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Ropes are widely used in various kinds of fishing gear. In this paper, a numerical method is presented which is competent in predicting the three-dimensional dynamic behavior of cables. The dynamic behavior of cables were simulated using dynamical equations and R language. The method was based on the lump-mass and spring model and the implicit algorithm was applied to solve the stiff differential equations. The mathematics-mechanical analysis was provided on the basic characteristics of the model. The R platform was used in programming, simulation, and visualization of the results. The three-dimensional dynamic behavior of cables in different examples were presented successfully and verify the effectiveness of this method, and the motion process was consistent with the observed data in the ocean.

The simplest two and three stage explicit Runge-Kutta methods are examined by a conveniently adapted form of the exponential fitting approach. The unusual feature is that the coefficients of the new versions are no longer constant, as in standard versions, but depend on the equation to be solved. Some valuable properties emerge from this. Thus, in the case of three-stage versions, although in general the order is three, that is the same as for the standard method, this is easily increased to four by a suitable choice of the position of the stage abscissas. Also, the stability properties are massively enhanced. Two particular versions of order four are A-stable, a fact which is quite unusual for explicit methods. This recommends them as efficient tools for solving stiff differential equations.

This chapter deals with stiff initial value problems for ODEs $$\dot{x}=F(x), \,x(0)={x_0}$$ The discretization of such problems is known to involve the solution of nonlinear systems per each discretization step–in one way or the other.

This paper contains an overview of a self-adjusting multirate method. A simple extension which allows the improvement of the efficiency of the method is introduced. The performance of the extended and the original method is compared for a test problem.

We propose to derive the explicit multistage methods of the Runge–Kutta type for ordinary differential equations (ODEs) with the aid of the expansion of grid functions into the Lagrange–Burmann series. New explicit first- and second-order methods are derived, which are applied to the numerical integration of the Cauchy problem for a moderately stiff ODE system. It turns out that the L 2 norm of the error of the solution obtained by the new numerical second-order method is 50 times smaller than in the case of the classical second-order Runge–Kutta method.

The dynamics of two types of gas filled microbubbles encapsulated with an elastic solid surface layer and immersed in water is examined by numerical evaluation of the numerically stiff Church non-linear differential equation for different state variable representation. The state variables influence the simulation results severely. The maximum amplitude of the oscillating shell radius is more sensitive to pressure and frequency of the driving sound wave than the surface tension. Chirp insonation is superior to broadband insonation for resonance excitation ensembles of microbubbles.

In this paper we analyze numerical optimization procedures in the context of level set based image segmentation. The Chan-Vese functional for image segmentation is a general and popular variational model. Given the corresponding Euler-Lagrange equation to the Chan-Vese functional the region based segmentation is usually done by solving a differential equation as an initial value problem. While most works use the standard explicit Euler method, we analyze and compare this method with two higher order methods (second and third order Runge-Kutta methods). The segmentation accuracy and the dependence of these methods on the involved parameters are analyzed by numerous experiments on synthetic images as well as on real images. Furthermore, the performance of the approaches is evaluated in a segmentation benchmark containing 1023 images. It turns out, that our proposed higher order methods perform more robustly, more accurately and faster compared to the commonly used Euler method.

Hardware-In-the-Loop (HIL) test rigs are more and more used to develop and/or improve modern control units. Due to the increasing complexity of these systems detailed vehicle models running in real-time are required. The main task in real time applications is to achieve a stable but still sufficiently accurate numerical solution under all operating conditions. Complex vehicles modeled by multi body systems result in stiff differential equations. In particular, taking into account the effects of dry friction requires sophisticated modeling techniques and simple but robust numerical solvers.

Central collisions between heavy nuclei in the relativistic regime have been systematically investigated at the Bevalac in Berkeley by a joint GSI-LBL-collaboration during the last years with 4π-detector devices, the LBL-Streamer Chamber and the GSI-Plastic Ball. These investigations have provided strong evidence for the compression of nuclear matter by a factor of up to 4. Moreover they have shown a collective flow of nuclear matter as predicted by hydrodynamical models. Many features of the space-time development of these reactions have been investigated and compared with theoretical predictions. The analysis of recent results give evidence for a rather stiff equation of state in contrast to considerations from astrophysical models for neutron star formation.

A numerical procedure is described for the steady, inviscid flow of a moving disturbance at the free surface. The method is valid for application in the case of a moving pressure distribution at the free surface. The solution is obtained by collocation where Laplace's equation is transformed into a system of differential equations. In the diagonal dominant structure one equation can be distinguished of the hyperbolic type and the other equations of the elliptic type.

