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A numerical study of hypersonic forebody/inlet integration problem is presented in the form of the view-graphs. The following topics are covered: physical/chemical modeling; solution procedure; flow conditions; mass flow rate at inlet face; heating and skin friction loads; 3-D forebogy/inlet integration model; and sensitivity studies.

We study the numerical computation of an expectation of a bounded function with respect to a measure given by a non-normalized density on a convex body. We assume that the density is log-concave, satisfies a variability condition and is not too narrow. We consider general convex bodies or even the whole $\R^d$ and show that the integration problem satisfies a refined form of tractability. The main tools are the hit-and-run algorithm and an error bound of a multi run Markov chain Monte Carlo method.

We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We additionally, however, also consider invariant preservation over persistent, simultaneous and/or frequent boundary interactions. Appropriately formulating geometric methods to include such conditions has long-remained challenging due to the inherent nonsmoothness they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Leveraging discrete variational techniques, we construct a family of geometric numerical integration methods that not only obtain the usual desirable properties of momentum preservation, approximate energy conservation and equality constraint preservation, but also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.

Using a spherical finite-difference grid system covering the hemisphere, numerical solutions of the non-divergent barotropic vorticity equation have be n obtained for periods up to 10 days by rela ation methods. In the case of analytic initial conditions, the solutions are sensitive to abrupt changes of the longitudinal mesh size in the regions of appreciable amplitude of the streamfunction tendency. A doubling of the longitudinal mesh size from 5 to 10 deg at 45 deg N produces a progressive tearing or shearing in the stream function which is quite noticeable at 10 days. When the grid variation is removed to 70 degrees N, however, the solutions at 10 day are free of this defect, although exhibiting the expected phase lag relative to the analytic solution. When applied to hemispherical barotropic prediction with actual 500 mb data, this spherical grid scheme appears to perform as satisfactorily as do conventional (rectangular) grid formulations.

We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.

Orbit equations with a set of conservative and a set of nonconservative perturbing potentials were considered. Scheifele's DS formulation of these equations has dependent variables similar to Delaunay's orbital elements with the true anomaly as the independent variable. Efficiency curves of computing cost v.s. accuracy were constructed for Adams integrators of order of 2 through 15 with several correcting algorithms and for a Runga-Kutta integrator. Considering stability regions, choices were made for the optimally efficient integration modes for the DS elements. Integrating in these modes reduces computing costs for a specified accuracy.

A converse theorem using Liapunov functions is used to obtain upper error bounds in optimal estimates of perturbed two body problems.

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.

A quasi-one dimensional, nonequlibrium, streamtube model and finite difference computer code has been developed to study nonisentropic supercritical, real gas flows. A system of unsteady Euler equations is integrated to a steady state solution utilizing MacCormack's explicit/implicit numerical scheme. Implcit characteristic boundary conditions are employed at the subsonic inlet to enable aut omatic solution of inlet conditons consistent with the critical mass flow rate. A numerical damping technique for the explicity evaluated reaction rate has been developed such that nonequilibrium flows with large reaction rate coefficients have been solved numerically with significant reduction in computation time.

We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We study the consistency of an alternative estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the right-hand side of the system applied to a nonparametrically estimated solution. This smooth and match estimator (SME) bypasses numerical integration altogether and reduces the amount of computational time drastically compared to ordinary least squares. Moreover, we show that under suitable regularity conditions this smooth and match estimation procedure leads to a $\sqrt{n}$-consistent estimator of the parameter of interest.

The numerical integration of the equations of motion for missiles and shell is a frequently occurring computational problem in government laboratories and in the aerospace industry. Because of the repetitive nature of such computations and the continuing requirement, it is desirable to make these computations as efficiently as possible in order to minimize the computer time spent in their solution. The cost in terms of time required to solve representative problems to a specified accuracy is a measure of the relative efficiency of numerical integration methods and can be determined only experimentation. This report describes the results of such an experimental study and finds that the Kutta-Merson procedure with automatic and continuous interval adjustment is far superior to the predictor-corrector techniques. (Author)

In the field of numerical integration, methods specially tuned on oscillating functions, are of great practical importance. Such methods are needed in various branches of natural sciences, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. Among others, probably the most important tool used to construct efficient methods for oscillatory problems is the exponential (trigonometric) fitting. The basic characteristic of these methods is that their phase lag vanishes at a predefined frequency. In this work, we introduce a new tool which improves the behavior of exponentially fitted numerical methods. The new technique is based on the vanishing of the first derivatives of the phase lag function at the fitted frequency. It is proved in the text that these methods present improved characteristics in oscillatory problems.

