Explicit Finite Difference Approximation Research Articles

Analysis and Computation with Stratified Fluid Models

Vertical averaging of the three-dimensional incompressible Euler equations leads to several reduced dimension models of flow over topography, including the one-layer and two-layer classic shallow water equations, and the one-layer and two-layer nonhydrostatic Green–Naghdi equations. These equations are derived and their well-posedness is discussed. Several implicit and explicit finite difference approximations of both the shallow water and Green–Naghdi models are presented, but for Green–Naghdi these are obtained using automatic code generation software. Numerical results are given in both well-posed and ill-posed regimes and compared with computations obtained by others.

Dynamic Thermal Stresses in a Finite Circular Plate with a Penny-Shaped Crack Generated by Impulsive Heating

A solution is presented for the stress-wave response of a partially transparent finite elastic circular plate with a penny-shaped crack subjected to impulsive electromagnetic radiation. The radiation is assumed to occur at a constant rate for the duration of the pulse, to be deposited with a radial Gaussian distribution and to diminish exponentially with distance from the exposed surface of the plate. The development of the analysis is based on the equations of uncoupled dynamic thermoelasticity with heat conduction neglected. The numerical procedure employs explicit finite difference approximations with second-order accuracy based on the integration of the governing equations along the bicharacteristics. Numerical calculations are carried out for the dynamic behavior of the thermal stresses and the stress intensity factors, and the results are shown in figures.

Instability and stability of numerical approximations to discrete velocity models of the Boltzmann equation

We study a standard, explicit finite difference approximation of the 2-D Broadwell model and construct a numerical solution with the sum-norm growing in time faster than any polynomial. Our construction is based on a structure of a self-similar fractal!

Asymmetric Dynamic Thermal Stresses in a Hollow Circular Cylinder Caused by Sudden Heating.

Two-dimensional asymmetric dynamic thermal stresses in a hollow circular cylinder subjected to a sudden time-dependent temperature field are studied numerically. The method employs the explicit finite difference approximations with second-order accuracy based on the integration of the governing equations along the bicharacteristics. Numerical calculations are carried out to analyze the propagation and reflection of thermal stress waves in a hollow circular cylinder subjected to sudden internal heat generation caused by the absorption of gamma-ray or electromagnetic radiation. The rate of heat generation is assumed to be constant for the duration of the pulse and to diminish exponentially with distance from the surface of cylinder and is an arbitary function of the angle around the axis of the cylinder. Heating time is accounted for ; however, heat conduction is ignored. Several numerical results are presented that illustrate the significance of the dynamic effects and their dependence on the coordinates and heating time.

A finite-difference solution to a two-dimensional vibrating membrane problem using a spreadsheet program

A complete 123 macro that finds the numerical solution of a two-dimensional vibrating membrane problem is described. The numerical method is based on an explicit finite difference approximation. The 123 program allows initial and boundary conditions to be entered directly into the spreadsheet and the numerical solution is obtained by pressing a single key. This important feature will allow the user to easily experiment the model problem with various initial and boundary conditions.

Dynamic Thermal Stresses in a Finite Circular Plate with a Penny-Shaped Crack Generated by Impulsive Heating.

A solution is presented for the stress-wave response of a partially transparent finite elastic circular plate with a penny-shaped crack subjected to electromagnetic radiation. The radiation is assumed to occur at a constant rate for the duration of the pulse and to be deposited with a radial Gaussian distribution and to diminish exponentially with distance from the surface of the plate. The development is based on the equations of uncoupled dynamic thermoelasticity with heat conduction neglected. The numerical procedure employs the explicit finite-difference approximations with second-order accuracy based on the integration of the governing equations along the bicharacteristics. Numerical calculations are carried out for the dynamic behavior of the thermal stresses and the stress intensity factors, and the results are shown in figures.