A new method of decomposition of multiscale systems of ordinary differential equations is suggested. The suggested approach is based on the comparative analysis of the magnitudes of the eigenvalues of the matrix JJ*, where J is the local Jacobi matrix of the system under consideration. The proposed approach provides with the separation of the variables into fast and slow ones. The hierarchy of the decomposition is subject of variation with time, therefore, this decomposition is called dynamic. Equations for fast variables are solved by a stiff ODE system solver with the slow variables taken at the beginning of the time step. This is considered as a zeroth order solution for these variables. The solution of equations for slow variables is presented in a simplified form, assuming linearised variations of these variables during the time evolution of the fast variables. This is considered as the first order approximation for the solution for these variables or the first approximation for the fast manifold. The new approach is applied to numerical simulation of diesel fuel spray heating, evaporation and the ignition of fuel vapour/ air mixture. The results show advantages of the new approach when compared with the one proposed by the authors earlier and the conventional CFD approach used in computational fluid dynamics codes, both from the point of view of accuracy and CPU efficiency.

In this paper we analyze the behaviour of two stiff ODE solvers in the solution of chemical kinetics systems arising from detailed models of Diesel combustion. We consider general-purpose solvers, based on Backward Differentiation Formulas or Runge-Kutta methods and compare their impact, in terms of reliability and efficiency, on the solution of two different chemical kinetics systems, modeling combustion in the context of realistic simulations of Common Rail Diesel Engines. Numerical experiments have been carried out by using an improved version of KIVA3V, interfacing a parallel combustion solver. The parallel combustion solver is based on the CHEMKIN package for evaluating chemical reaction rates and on the general-purpose stiff ODE solvers for solving chemical kynetic equations.

In chapter 2, the structure of FEM, i.e., the derivation process of stiffness equation and element stiffness matrix, was explained. It might be needlessly time consuming, if we had to follow these processes every time we solve a problem. Fortunately, there is no need for it. Because the stiffness equation is, as will be understood from the derivation process in chapter 2, basically invariable whether it belongs to two-dimensional problem or to three-dimensional problem. FEM is formulated and programmed so as to be applicable to a variety of problems. Several kinds of element shapes are prepared as a standard for the variety of modeling which requires different geometry. The element stiffness matrix for each element is also predetermined as a standard. Our effort is limited to select — according to the kind of the problem or the geometry of the object — the appropriate element from a pool of prepared elements. If we cannot find the appropriate element from the standard, new elements must be developed. In this case, the stiffness matrix of the new element will be gained by following derivation process in chapter 2. It is admittedly rare that we face a problem that needs element derivation, because a fairly abundant number of elements are usually prepared for the general purpose FEM software. The situation is quite similar in the case of spring calculation by FEM, and it is important to select appropriate elements corresponding to the type of spring.

Kepler's generalized equation is a transcendental nonlinear equation that must be solved in order to compute the position and velocity of an artificial satellite at any instant t. In this paper, we propose a method to solve analytically that equation. The method is based on the properties of non canonical Lie transformations and, under certain conditions, it is shown that it gives better results than numerical methods.

The task of providing aa optimal analysis of the state of the atmosphere requires the development of novel computational tools that facilitate an efficient integeration of observational data into models. In this paper we discuss some of the computational tools developed for the assimilation of chemical data imo atmospheric models. We perform a theoretical analysis of discrete and cominuous adjoints for stiff differential equation solvers. Software tools particularly tailored for direct and adjoint sensitivity analysis of chemical systems are presented. The adjoint of the state-of-the-art model STEM-III is discussed, together with ozone assimilation results for a realistic test problem.

A typical stage in the manufacturing of container glass, such as bottles and jars, is the pressing phase. At this stage a gob of hot glass (1000 C or higher) is pressed in a mechanical construction by a so called plunger. This paper describes an important part of the simulation, the motion of the plunger. The problem is to determine the velocity of the latter which appears as a (kinematic) boundary condition of the glass flow problem. We show how to effectively uncouple these and to solve the resulting stiff differential equation numerically.

In this note we describe a novel approach to the numerical solution of the Fokker-Planck-Landau equation in the non-homogeneous case. The method couples, through a time splitting algorithm, a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator.

A class of higher order methods is investigated which can be viewed as implicit Taylor series methods based on Hermite quadratures. Improved automatic differentiation techniques for the claculation of the Taylor-coefficients and their Jacobians are used. A new rational predictor is used which can allow for larger step sizes on stiff problems.