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$.

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to the non-integrable case.

Bell System Technical Journal, 60: 8. October 1981 pp 1927-1940. Numerical Integration of Stochastic Differential Equations - II. (Greenside, H.S.; Helfand, E. )

We consider a continuum phase field model for crystal growth via molecular beam epitaxy, with the goal of determining stable numerical time integration methods for the dynamics. We parametrize a class of semi-implicit methods that are linear in the updated field, which allows for efficient implementation with fast Fourier transforms. We perform unconditional von Neumann stability analysis to identify the region of stability in parameter space, and then test these predictions numerically for gradient stability. We find strong agreement between the approaches.

We develop a numerical method for pricing multidimensional vanilla options in the Black-Scholes framework. In low dimensions, we improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. In higher dimensions, this new algorithm is used as a control variate after a dimension reduction based on principal component analysis. Numerical tests are performed on the pricing of basket, put on minimum and digital options in dimensions up to ten.

The objective of the present research program is the development of fast numerical multidimensional Fourier techniques applied to a wide range of wave modeling and data analysis problems. Important progress made in the past year has been the development of new methods for extending the nonlinear Fourier approach to arbitrary order. Thus one can now push toward the solution of the Euler and other higher order equations in a more systematic way that requires very little additional central processor time.

In the finite difference method which is commonly used in computational micromagnetics, the demagnetizing field is usually computed as a convolution of the magnetization vector field with the demagnetizing tensor that describes the magnetostatic field of a cuboidal cell with constant magnetization. An analytical expression for the demagnetizing tensor is available, however at distances far from the cuboidal cell, the numerical evaluation of the analytical expression can be very inaccurate. Due to this large-distance inaccuracy numerical packages such as OOMMF compute the demagnetizing tensor using the explicit formula at distances close to the originating cell, but at distances far from the originating cell a formula based on an asymptotic expansion has to be used. In this work, we describe a method to calculate the demagnetizing field by numerical evaluation of the multidimensional integral in the demagnetization tensor terms using a sparse grid integration scheme. This method improves the accuracy of computation at intermediate distances from the origin. We compute and report the accuracy of (i) the numerical evaluation of the exact tensor expression which is best for short distances, (ii) the asymptotic expansion best suited for large distances, and (iii) the new method based on numerical integration, which is superior to methods (i) and (ii) for intermediate distances. For all three methods, we show the measurements of accuracy and execution time as a function of distance, for calculations using single precision (4-byte) and double precision (8-byte) floating point arithmetic. We make recommendations for the choice of scheme order and integrating coefficients for the numerical integration method (iii).

The 3-level leapfrog time integration algorithm is an attractive choice for numerical relativity simulations since it is time-symmetric and avoids non-physical damping. In Newtonian problems without velocity dependent forces, this method enjoys the advantage of long term stability. However, for more general differential equations, whether ordinary or partial, delayed onset numerical instabilities can arise and destroy the solution. A known cure for such instabilities appears to have been overlooked in many application areas. We give an improved cure ("deloused leapfrog") that both reduces memory demands (important for 3+1 dimensional wave equations) and allows for the use of adaptive timesteps without a loss in accuracy. We show both that the instability arises and that the cure we propose works in highly relativistic problems such as tightly bound geodesics, spatially homogeneous spacetimes, and strong gravitational waves. In the gravitational wave test case (polarized waves in a Gowdy spacetime) the deloused leapfrog method was five to eight times less CPU costly at various accuracies than the implicit Crank-Nicholson method, which is not subject to this instability.

OUTLINE: * Linear non-autonomous ODEs with parameters * Toy problem: modified Airy equation * WKB asymptotics and error analysis * The Evans function.

The field method is developed for arbitrary open branch domains subjected to general linear boundary conditions. Although closed branches are within the scope of the method, they are not treated here. The numerical feasibility of the method has been demonstrated by implementing it in a computer program for the linear static analysis of open branch shells of revolution under asymmetric loads. For such problems the field method eliminates the well-known numerical problem of long subintervals associated with the rapid growth of extraneous solutions. Also, the method appears to execute significantly faster than other numerical integration methods.

