Runge-Kutta Methods

The fourth-order Runge–Kutta method is used to
numerically integrate the equations of motion for a fastpitch
softball pitch and to create a model from which the trajectories
of drop balls, rise balls and curve balls can be computed and
displayed.  By requiring these pitches to pass through the strike
zone, and by assuming specific values for the initial speed,
launch angle and height of each pitch, an upper limit on the lift
coefficient can be predicted which agrees with experimental
data. This approach also predicts the launch angles necessary
to put rise balls, drop balls and curve balls in the strike zone, as
well as a value of the drag coefficient that agrees with experimental
data. Finally, Adair’s analysis of a batter’s swing is
used to compare pitches that look similar to a batter starting
her swing, yet which diverge before reaching the home plate,
to predict when she is likely to miss or foul the ball.

The study of the steady laminar mixed convection boundary layer flow of an incompressible viscous fluid, with heat and mass transfer along vertical thin needles with variable heat flux has been considered. The governing boundary layer equations are first transformed into non dimensional form and then after using similarity transformations converted into set of ordinary differential equations. The set of ordinary differential equations are solved with the help Runge-Kutta method with shooting technique. The value of skin friction coefficient and the surface temperature, wall concentration, velocity profile as well as temperature profile are obtained for different values of dimensionless parameters with m=0, which is case for a blunt nosed needle. The magnitude of velocity decreases with increasing needle size and increases with increasing local buoyancy parameter. The graphical and tabulated results are presented.

This is a one page reference sheet for several common numerical methods.  It lists and/or describes several methods for root finding, linear & non-linear systems of equations, quadrature, and solving differential equations. It also  includes certain Mat-lab built in functions. As it is only one page, some numerical methods are only mentioned in order to provide a starting point for further research. A special thanks goes to my professor for the Mat-lab Programming & Numerical Methods class in Howard County Community College, Professor Mark Edelen, who was an excellent teacher for this class. He also looked over an earlier version of this paper and proofread it.

Numerical determination of the temperature distribution T(x) and fin heat transfer rate qf of a cylindrical pin fin is presented. The classical Runge-Kutta 4th-order method is used to solve a 2nd-order ODE governing the temperature profile. The shooting method converts the boundary value problem into an initial value problem. Multiple iterations are performed until the solution converges to an adiabatic condition at the fin tip. Numerical results at four different step sizes Delta x are compared to the analytical solution. The R-K 4th-order method produces a fin heat transfer rate qf accurate to 3 decimal digits at the largest step size Delta x = 0.2. The use of four appropriately weighted derivative functions permits the extremely high accuracy of R-K 4th-order method in predicting the y-value jumps over each step Delta x. The major downside is that the shooting method/RK4 method is computationally expensive, which is chiefly due to the requirement of conducting multiple iterations before convergence.

De nombreux problèmes de sciences physiques, biologiques, technologiques ou des problèmes issus de modèles économiques et sociaux sont traités par les méthodes numériques. La partie théorique des méthodes numériques relève des mathématiques, et la partie mise en pratique aboutit généralement à l’implémentation d’algorithmes sur ordinateur. Ces méthodes se fondent à la fois sur la recherche de solutions exactes comme dans le cas de l’analyse matricielle ou du calcul symbolique, sur des solutions approchées qui résultent le plus souvent de processus de discrétisation comme dans le traitement des équations diérentielles.

A direct explicit Runge-Kutta type (RKT) method via shooting technique to approximate analytical solutions to the third-order two-point boundary value problems (BVPs) with boundary condition type I and II are proposed. In this paper first, a three-stage fourth-order direct explicit Runge-Kutta type method denoted as RKT3s4 is constructed. A new algorithm of shooting technique for solving two-point BVPs for third-order ordinary differential equations (ODEs) is presented.

The study of mixed convection evaporation of an inclined wet flat plate was carried out. As the resolution of partial differential equations occupies an important place in the world of research, our study can serve as a reference. We have proposed two methods for understanding and solving the physical phenomena interfering in mixed convection. The first being the semi-analytical resolution allowed us to analyse and to study the preponderance of natural convection and then forced convection. The other is the numerical resolution by the finite difference implicit method which displays results similar to the first, just by varying the Richardson number. The influence of the inclination is also presented in each case. Mass and heat transfers correspond to forced convection and natural convection, respectively. The combination of the two gives rise to mixed convection. In our case, we chose air but the equations can be used for other fluids for heat exchange with a solid surface.

Les méthodes de Runge-Kutta (ou RK), l'ordre 2 ou 4, sont très couramment utilisées pour la résolution d'équations différentielles ordinaires (EDO). Ce sont des méthodes à pas unique, directement dérivées de la méthode d'Euler, qui est une méthode RK1.

Se formula un método Runge-Kutta implícito de cuatro etapas basado en la cudratura de Lobatto y que es de sexto orden. El método se empotra con un método explícito de tercer orden de igual número de etapas con los mismos puntos de colocación basado en Ralston. Se implementa un algoritmo con ambas soluciones que permite controlar el paso de integración. Finalmente se hace un estudio de la estabilidad del método. Se muestra por último un ejemplo de aplicación a las órbitas de Arenstorf donde se observan las bondades del método comparándolo con otros.

