Relevant bibliographies by topics / RUNGE-KUTTA FOURTH ORDER METHOD

Academic literature on the topic 'RUNGE-KUTTA FOURTH ORDER METHOD'

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Published: 16 January 2024

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Journal articles on the topic "RUNGE-KUTTA FOURTH ORDER METHOD"

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Suryani, Irma, Wartono Wartono, and Yuslenita Muda. "Modification of Fourth order Runge-Kutta Method for Kutta Form With Geometric Means." Kubik: Jurnal Publikasi Ilmiah Matematika 4, no. 2 (February 25, 2020): 221–30. http://dx.doi.org/10.15575/kubik.v4i2.6425.

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This paper discuss how to modified Fourth order Runge-Kutta Kutta method based on the geometric mean. Then we have parameters and however by re-comparing the Taylor series expansion of and up to the 4th order. For make error term re-compering of the Taylor series expansion of and up to the 5th order. In the error term an make substitution for the values of and into the Taylor seriese expansion up to the 5th order. So that we have error term modified Fourth Order Runge-Kutta Kutta based on the geometric mean. Modified Fourth Order Runge-Kutta Kutta based on the geometric mean that usually used to solved ordinary differential equations.
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Trifina, Leonora L. R., Ali Warsito, Laura A. S. Lapono, and Andreas Ch Louk. "VISUALISASI FENOMENA HARMONIS DAN CHAOS PADA GETARAN TERGANDENG BERBASIS KOMPUTASI NUMERIK RUNGE KUTTA." Jurnal Fisika : Fisika Sains dan Aplikasinya 8, no. 1 (April 27, 2023): 11–20. http://dx.doi.org/10.35508/fisa.v8i1.11817.

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Research has been carried out on the visualization of harmonic and chaos phenomenont on coupled vibration physcal case using the Runge Kutta numerical computation method with the aim of applying the first to fourth order Runge Kutta computation method to obtain a second order differential equation solution on coupled vibration system, calculating the displacement value of objects using computation method Runge Kutta order first to fourth, obtained a graph of the displacement of objects againts time in case of coupled vibration for harmonic and chaos states at certain step width values and compare the convergence of the Runge Kutta method from first to fourth order with the special analytical method. The solution of coupled vibration equation which is classified as a second order differential equation was quite difficulted to solve analytically, so the Runge Kutta computation method was used to solve it as an alternative solution. The results of the research showed that the harmonic state of the system was obtained when the displacement graph showed the motion of each pendulum which was constant with the pendulum displacement position with respect to time in the form of a sinusoidal graph at a value of C1 = 40 N/m, C2 = 30 N/m, C = 10 N/m, C = 0 N/m and the chaotic state was represented by a graph of the displacement of the pendulum with respect to time with an irregular pattern. In this case, it was found that the fourth order Runge Kutta method converged faster than the first to third order Runge Kutta method with the best results obtained at a step width value of 0,001. The fourth order Runge Kutta method also has a smaller approximation average error value from first to third order Runge Kutta method was on the fourth order Runge Kutta method and the avarage error values are , and on the Runge Kutta method of first to third order.
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Husin, Nurain Zulaikha, Muhammad Zaini Ahmad, and Mohd Kamalrulzaman Md Akhir. "Incorporating Fuzziness in the Traditional Runge–Kutta Cash–Karp Method and Its Applications to Solve Autonomous and Non-Autonomous Fuzzy Differential Equations." Mathematics 10, no. 24 (December 8, 2022): 4659. http://dx.doi.org/10.3390/math10244659.

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The study of the fuzzy differential equation is a topic that researchers are interested in these days. By modelling, this fuzzy differential equation can be used to resolve issues in the real world. However, finding an analytical solution to this fuzzy differential equation is challenging. Thus, this study aims to present the fuzziness in the traditional Runge–Kutta Cash–Karp of the fourth-order method to solve the first-order fuzzy differential equation. Later, this method is referred to as the fuzzy Runge–Kutta Cash–Karp of the fourth-order method. There are two types of fuzzy differential equations to be solved: autonomous and non-autonomous fuzzy differential equations. This fuzzy differential equation is divided into the (i) and (ii)–differentiability on the basis of the characterization theorem. The convergence analysis of the fuzzy Runge–Kutta Cash–Karp of the fourth-order method is also presented. By implementing the fuzzy Runge–Kutta Cash–Karp of the fourth-order method, the approximate solution is compared with the analytical and numerical solutions obtained from the fuzzy Runge–Kutta of the fourth-order method. The results demonstrated that the approximate solutions of the proposed method are accurate with an analytical solution, when compared with the solutions of the fuzzy Runge–Kutta of the fourth-order method.
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Rijoly, Monalisa E., and Francis Yunito Rumlawang. "Penyelesaian Numerik Persamaan Diferensial Orde Dua Dengan Metode Runge-Kutta Orde Empat Pada Rangkaian Listrik Seri LC." Tensor: Pure and Applied Mathematics Journal 1, no. 1 (May 28, 2020): 7–14. http://dx.doi.org/10.30598/tensorvol1iss1pp7-14.

