Variable mesh multistep methods for ordinary differential equations
Variable mesh multistep predictor-corrector method for iterative solution of ordinary differential equations, considering numerical stability and algorithm efficiency
Van Wyk, R.↗
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Variational differential equations for engineering type trajectories close to a planet with an atmosphere
The differential equations for the adjoint variables are derived and coded in FORTRAN. The program is written in a form to either take into account or neglect thrust, aerodynamic forces, planet rotation and oblateness, and altitude dependent winds.
Dickmanns, E. D.↗
Bifurcation and stability for a nonlinear parabolic partial differential equation
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
Chafee, N.↗
Parallel Algorithm Solves Coupled Differential Equations
Numerical methods adapted to concurrent processing. Algorithm solves set of coupled partial differential equations by numerical integration. Adapted to run on hypercube computer, algorithm separates problem into smaller problems solved concurrently. Increase in computing speed with concurrent processing over that achievable with conventional sequential processing appreciable, especially for large problems.
Hayashi, A.↗
Differential geometry techniques for sets of nonlinear partial differential equations
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Estabrook, Frank B.↗
On the removal of boundary errors caused by Runge-Kutta integration of non-linear partial differential equations
It has been previously shown that the temporal integration of hyperbolic partial differential equations (PDE's) may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic (PDE's) (linear and non-linear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.
Abarbanel, Saul↗
Spurious Numerical Solutions Of Differential Equations
Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.
Lafon, A.↗
Stability theory of nonlinear operational differential equations in Hilbert space
Existence, uniqueness, and stability criteria for linear and nonlinear operational differential equations in Banach and Hilbert spaces
Pao, C.-V.↗
Monte Carlo solution of partial differential equations using a hybrid computer
Analog-hybrid Monte Carlo method for solving partial differential equation on hybrid computer
Handler, H.↗
Averaging method for the solution of nonlinear differential equations with periodic nonharmonic solutions
Averaging technique using Jacobi elliptic functions for solving nonlinear differential equations with periodic nonharmonic solutions
Cap, F. F.↗
Control of functional differential equations to target sets in function space
Optimal control of systems governed by functional differential equations of retarded and neutral type is considered. Problems with function space initial and terminal manifolds are investigated. Existence of optimal controls, regularity, and bang-bang properties are discussed. Necessary and sufficient conditions are derived, and several solved examples which illustrate the theory are presented.
Banks, H. T.↗
A comparison of digital computer programs for the numerical solution of ordinary differential equations
Recently the determination of the best technique for numerically solving systems of ordinary differential equations on a digital computer has received much attention. The use of these formulas in conjunction with a stepsize control developed is explained, and one of the formulas is chosen for comparison with other integration techniques. This comparison of one of the best of Fehlberg's formulas with the different numerical techniques described in previous studies on a variety of test problems clearly shows the superiority of Fehlberg's formula. That is, on each of the test problems, the chosen Fehlberg formula is able to achieve a given accuracy in less computer time than any of the other techniques tested. Also, the computer program for the chosen Fehlberg formula is less complex and easier to use than the computer programs for most of the other techniques. To illustrate the use of the chosen Fehlberg formula, a computer listing of its application to several example problems is included along with a sample of the computer output from these applications.
Ingram, H. L.↗
On approximating hereditary dynamics by systems of ordinary differential equations
The paper deals with methods of obtaining approximate solutions to linear retarded functional differential equations (hereditary systems). The basic notion is to project the infinite dimensional space of initial functions for the hereditary system onto a finite dimensional subspace. Within this framework, two particular schemes are discussed. The first uses well-known piecewise constant approximations, while the second is a new method based on piecewise linear approximating functions. Numerical results are given.
Cliff, E. M.↗
Applying integrals of motion to the numerical solution of differential equations
A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
Jezewski, D. J.↗
Applying integrals of motion to the numerical solution of differential equations
A method is developed for using the integrals of systems of nonlinear, ordinary, differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scalar or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
Vezewski, D. J.↗
Solution of partial differential equations on vector and parallel computers
The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.
Ortega, J. M.↗
Solution of partial differential equations on vector and parallel computers
The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.
Ortega, J. M.↗
A method for exponential propagation of large systems of stiff nonlinear differential equations
A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.
Friesner, Richard A.↗