The propagation and reflection of thermal stress waves in anisotropic nonhomogeneous hollow cylinders and spheres

The propagation and reflection of thermal stress waves in anisotropic nonhomogeneous hollow cylinders and spheres subjected to radially symmetric time-dependent temperature fields are investigated. The material of the structure is assumed to be orthotropic with cylindrical or spherical anisotropy and, in addition, is continuously nonhomogeneous with thermal and mechanical properties varying along the radius. The curvilinear characteristics in the space-time plane are transformed into straight lines of equal slope so that the numerical errors can be minimized. The problem is then solved using appropriate characteristic relations on boundaries while using more convenient explicit finite-difference approximations at all other intermediate points in the transformed space-time plane. The physically interesting problem of an anisotropic nonhomogeneous hollow cylinder subjected to sudden uniform change in temperature of its internal boundary is considered in detail. The numerical results for the dynamic thermal stresses are shown in figures, and the effects of anisotropic nonhomogeneous material properties on the dynamic thermal stresses are discussed briefly.

The Hopscotch Algorithm for Three‐dimensional Simulation

This paper presents results obtained for the solution of the three‐dimensional ground‐water flow and mass‐transport equations using the Hopscotch algorithm, an iterative technique that alternates explicit and implicit finite difference approximations of the governing equations. While the method is not new, it has been mostly ignored, particularly in three dimensions. Three application examples are presented: (1) Contaminant transport in a hypothetical, steady‐state, saturated ground‐water flow system; (2) infiltration of a solute in an unsaturated soil; and (3) advection and advection‐diffusion of a Gaussian hill in rotational flow fields. The results indicate that the Hopscotch algorithm is an intriguing and powerful alternative to SIP, SOR, and conjugate gradient techniques in three dimensions. Certainly it warrants more research.

Dynamic Thermal Stresses in Anisotropic Nonhomogeneous Hollow Circular Cylinder.

A numerical study is made of the transient, uncoupled dynamic thermal stresses in a hollow circular cylinder subjected to a sudden change in temperature on its internal boundary. The material of the structure is orthotropic with cylindrical anisotropy and, in addition, is continuously in-homogeneous with mechanical properties varying along the radius. The curvilinear characteristics in the space-time plane are transformed into straight lines of equal slope so that the numerical errors can be minimized. The problem is then solved using appropriate characteristic relations on boundaries while using more convenient explicit finite-difference approximations at all other points in the transformed space-time plane. The numerical results show clearly the propagation of discontinuities in stresses due to step heating on the internal boundary. The effects of the anisotropic non-homogeneous material properties on the dynamic stress distributions are discussed briefly.

Dynamic Thermal Stresses in Composite Hollow Spheres and Circular Cylinders.

A numerical study is made of the transient, uncoupled dynamic thermal stresses in spherical and cylindrical bodies subjected to arbitrarily prescribed symmetric temperature fields. The numerical procedure employs appropriate characteristic relations on boundaries and on interfaces between media with different material properties while using more convenient explicit finite difference approximations at all other points. The solutions of the specific examples calculated show excellent agreement with existing solutions by other methods. Numerical examples are also given for the stress wave propagation in composite hollow spheres and circular cylinders subjected to step heating on the inner boundary.

Low Rayleigh number flow in a heat generating porous media

A numerical analysis has been performed for the steady-state temperature and stream function distributions in a short cylinder, having an isothermal side and top, an insulated bottom, for a uniform heat generating porous medium. The analysis uses the stream function formulation of Darcy's equation in cylindrical coordinates and the Boussinesq approximation. A single energy equation was used for the fluid and solid, since conduction was the expected mode of heat transfer at low heat generation rates for a lead sphere air porous media system. The solution of the non-dimensionalized momentum and energy equations resulted in small Rayleigh numbers (2×10 −6 to 0.2) indicating the heat transfer is by conduction. Solutions for the stream lines and isotherms were obtained using a transient explicit finite-difference approximation using a mean bed thermal conductivity.

Soil Temperature in a Row Crop with Incomplete Surface Cover

AbstractA method is presented to predict soil temperature distribution between rows of crops that partially shade the soil surface. The method combines a model for predicting time and location of soil surface shading with a model for two‐dimensional soil heat transfer. The track of shadow model is general, and requires only plant shape parameters, field location coordinates, row spacing, and row direction to predict soil surface shading as a function of time. The soil heat transfer model is an explicit finite difference approximation of the transient conduction equation, allowing for time and space dependence of soil thermal properties. The surface boundary temperatures are computed by partitioning the surface net radiation into the sensible air, latent, and ground heat flux components. Standard meteorological inputs required to compute the energy partitioning are global radiation, air temperature and humidity, and windspeed. Observed and computed temperature distributions show agreement for a 1‐d simulation. To predict surface energy partitioning and soil temperature accurately for longer periods of time, the method must be expanded to account for water transport.

