Generalized relaxation methods applied to problems in transonic flow
Generalized relaxation methods application to transonic flow problems, combining with numerical integration theory for ordinary differential equations
Lomax, H.↗
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Dynamic characteristics of a variable-mass flexible missile
The general motion of a variable mass flexible missile with internal flow and aerodynamic forces is considered. The resulting formulation comprises six ordinary differential equations for rigid body motion and three partial differential equations for elastic motion. The simultaneous differential equations are nonlinear and possess time-dependent coefficients. The differential equations are solved by a semi-analytical method leading to a set of purely ordinary differential equations which are then solved numerically. A computer program was developed for the numerical solution and results are presented for a given set of initial conditions.
Meirovitch, L.↗
On the stability of numerical integration routines.
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
Glover, K.↗
Continuation Methods for Qualitative Analysis of Aircraft Dynamics
A class of numerical methods for constructing bifurcation curves for systems of coupled, non-linear ordinary differential equations is presented. Foundations are discussed, and several variations are outlined along with their respective capabilities. Appropriate background material from dynamical systems theory is presented.
Cummings, Peter A.↗
Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study
One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.
Turpin, Jason B.↗
Legendre-Tau approximations for functional differential equations
The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.
Ito, K.↗
Legendre-tau approximations for functional differential equations
The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.
Ito, K.↗
Stable boundary approximations for a class of implicit schemes for the one-dimensional inviscid equations of gas dynamics
The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations for initial-boundary-value problems of the hyperbolic type. For the numerical experiments, select the one-dimensional inviscid gas-dynamic equations in conservation-law form is selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit versus implicit boundary schemes, unconditionally stable schemes, and underspecification or overspecification of boundary conditions.
Yee, H. C.↗
Error behavior of multistep methods applied to unstable differential systems
The problem of modeling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulas. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulas and two different stepsizes to illustrate the appropriateness of the criteria.
Brown, R. L.↗
Error behaviour of multistep methods applied to unstable differential systems
The problem of modelling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulae. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulae and two different stepsizes to illustrate the appropriateness of the criteria.
Brown, R. L.↗
Neural network error correction for solving coupled ordinary differential equations
A neural network is presented to learn errors generated by a numerical algorithm for solving coupled nonlinear differential equations. The method is based on using a neural network to correctly learn the error generated by, for example, Runge-Kutta on a model molecular dynamics (MD) problem. The neural network programs used in this study were developed by NASA. Comparisons are made for training the neural network using backpropagation and a new method which was found to converge with fewer iterations. The neural net programs, the MD model and the calculations are discussed.
Shelton, R. O.↗
A deterministic particle method for one-dimensional reaction-diffusion equations
We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.
Mascagni, Michael↗
Multistep integration formulas for the numerical integration of the satellite problem
The use of two Class 2/fixed mesh/fixed order/multistep integration packages of the PECE type for the numerical integration of the second order, nonlinear, ordinary differential equation of the satellite orbit problem. These two methods are referred to as the general and the second sum formulations. The derivation of the basic equations which characterize each formulation and the role of the basic equations in the PECE algorithm are discussed. Possible starting procedures are examined which may be used to supply the initial set of values required by the fixed mesh/multistep integrators. The results of the general and second sum integrators are compared to the results of various fixed step and variable step integrators.
Lundberg, J. B.↗
Numerical integration of ordinary differential equations of various orders
Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.
Gear, C. W.↗
Finite element or Galerkin type semidiscrete schemes
A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.
Durgun, K.↗
Nonstandard finite difference schemes
The major research activities of this proposal center on the construction and analysis of nonstandard finite-difference schemes for ordinary and partial differential equations. In particular, we investigate schemes that either have zero truncation errors (exact schemes) or possess other significant features of importance for numerical integration. Our eventual goal is to bring these methods to bear on problems that arise in the modeling of various physical, engineering, and technological systems. At present, these efforts are extended in the direction of understanding the exact nature of these nonstandard procedures and extending their use to more complicated model equations. Our presentation will give a listing (obtained to date) of the nonstandard rules, their application to a number of linear and nonlinear, ordinary and partial differential equations. In certain cases, numerical results will be presented.
Mickens, Ronald E.↗
On testing a subroutine for the numerical integration of ordinary differential equations
This paper discusses how to numerically test a subroutine for the solution of ordinary differential equations. Results obtained with a variable order Adams method are given for eleven simple test cases.-
Krogh, F. T.↗
On testing a subroutine for the numerical integration of ordinary differential equations
This paper discusses how to numerically test a subroutine for the solution of ordinary differential equations. Results obtained with a variable order Adams method are given for eleven simple test cases.-
Krogh, F. T.↗