic expressions for electron-hole (e-h) and carrier-LO (c-LO) phonon scatterings are directly used to derive the momentum and energy relaxation rates. These rates expressed as functions of temperatures and densities lead to microscopic expressions for self- and mutual-diffusion coefficients in the coupled density-temperature diffusion equations. Approximations for reducing the general two-component description of the electron-hole plasma (EHP) to a single-component one are discussed. In particular, we show that a special single-component reduction is possible when e-h scattering dominates over c-LO phonon scattering. The ambipolar diffusion approximation is also discussed and we show that the ambipolar diffusion coefficients are independent of e-h scattering, even though the diffusion coefficients of individual components depend sensitively on the e-h scattering rates. Our discussions lead to new perspectives into the roles played in the single-component reduction by the electron-hole correlation in momentum space induced by scatterings and the electron-hole correlation in real space via internal static electrical field. Finally, the theory is completed by coupling the diffusion equations to the lattice temperature equation and to the effective optical polarization which in turn couples to the laser field.

Li, Jianzhong↗
NASA NTRS1992

A new nonlinear diffusion formalism in a magnetized plasma - Application to space physics and astrophysics

A novel diffusion formalism that takes into account the finite width of resonances is presented. The resonance diagram technique is shown to reproduce the details of the particle orbits very accurately, and can be used to determine the acceleration/scattering in the presence of a given wave spectrum. Ways in which the nonlinear orbits can be incorporated into the diffusion equation are shown. The resulting diffusion equation is an extension of the Q-L theory to cases where the waves have large amplitudes and/or are coherent. This new equation does not have a gap at 90 deg in cases where the individual orbits can cross the gap. The conditions under which the resonance gap at 90-deg pitch angle exits are also examined.

Karimbadi, H.↗