In Sect. II.7 the first implicit Runge-Kutta methods were introduced. As we saw in Sect. IV.3, not all of them are suitable for the solution of stiff differential equations. This section is devoted to the collection of several classes of fully implicit Runge-Kutta methods possessing good stability properties.

A theory [1,2] has been developed to model the effects of inelastic deformation of metals at high temperatures based on the renewal theory of statistics. The theory successfully combines elastic, creep and plastic effects into a single viscoplastic model. The resulting expression for the uniaxial case consists of two coupled, first order, non-linear, ordinary differential equations. The solution of this equation is free of many of the numerical problems faced when dealing with other theories [3,4] and numerical solutions are readily carried out using a fourth order Runge-Kutta algorithm. Implicit methods developed for stiff differential equations are not required.

A new method of homology design based on finite element sensitivity analysis is proposed in static problems. The stiffness equation is separated into two equations by introducing homologous constraint. The governing equation of the design variables is derived from the condition for the two equations to hold. The Moore-Penrose generalized inverse is employed to determine the design variables, as the governing equation is expressed in a rectangular matrix form. The validity of the proposed method is demonstrated by the numerical example concerning with out-of-plane bending of a planar frame structure, for which the homologous constraint is set to keep a member straight.

Multistep methods (BDF) were the first numerical methods to be proposed for stiff differential equations (Curtiss & Hirschfelder 1952) and since Gear’s book (1971) computer codes based on these methods have been the most prominent and most widely used for all stiff computations.

Differential equations which describe biological systems can generally only be solved, if at all, by assuming idealizing initial and boundary conditions. The treatment of such equations by simulation methods is thus of growing imprtance. Block-oriented simulation systems, which invole much less work than the writing of special programs in a higher programming language, are convenient tools for modelling continuous systems, especially with respect to non-linear components and delays. In this contribution the advantages of a particular block-oriented simulation system (improved version of MISS as described in GOTT76) using a robust stiff integrator are discussed. Here it is applied to two biological processes: one simple example from population dynamics and one from molecular biology.

To simulate the model \( \dot x = Ax + b,\;t \geq 0,\;x\left( 0 \right) = x_0 \), will, for us, mean to solve the differential equation system numerically. To do this, we must select the input b, initialize the levels in each compartment by choosing x0, and then determine the model parameters aij in A to be used. The identification of these fractional transfer coefficients usually comes about in one of two ways. Either data are collected from the experiment in question and numerical methods are applied to estimate the aij s (see the previous section), or if observations are not possible, then we can talk to some of the experts in the field and put together educated guesses for the values of the a... We may only need “reasonable” values for these coefficients if we are not too sure of the validity of the model equations. By trying different sets of transfer parameters {aij}, a variety of scenarios can be created and thus the true physical processes can be simulated by the mathematical model. We may wish to test some basic assumption that is present--for example, that the inflow or outflow of material in the system is really proportional to the amount of material in the donor compartment.

We have mentioned some items relating to the appropriate choice of integration scheme and to the aspect of structuring and designing initial value codes for better applicability in solving boundary value problems. The capabilities needed for the various boundary problem techniques seem to be sufficiently different so as to warrant treatment on an individual basis. The best available software for the initial value problem should be utilized to the extent for which it is appropriate. In some instances, modifications or different methods will be necessary or else some inefficiency will result. (See also Anderson's paper [this proceedings].)

An approach of B. Lindberg [3] to the estimation and control of a norm of the global error is modified and applied to systems of stiff ODE's in partitioned form.

The main advantage of using potential methods for generating numerical solutions to elliptic partial differential equations is that the dimension of the problem is reduced by one. This feature reduces, not only the number of algebraic equations to be solved, but also alleviates the problems of mesh generation. Both of these factors combine to reduce considerably the cost of computing acceptable numerical solutions. In this paper a potential method is developed which essentially extends to the diffusion equation most of the advantages currently afforded to elliptic equations.

Systems of difference equations containing small parameters are studied by a constructive perturbation scheme analogous to the one developed by the authors for the study of differential equations. The method results in an averaging procedure for difference equations, and it is particularly well suited to certain highly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, regression analysis, and the numerical analysis of stiff differential equations.

In this note we consider a boundary control problem for the heat equation in arbitrary dimensions. For simplicity we treat the L2-minimum norm case only and discuss the results which we proved in [1],[4]. We show how controllability- and normality-properties of a certain linear operator enter into the proof of a bang-bang theorem.