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Approximation schemes employing cubic splines in the context of a linear semigroup framework are developed for both parabolic and hyperbolic second-order partial differential equation parameter estimation problems. Convergence results are established for problems with linear and nonlinear systems, and a summary of numerical experiments with the techniques proposed is given.
Three second-order accurate numerical methods - MacCormack's method, Lambda scheme and Gabutti scheme - are introduced to solve the quasi-linear, hyperbolic partial differential equations describing transient flows in closed conduits. The details of these methods and the treatment of boundary conditions are presented and the results computed by using these methods for a typical piping system are compared. It is shown that for the same accuracy, second-order methods require considerably lesser number of computational nodes and computer time as compared to those required by the first-order methods.
A one dimensional model for studying the thermal dynamics of cooling lakes was developed and verified. The model is essentially a set of partial differential equations which are solved by finite difference methods. The model includes the effects of variation of area with depth, surface heating due to solar radiation absorbed at the upper layer, and internal heating due to the transmission of solar radiation to the sub-surface layers. The exchange of mechanical energy between the lake and the atmosphere is included through the coupling of thermal diffusivity and wind speed. The effects of discharge and intake by power plants are also included. The numerical model was calibrated by applying it to Cayuga Lake. The model was then verified through a long term simulation using Lake Keowee data base. The comparison between measured and predicted vertical temperature profiles for the nine years is good. The physical limnology of Lake Keowee is presented through a set of graphical representations of the measured data base.
A finite element algorithm is derived for the numerical solution of a three-dimensional flow field described by a system of initial-valued, elliptic boundary value partial differential equations. The familiar three-dimensional boundary layer equations belong to this description when diffusional processes in only one coordinate direction are important. The finite element algorithm transforms the original description into large order systems of ordinary differential equations written for the dependent variables discretized at node points of an arbitrarily irregular computational lattice. The generalized elliptic boundary conditions is piecewise valid for each dependent variable on boundaries that need not explicitly coincide with coordinate surfaces. Solutions for sample problems in laminar and turbulent boundary flows illustrate favorable solution accuracy, convergence, and versatility.
Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.
There is considerable interest in developing a numerical scheme for solving the time dependent viscous compressible three dimensional flow to aid in the design of helicopter rotors. Numerical algorithms are examined to determine their overall suitability for the efficient and routine solution of an appropriate system of partial differential equations. It is concluded that a consistently split time linearized block implicit scheme using either quintic B spline collocation or the generalized operator compact implicit approach to generate a fourth order accurate algorithm is particularly well suited for use on the present problem. High cell Reynolds number behavior leads to favoring the generalized operator compact implicit approach over the quintic B spline collocation method.
A parallel multigrid algorithm for solving elliptic partial differential equations is developed and evaluated. A V-cycle multigrid method is altered to increase the degree of parallelism. A numerical analysis of the resulting concurrent-iteration multigrid algorithm is performed; its architectural implications are considered; highly parallel systems without shared memory are examined (including mesh-connected arrays, mesh-shuffle-connected systems, permutation networks, and direct VLSI embeddings); and the results of numerical experiments are presented in tables and graphs.
It is shown how the complete field of the electron beam may be incorporated into the transmission line model theory of the traveling wave tube (TWT). The fact that the longitudinal component of the field due to the bunched beam is not used when formulating the beam-to-circuit coupling equation is not well-known. The fundamental partial differential equation for the traveling wave field is developed and compared with the older (now standard) one. The equation can be solved numerically using the same algorithms, but now the coefficients can be updated continuously as the calculation proceeds down the tube. The coefficients in the older equations are primarily derived from preliminary measurements and some trial and error. The newer coefficients can be found by a recursive method, since each has a well defined physical interpretation and can be calculated once a reasonable first trial solution is postulated. The results of the new expression were compared with those of the older forms, as well as to a field theory model to show the ease in which a reasonable fit to the field prediction is obtained. A complete summary of the existing transmission line modeling of the TWT is given to explain the somewhat vague ideas and techniques in the general area of drifting carrier-traveling circuit wave interactions. The basic assumptions and inconsistencies of the existing theory and areas of confusion in the general literature are examined and hopefully cleared up.
Particle-in-cell (PIC) simulations and Zakharov's partial differential equations (PDEs) are used to investigate the collapse of isolated Langmuir wave packets in two dimensions. Collapse thresholds are determined numerically, and the roles of enhanced Langmuir wave damping and nonlinearities not included in the standard Zakharov equations are discussed. The Langmuir wave and ion dynamics in PIC simulations are compared with the predictions of PDE simulations incorporating enhanced Langmuir damping. Electron heating and coherent acceleration in the PIC simulations are discussed and compared with predictions of the transit-time theory.
Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eignevalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: the interior of a spherical triangle, and the exterior of a great circle arc extending for less than pi radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.
A new numerical method is presented for the analysis of combined natural convection and radiation heat transfer with applications in many engineering situations such as materials processing, combustion and fire research. Because of the recent interest in the low gravity environment of space, attention is devoted to both 1-g and low-g applications. The two-dimensional mathematical model is represented by a set of coupled nonlinear integro-partial differential equations. Radiative exchange is formulated using the Discrete Exchange Factor method (DEF). This method considers point to point exchange and provides accurate results over a wide range of radiation parameters. Numerical results show that radiation significantly influences the flow and heat transfer in both low-g and 1-g applications. In the low-g environment, convection is weak, and radiation can easily become the dominant heat transfer mode. It is also shown that volumetric heating by radiation gives rise to an intricate cell pattern in the top heated enclosure.
The study of ray trajectories of plasma waves in a torodial geometry using MACSYMA is an example of how symbolic, numerical, and graphical facilities can be used in concert to accomplish a complex computational goal. Computational features of this study which are of particular significance include: the derivation of code (i.e. writing functions to generate program fragments), the use of array functions to simplify the specification of a numerical iteration scheme, and the graphical presentation of the results. Mathematically, this study originates in the solution of a linear inhomogeneous partial differential equation in 3 dimensions by the method of characteristics. While it is possible to describe this equation compactly by using vector notation, and by specifying the spatial variation of the coefficients in terms of intermediate parameters, the transformation of the equation into a form amenable to solution is very tedious. A MACSYMA program is presented for obtaining description of the rf field structure excited by a waveguide located at the edge of a toroidal plasma confinement device.
A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.
The solution of partial differential equations by means of either implicit or conservative explicit finite-difference methods, on the one hand, or multicomponent overlapping grid schemes on the other, is presently subjected to interpolation at grid boundaries in an attempt to eliminate the time lag. The accuracy of the transient solution of a hyperbolic or parabolic partial differential equation is dependent on the procedures used to transfer information between grids in a composite grid system. The present techniques reduce numerical solution errors.
This report compiles the various research activities conducted under the auspices of the NASA Grant NAG3-1026, "Numerical Investigation of Hot Gas Ingestion by STOVL Aircraft" during the period of April 1989 to April 1994. The effort involved the development of multigrid based algorithms and computer programs for the calculation of the flow and temperature fields generated by Short Take-off and Vertical Landing (STOVL) aircraft, while hovering in ground proximity. Of particular importance has been the interaction of the exhaust jets with the head wind which gives rise to the hot gas ingestion process. The objective of new STOVL designs to reduce the temperature of the gases ingested into the engine. The present work describes a solution algorithm for the multi-dimensional elliptic partial-differential equations governing fluid flow and heat transfer in general curvilinear coordinates. The solution algorithm is based on the multigrid technique which obtains rapid convergence of the iterative numerical procedure for the discrete equations. Initial efforts were concerned with the solution of the Cartesian form of the equations. This algorithm was applied to a simulated STOVL configuration in rectangular coordinates. In the next phase of the work, a computer code for general curvilinear coordinates was constructed. This was applied to model STOVL geometries on curvilinear grids. The code was also validated in model problems. In all these efforts, the standard k-Epsilon model was used.
The mathematical background regarding the moving finite element (MFE) method of Miller and Miller (1981) is discussed, taking into account a general system of partial differential equations (PDE) and the amenability of the MFE method in two dimensions to code modularization and to semiautomatic user-construction of numerous PDE systems for both Dirichlet and zero-Neumann boundary conditions. A description of test problem results is presented, giving attention to aspects of single square wave propagation, and a solution of the heat equation.
Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergence rate of spectral schemes. The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems, as well as the efficient solution of the resulting discrete equations by rapidly convergent tensor-product-based iterative procedures. Several examples of spectral element simulation of moderate Reynolds number unsteady flow in complex geometry are presented.
A mathematical theory and an accompanying numerical scheme have been developed for predicting the oxidation behavior of C/SiC composite structures. The theory is derived from the mechanics of the flow of ideal gases through a porous solid. Within the mathematical formulation, two diffusion mechanisms are possible: (1) the relative diffusion of one species with respect to the mixture, which is concentration gradient driven and (2) the diffusion associated with the average velocity of the gas mixture, which is total gas pressure gradient driven. The result of the theoretical formulation is a set of two coupled nonlinear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations must be solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of space and time. The local rate of carbon oxidation is determined as a function of space and time using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The nonlinear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual method, allowing for the solution of the differential equations numerically. The end result is a numerical scheme capable of determining the variation of the local carbon oxidation rates as a function of space and time for any arbitrary C/SiC composite structures.