From the bitsavers.org collectio n, a scanned-in computer-related document. components :: motorola :: appNotes :: AN-0541 Medium Scale Integration In The Numerical Control Field

In the present paper we investigate the performance of explicit splitting schemes and related techniques applied to a rigid body model subject to a stochastic torque and random perturbations in the inertia tensor. Results are discussed and compared with traditional solvers for such model.

The design of a very fast, automatic black-box code for homogeneous, gas-phase chemical kinetics problems requires an understanding of the physical and numerical sources of computational inefficiency. Some major sources reviewed in this report are stiffness of the governing ordinary differential equations (ODE's) and its detection, choice of appropriate method (i.e., integration algorithm plus step-size control strategy), nonphysical initial conditions, and too frequent evaluation of thermochemical and kinetic properties. Specific techniques are recommended (and some advised against) for improving or overcoming the identified problem areas. It is argued that, because reactive species increase exponentially with time during induction, and all species exhibit asymptotic, exponential decay with time during equilibration, exponential-fitted integration algorithms are inherently more accurate for kinetics modeling than classical, polynomial-interpolant methods for the same computational work. But current codes using the exponential-fitted method lack the sophisticated stepsize-control logic of existing black-box ODE solver codes, such as EPISODE and LSODE. The ultimate chemical kinetics code does not exist yet, but the general characteristics of such a code are becoming apparent.

This paper describes a new class of algorithms for integrating linear second order equations and those containing smooth nonlinearities. The algorithms are based on a combination of ideas from standard Newmark integration methods and extrapolation techniques. For the algorithm to work the underlying Newmark method must be stable second order accurate, and produce asymptotic error expansions for response quantities containing only even ordered terms. It is proved that setting the Newmark parameter Gamma to 1/2 gives a desirable asymptotic expansion, irrespective of the setting for Beta. Numerical experiments are conducted for two linear and two nonlinear applications.

This paper uses logic programming to describe numerical integration in a general way. Axioms are added to the basic logic programs to define specific known numerical integration algorithms. Using a slightly different formalization, we construct logic programs for adaptive Romberg integration and for a new algorithm for adaptive integration. The logic programs provide a formal basis for classifying numerical quadrature algorithms (recall Rice's comment that there are 1,000,000 useful ones). The logic programs are also more understandable than the corresponding programs in conventional programming languages. In terms of logic programming there are several novel points in this paper. The data type, real number, is not definded by a recursive constructor function. Termination is decided by error bounds, rather than by reaching some basis case of the data constructor. The problem itself seems to demand concurrent execution and global variables but in fact is solved in a linear fashion. (Author)

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

We find estimates for the error in replacing an integral $\int f d\mu$ with respect to a fractal measure $\mu$ with a discrete sum $\sum_{x \in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $\Delta f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpi\'{n}ski gasket, both for the standard self-similar measure and energy measures, and for other fractals.

An algorithm is presented for adaptively partitioning a multidimensional coordinate space based on optimization of a scalar function of the coordinates. The goal is to construct a set of hyperrectangular regions, such that the variation of function values within each region is small. These regions are then used as the basis for a stratified sampling estimate of the definite integral of the function. (Author)

We are interested in the following problem: given an open, bounded domain $\Omega \subset \mathbb{R}^2$, what is the largest constant $\alpha = \alpha(\Omega) > 0$ such that there exist an infinite sequence of disks $B_1, B_2, \dots, B_N, \dots \subset \mathbb{R}^2$ and a sequence $(n_i)$ with $n_i \in \left\{1,2\right\}$ such that $$ \sup_{N \in \mathbb{N}}{N^{\alpha}\left\| \chi_{\Omega} - \sum_{i=1}^{N}{(-1)^{n_i}\chi_{B_i}}\right\|_{L^1(\mathbb{R}^2)}} < \infty,$$ where $\chi$ denotes the characteristic function? We prove that certain (somewhat peculiar) domains $\Omega \subset \mathbb{R}^2$ satisfy the property with $\alpha = 0.53$. For these domains there exists a sequence of points $(x_i)_{i=1}^{\infty}$ in $\Omega$ with weights $(a_i)_{i=1}^{\infty}$ such that for all harmonic functions $u:\mathbb{R}^2 \rightarrow \mathbb{R}$ $$ \left|\int_{\Omega}{u(x)dx} - \sum_{i=1}^{N}{a_i u(x_i)}\right| \leq C_{\Omega}\frac{\|u\|_{L^{\infty}(\Omega)}}{N^{0.53}},$$ where $C_{\Omega}$ depends only on $\Omega$. This gives a Quasi-Monte-Carlo method for harmonic functions which improves on the probabilistic Monte-Carlo bound $\|u\|_{L^{2}(\Omega)}/N^{0.5}$ \textit{without} introducing a dependence on the total variation. We do not know which decay rates are optimal.