The FORTRAN code and its execution (first and last page) for solving Lane-Emden differential equation for a differentially rotating polytropic star.
For theoretical definitions see:
http://dx.doi.org/10.1007/BF00572408
(Equation (1.6) and its description)
Work done for Numerical Analysis course, Department of Physics, University of Patras at Spring-Summer 1988 under the supervision of V. S. Geroyannis.
Details:
*341 lines of code
*execution using a UNIVAC 1100 computer and CTS operational system

The second order ordinary differential equation cited in the title of this article, is then solved using the fourth order Runge-Kutta method (RK4), programmed in FORTRAN90. The solutions were graphed with the help of GNUPLOT, the result shows oscillatory and convergent solutions.

In spite of Runge-Kutta method is the most used by scientists and engineers, it is not the most powerful method. In this paper, a comparative study between Piece-wise Analytic Method (PAM) and Runge-Kutta Methods is introduced. The result of comparative study shows that PAM is more powerful and gives results better than Runge-Kutta methods. PAM can be considered as a new step in the evolution of solving nonlinear differential equations.

This is a presentation that discusses the simulation of a projectile (potentially a rocket) being fired from the Earth with some initial velocity and directed towards the Moon. It explores the use of different iterative methods to model the differential equations governing the trajectory, particularly Euler's method, the classical fourth-order Runge-Kutta method, and Mat-lab's built-in function for differential equations, ode45.

There are two main sections of the presentation. Slides 1 through 8 and the final two slides (slides 20 and 21). Slides 1 through 8 contain a video showing a Mat-lab program simulating the motion of the projectile when governed by Euler's method and the fourth-order Runge-Kutta method when a fixed time step is used. It also contains the assumptions necessary for creating the model and provides an intuition of some of the Math behind the Mat-lab programs hopefully understandable for someone with less of a background in math. A picture of a simulation of the path of the projectile when governed by Euler's method, the Runge-Kutta method, and ode45 when a "distance-step" is used is also shown. Slides 20 and 21 contain the acknowledgments and links to the Mat-lab programs used to create the presentation.

The other section (slides 9 through 19) contain the derivation of orbital equations and the math used in setting up Euler's method, the Runge-Kutta method, and ode45. The math required to understand the derivations and the implementation of the numerical methods requires a basic understanding of vector algebra and some knowledge in differential equations & the numerical methods used for differential equations.

This presentation was created for BCCC's Math Awareness Week Events.

In this paper, a new method is introduced for engineers and scientists which can be used for solving highly nonlinear differential equations. The method is called Piecewise Analytic Method (PAM). PAM is used to solve problems which other methods can't solve. the paper also shows how the accuracy and error can be controlled according to our needs.

In this paper, a new interval version of Runge-Kutta methods is proposed for time discretization and solving of optimal control problems (OCPs) in the presence of uncertain parameters. A new technique based on interval arithmetic is introduced to achieve the confidence bounds of the system. The proposed method is based on the new forward representation of Hukuhara interval difference and combining it with Runge-Kutta method for solving the OCPs with interval uncertainties. To perform the proposed method on OCPs, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, using some algebraic manipulations, they are converted to an ordinary differential equation to achieve the interval optimal solution for the considered OCP with uncertain parameters. Shooting method is also employed to cover the Runge-Kutta methods restrictions in solving the OCPs with boundary values. The simulation results are applied to some practical case studies for demonstrating the effectiveness of the proposed method.

In this study, a class of direct numerical integrators for solving special second-order ordinary differential equations (ODEs) is proposed and studied. The method is multistage and multistep in nature. This class of integrators is called "two-step Runge-Kutta-Nyström", denoted by TSRKN. The direct approach to higher-order ODEs is desirable to avoid tedious computational work caused by converting the higherorder ODEs into the system of first-order equations. The order conditions for the TSRKN are derived using Taylors series expansion and according to the order conditions, a three-stage TSRKN method which is convergent of order four is constructed. The convergence analysis of the method is discussed and the performance of the newly derived method is compared with existing methods. The numerical results show the superiority of the TSRKN method in terms of number of function evaluations and demonstrate that the TSRKN can also be used to solve linear second-order boundary value problems (BVPs) since Runge-Kutta-Nyström (RKN) approach is practically used to only solve higher-order initial value problems (IVPs) directly.

Predator-prey models are the building blocks of the ecosystems as biomasses are grown out of their resource masses. Different relationships exist between these models as different interacting species compete, metamorphosis occurs and migrate strategically aiming for resources to sustain their struggle to exist. To numerically investigate these assumptions, ordinary differential equations are formulated, and a variety of methods are used to obtain and compare approximate solutions against exact solutions, although most numerical methods often require heavy computations that are time-consuming. In this paper, the traditional differential transform (DTM) method is implemented to obtain a numerical approximate solution to prey-predator models. The solution obtained with DTM is convergent locally within a small domain. The multi-step differential transform method (MSDTM) is a technique that improves DTM in the sense that it increases its interval of convergence of the series expansion. One predator-one prey and two-predator-one prey models are considered with a quadratic term which signifies other food sources for its feeding. The result obtained numerically and graphically showspoint DTM diverges. The advantage of the new algorithm is that the obtained series solution converges for wide time regions and the solutions obtained from DTM and MSDTM are compared with solutions obtained using the classical Runge-Kutta method of order four. The results demonstrated is that MSDTM computes fast, is reliable and gives good results compared to the solutions obtained using the classical Runge-Kutta method.