A new explicit method for the solution of

An explicit finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, non-homogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems were compared with the exact solution and agreed closely with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in satisfactory agreement with the exact solution.

Solution for Radial-Flow with Nonlinear Adsorption

A nonlinear convective diffusion equation is obtained for a radial flow with Freundlich equilibrium adsorption. Numerical solution by explicit or implicit finite difference approximation requires large computation time and suffers from the error due to numerical dispersion. The equation is solved by an efficient and higher order finite difference approximations, i.e., the Crank-Nicolson method. The solution is presented for the movement of pollutants from a point source. The numerical solution is useful for studying the spatial and temporal movement of the chemicals from contaminated water.

Stability analysis of rotational Couette flow of stratified fluids

A linear stability analysis is used to investigate the stability of rotational Couette flow of sbratified fluids. The linearized time-dependent perturbation equations are solved using explicit finite-difference approximations. Small random axisymmetric perturbations of a given wavelength are initially distributed in the flow field, and their development in time is obtained by numerical integration. It is found that the kinetic energy of the perturbations oscillates in time owing to the periodic transformation of the disturbance flow field from a one-vortex system to a two-vortex system and vice versa. The neutral condition is defined as the state in which the maxima of the perturbation kinetic energy curve no longer change in time. A neutral-stability curve is obtained using the experimentally observed critical wavelengths. It is in general agreement with the experimental data, and it confirms the experimental result that stable density stratification enhances stability.

Numerical experiments on time-dependent rotational Couette flow

The flow induced by impulsively starting the inner cylinder in a Couette flow apparatus is investigated by using a nonlinear analysis. Explicit finite-difference approximations are used to solve the Navier–Stokes equations for axisymmetric flows. Random small perturbations are distributed initially and periodic boundary conditions are applied in the axial direction over a length which, in general, is chosen to be the critical wavelength observed experimentally. Simultaneous occurrence of Taylor vortices is obtained at supercritical Reynolds numbers. The development of streamlines, perturbation velocity components and the kinetic energy of the perturbations is examined in detail. Many salient features of the physical flow are observed in the numerical experiments.

QNH: A Portable, Parallel, Multi-scale Quasi Non-hydrostatic Meteorological Model

QNH is a Quasi Non-Hydrostatic meteorological model which has been created by A. E. MacDonald and under development for three years at the NOAA Forecast Systems Laboratory. QNH is well-posed and therefore runs with a minimum of dissipation and it is applicable to all atmospheric scales, from the small scale to the planetary scale. Since QNH uses explicit finite difference approximations in all three spatial dimensions, it is readily amenable to massively parallel processing. We have written the code in such a way that it is portable and achieves high performance on a wide range of serial, vector and parallel computers. In this paper we describe the QNH model and code. We give performance results for the generation of a hurricane meteorological test case, on several massively parallel processors: the Intel Paragon, Cray T3D and IBM SP2.

On the Solution of the Diffusion Equations by Numerical Methods

An explicit-finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, nonhomogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems are compared with the exact solution and with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in closer agreement with the exact solution than are those obtained by the other methods.

A SURVEY OF FINITE-DIFFERENCE APPROXIMATIONS TO THE PRIMITIVE EQUATIONS

The linearized primitive equations for a viscous barotropic fluid in a nonrotating frame are used to investigate the stability properties and the accuracy of several explicit finite difference approximations. Four different schemes are described and their qualities examined. For two schemes the exact numerical solution was derived and compared with the true solution of the differential system. Actual computations are performed and the errors in phase and amplitude evaluated to test the theoretical results.

On the stability of certain finite difference approximations to parabolic systems of differential equations

Abstract : Certain simple explicit finite difference approximations to the solution of the initial value problem are considered for a uniformly parabolic system of linear differential equations with variable coefficients. The problem is to determine conditions which will guarantee that the approximating difference problem is well-posed. Since explicit difference equations involving a finite number of lattice points are used, the existence of a unique solution of the difference problem is trivial. Primarily, the concern is with the dependence of the solution of the difference problem on its data. (Author)