The double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and Fourier integral. The double exponential transformation is not only useful for numerical computations but it is also used in different methods of Sinc theory. In this paper we give an upper bound estimate for the error of double exponential transformation. By improving integral estimates having singular final points, in theorem 1 we prove that the method is convergent and the rate of convergence is $\mathcal{O}(h^2)$ where h is a step size. Our main tool in the proof is DE formula in Sinc theory. The advantage of our method is that the time and space complexity is drastically reduced. Furthermore, we discovered upper bound error in DE formula independent of N truncated number, as a matter of fact we proved stability. Numerical tests are presented to verify the theoretical predictions and confirm the convergence of the numerical solution.

The error scaling for Markov-Chain Monte Carlo techniques (MCMC) with $N$ samples behaves like $1/\sqrt{N}$. This scaling makes it often very time intensive to reduce the error of computed observables, in particular for applications in lattice QCD. It is therefore highly desirable to have alternative methods at hand which show an improved error scaling. One candidate for such an alternative integration technique is the method of recursive numerical integration (RNI). The basic idea of this method is to use an efficient low-dimensional quadrature rule (usually of Gaussian type) and apply it iteratively to integrate over high-dimensional observables and Boltzmann weights. We present the application of such an algorithm to the topological rotor and the anharmonic oscillator and compare the error scaling to MCMC results. In particular, we demonstrate that the RNI technique shows an error scaling in the number of integration points $m$ that is at least exponential.

A method for integrating numerically through a saddle-point singularity of an ordinary differential equation is presented. The solution of the differential equation is assumed to depend also on the value of a parameter, such as the mass flow. The method is thus applicable to a wide assortment of gas-dynamic problems including one-dimensional nonequilibrium nozzle flow and two-phase nozzle flow. Specific application is made to nonequilibrium nozzle flow, and the results of this application are presented and discussed. The method proved to be numerically accurate without requiring an exceedingly precise estimate for the critical mass flow. The work also includes a modification of a method for calculating approximate equilibrium nozzle flows. (Author)

Calculations of observables in quantum chromodynamics are typically performed using a method that combines numerical integrations over the momenta of final state particles with analytical integrations over the momenta of virtual particles. I describe the most important steps of a method for performing all of the integrations numerically.

In this thesis we present an algorithm for the precise determination of Fermi-Dirac (FD) integral functions, for arbitrary values of the parameter and the argument. The FD integrals are a class of functions that are used extensively in the modeling of semiconductor devices, e.g., when the charge carriers are in a strongly quantum, degenerate regime, such as in heavily doped semiconductors. The determination of FD integrals has a long history. Our approach to evaluating these functions is two-fold. First, we develop exact power series expansions of the integral. These series, however, converge too slowly to be a practical means of evaluating the integral. The second aspect of our approach is to apply numerical series acceleration methods to improve significantly the rate of convergence of these series expansions. The result is a computer program that provides efficient, accurate values of the FD integral.

QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $\alpha$. Using Parseval's identity, this smoothness can be defined for all real numbers $\alpha > 1/2$. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $\alpha$ arbitrarily close to $1/2$. To obtain such estimates we introduce a $\log$-scale for functions with smoothness close to $1/2$, which we call $\log$-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-\mu(1-\lambda)/2})$ for any $1/\mu 1$ is a power in the $\log$-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(\gamma_j)_{j \in \mathbb{N}}$. It is known that if $\sum_{j=1}^\infty \gamma_j^\tau < \infty$ for some $1/(2\alpha) < \tau \le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $\tau$ is close to $1/(2 \alpha)$, namely we show dimension independent error bounds under the condition that $\sum_{j=1}^\infty \gamma_j \max\{1, \log \gamma_j^{-1}\}^{\mu(1-\lambda)} < \infty$ for some $1/\mu < \lambda \le 1$. The essential tool in our analysis is a $\log$-scale Jensen's inequality. The results described above also apply to integration in $\log$-cosine spaces using tent-transformed lattice rules.