In this work, we investigate just a few of the many computational issues involved in achieving autonomous control of unmanned air vehicles (UAVs). Of the many techniques used for UAV control, this research is vision based. We seek autonomous control and obstacle avoidance using a sequence of images from a camera. With the use of a camera, one part of this process is image processing, but it is not discussed here. We instead investigate several of the `post processing of the image' issues. Image processing seeks to determine the °ow of the image in the image plane. Using two consecutive images, image motion vectors are computed for a given collection of pixel position vectors.

We describe two methods used to recover the linear and angular velocity that the UAV (camera) underwent between the two camera frames. The only information needed is the pixel position vectors, the image motion vectors, and the speed of the UAV. The two methods discussed are the continuous eight-point algorithm and a direct minimization of a cost function. Without access to real data taken from a camera undergoing a known motion, we test the algorithms on random synthetic data and compare the results to the initial known linear and angular velocity.

The linear and angular velocity signals being computed may be noisy; as such, we also investigate a nonlinear observer to filter the possible noisy signal and test this observer on the equations of motion of a rigid body undergoing rotations and translations. As solving the observer equations involves numerical integration, we also investigate variable step-size selection methods for implicit integration schemes. The minimization of an efficiency function is the basis for the selection of the step-sizes for implicit Runge-Kutta and Runge-Kutta-Nystrom methods. The methods are tested on three well-known examples from the literature.

Finally, we consider the topic of obstacle avoidance. A novel approach based on Lyapunov theory is used to steer a UAV away from obstacles detected by the camera. We test the algorithm on a UAV simulation evolving in a two-dimensional space.

This research work presents a new evolutionary optimization algorithm, EVO-RUNGE-KUTTA in theoretical mathematics with applications in scientific computing. We illustrate the application of EVO-RUNGE-KUTTA, a two-phase optimization algorithm, to a problem of pure algebra, the study of the parameterization of an algebraic variety, an open problem in algebra. Results show the design and optimization of particular algebraic varieties, the Runge-Kutta methods of order q. The mapping between algebraic geometry and evolutionary optimization is direct, and we expect that many open problems in pure algebra will be modelled as constrained global optimization problems.

This paper discusses a weighted fourth order Runge-Kutta method based on contra-harmonic mean to
solve ordinary differential equations. The local truncation error and stability analysis for this method are presented.
The numerical experiment reveals that the proposed method is suitable for solving the stiff initial value problems.

In recent years, the derivation of Runge-Kutta methods with higher derivatives has been on the increase. In this paper, we present a new class of three stage Runge-Kutta method with first and second derivatives. The consistency and stability of the method is analyzed. Numerical examples with excellent results are shown to verify the accuracy of the proposed method compared with some existing methods.

This is a presentation that discusses the simulation of a projectile (potentially a rocket) being fired from the Earth with some initial velocity and directed towards the Moon. It explores the use of different iterative methods to model the differential equations governing the trajectory, particularly Euler's method, the classical fourth-order Runge-Kutta method, and Mat-lab's built-in function for differential equations, ode45.

There are two main sections of the presentation. Slides 1 through 8 and the final two slides (slides 20 and 21). Slides 1 through 8 contain a video showing a Mat-lab program simulating the motion of the projectile when governed by Euler's method and the fourth-order Runge-Kutta method when a fixed time step is used. It also contains the assumptions necessary for creating the model and provides an intuition of some of the Math behind the Mat-lab programs hopefully understandable for someone with less of a background in math. A picture of a simulation of the path of the projectile when governed by Euler's method, the Runge-Kutta method, and ode45 when a "distance-step" is used is also shown. Slides 20 and 21 contain the acknowledgments and links to the Mat-lab programs used to create the presentation.

The other section (slides 9 through 19) contain the derivation of orbital equations and the math used in setting up Euler's method, the Runge-Kutta method, and ode45. The math required to understand the derivations and the implementation of the numerical methods requires a basic understanding of vector algebra and some knowledge in differential equations & the numerical methods used for differential equations.

This presentation was created for BCCC's Math Awareness Week Events.

The study of the steady laminar mixed convection boundary layer flow of an incompressible viscous fluid, with heat and mass transfer along vertical thin needles with variable heat flux has been considered. The governing boundary layer equations are first transformed into non dimensional form and then after using similarity transformations converted into set of ordinary differential equations. The set of ordinary differential equations are solved with the help Runge-Kutta method with shooting technique. The value of skin friction coefficient and the surface temperature, wall concentration, velocity profile as well as temperature profile are obtained for different values of dimensionless parameters with m=0, which is case for a blunt nosed needle. The magnitude of velocity decreases with increasing needle size and increases with increasing local buoyancy parameter. The graphical and tabulated results are presented.