Senior secondary students cover numerical integration techniques in their mathematics courses. In particular, students would be familiar with the "midpoint rule," the elementary "trapezoidal rule" and "Simpson's rule." This article derives these techniques by methods which secondary students may not be familiar with and an approach that undergraduate students should be familiar with. Secondary students will also find interesting the "two-point Gauss rule," which is an extension of the trapezoidal rule. There are many applications of integral calculus and developing a deeper understanding of some of the numerical methods will increase understanding of the techniques. The methods chosen in this article have been investigated as secondary students will be familiar with their applications. However, secondary text books and teachers may not use the techniques covered in this article, and this alternate approach may increase the understanding of the importance and applications of the techniques, as well as increase an appreciation of the beauty of mathematics in general. The article also provides a detailed summary of the techniques that will be beneficial for undergraduate students. In this article the midpoint rule, the elementary trapezoidal rule, the two-point Gauss rule and Simpson's elementary rule are developed. (Contains 1 figure.)

The Cuba library provides new implementations of four general-purpose multidimensional integration algorithms: Vegas, Suave, Divonne, and Cuhre. Suave is a new algorithm, Divonne is a known algorithm to which important details have been added, and Vegas and Cuhre are new implementations of existing algorithms with only few improvements over the original versions. All four algorithms can integrate vector integrands and have very similar Fortran, C/C++, and Mathematica interfaces.

Self-starting multistep methods for numerical integration of ordinary differential equations

This letter summarizes some known properties and also presents several new properties of the Numerical Integration (NI) method for time-optimal trajectory planning along a specified path. The contribution is that rigorous mathematical proofs of these properties are presented, most of which cannot be found in existing literatures. We first give some properties regarding switch points and accelerating/decelerating curves of the NI method. Then, for the fact that when kinematic constraints are considered, the original version of NI which only considers torque constraints may result in failure of trajectory planning, we give the concrete failure conditions with rigorous mathematical proof. Accordingly, a failure detection algorithm is given in a run-and-test manner. Some simulation results on a unicycle vehicle are provided to verify those presented properties. Note that though those known properties are not discovered first, their mathematical proofs are given first in this letter. The detailed proofs make the theory of NI more complete and help interested readers to gain a thorough understanding of the method.

The paper studies several approaches to numerical integration over a domain defined implicitly by an indicator function such as the level set function. The integration methods are based on subdivision, moment--fitting, local quasi-parametrization and Monte-Carlo techniques. As an application of these techniques, the paper addresses numerical solution of elliptic PDEs posed on domains and manifolds defined implicitly. A higher order unfitted finite element method (FEM) is assumed for the discretization. In such a method the underlying mesh is not fitted to the geometry, and hence the errors of numerical integration over curvilinear elements affect the accuracy of the finite element solution together with approximation errors. The paper studies the numerical complexity of the integration procedures and the performance of unfitted FEMs which employ these tools.

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

This report discusses the computer simulation of the ATACMS fin actuators. These subroutines are one of many possible methods of representing actuator behavior in a 6DOF (six degree of freedom) or 5DOF missile simulation. Also included is a section to discuss the accuracy of a version of the Runge- Kutta integration method. This particular simulation was run on a Zenith personal computer with Ryan-McFarland FORTRAN.

In this paper, we present and analyze the Clenshaw-Curtis-Filon methods for computing two classes of oscillatory Bessel transforms with algebraic or logarithmic singularities. More importantly, for these quadrature rules we derive new computational sharp error bounds by rigorous proof. These new error bounds share the advantageous property that some error bounds are optimal on $\omega$ for fixed $N$, while other error bounds are optimal on $N$ for fixed $\omega$. Furthermore, we prove from the presented error bounds in inverse powers of $\omega$ that the accuracy improves greatly, for fixed $N$, as $\omega$ increases.

Power series expansions (with coefficients obtained by recurrence formulas) are more efficient than other integration procedures for computing concurrently an orbit and the resolvent matrix of its variational equations, in the Restricted Problem of Three Bodies. For the same requirements on accuracy, the series expansions use only about 30 per cent of the computing time of the multi-step procedures, and only 12 to 15 per cent of the computing time of the Runge-Kutta-Nystrom method.

The integration of the equations of motion in gravitational dynamical systems -- either in our Solar System or for extra-solar planetary system -- being non integrable in the global case, is usually performed by means of numerical integration. Among the different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie series has shown some advantages. In its original form (Hanslmeier 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the method by deriving an expression of the Lie-terms when other major forces are considered. As a matter of fact, previous studies had been made but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations: relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration of the equation of motions for typical Near-Earth objects and planet Mercury.