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One alternative to solve second order differential equations by numerical methods, specificallynon-liner differential equations is the Runge-Kutta fourth order method. The Runge-Kutta fourth ordermethod is a numerical method that has high degree of precision and accuracy when compared to othernumerical methods. In this paper we will discuss the numerical solution of second order differentialequations on LC series circuit problem using the Runge-Kutta fourth order method. The numericalsolution generated by the computational calculation using the MATLAB program, the strong current andcharge are obtaind from t = 0 and t =0,5 second and different step size values
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Hussain, Kasim, Fudziah Ismail, and Norazak Senu. "Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/893763.

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A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.
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Hussain, Kasim A., and Waleed J. Hasan. "Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach." Ibn AL-Haitham Journal For Pure and Applied Sciences 36, no. 1 (January 20, 2023): 345–54. http://dx.doi.org/10.30526/36.1.2963.

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This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions and for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency is used. The novel method is more accurate than the conventional Runge-Kutta method and IRK4. Several test problems for the system of first-order ordinary differential equations carry out numerically to demonstrate the effectiveness of this approach. The computational studies show that the TFIRK4 approach is more efficient than the existing Runge-Kutta methods.
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Zhou, Naying, Hongxing Zhang, Wenfang Liu, and Xin Wu. "A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes." Astrophysical Journal 927, no. 2 (March 1, 2022): 160. http://dx.doi.org/10.3847/1538-4357/ac497f.

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Abstract In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.
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Chauhan, Vijeyata, and Pankaj Kumar Srivastava. "Computational Techniques Based on Runge-Kutta Method of Various Order and Type for Solving Differential Equations." International Journal of Mathematical, Engineering and Management Sciences 4, no. 2 (April 1, 2019): 375–86. http://dx.doi.org/10.33889/ijmems.2019.4.2-030.

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The Runge-Kutta method is a one step method with multiple stages, the number of stages determine order of method. The method can be applied to work out on differential equation of the type’s explicit, implicit, partial and delay differential equation etc. The present paper describes a review on recent computational techniques for solving differential equations using Runge-Kutta algorithm of various order. This survey includes the summary of the articles of last decade till recent years based on third; fourth; fifth and sixth order Runge-Kutta methods. Along with this a combination of these methods and various other type of Runge-Kutta algorithm based articles are included. Comparisons of methods with own critical comments as remarks have been included.
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Christopher, Dr Esekhaigbe Aigbedion. "Consistency and Convergence Analysis of an 𝐹(𝑥,𝑦) Functionally Derived Explicit Fifth-Stage Fourth-Order Runge-Kutta Method." International Journal of Basic Sciences and Applied Computing 10, no. 4 (December 30, 2022): 10–13. http://dx.doi.org/10.35940/ijbsac.a1145.1210423.

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The purpose of this paper is to analyze the consistency and convergence of an explicit fifth-stage fourth-order Runge-Kutta method derived using 𝒇(𝒙,𝒚) functional derivatives. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in a previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.
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Ahmad, S. Z., F. Ismail, N. Senu, and M. Suleiman. "Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/136961.

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We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.
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Dissertations / Theses on the topic "RUNGE-KUTTA FOURTH ORDER METHOD"

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Boat, Matthew. "The time-domain numerical solution of Maxwell's electromagnetic equations, via the fourth order Runge-Kutta discontinuous Galerkin method." Thesis, Swansea University, 2008. https://cronfa.swan.ac.uk/Record/cronfa42532.

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This thesis presents a high-order numerical method for the Time-Domain solution of Maxwell's Electromagnetic equations in both one- and two-dimensional space. The thesis discuses the validity of high-order representation and improved boundary representation. The majority of the theory is concerned with the formulation of a high-order scheme which is capable of providing a numerical solution for specific two-dimensional scattering problems. Specifics of the theory involve the selection of a suitable numerical flux, the choice of appropriate boundary conditions, mapping between coordinate systems and basis functions. The effectiveness of the method is then demonstrated through a series of examples.
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Auffredic, Jérémy. "A second order Runge–Kutta method for the Gatheral model." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-49170.

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In this thesis, our research focus on a weak second order stochastic Runge–Kutta method applied to a system of stochastic differential equations known as the Gatheral Model. We approximate numerical solutions to this system and investigate the rate of convergence of our method. Both call and put options are priced using Monte-Carlo simulation to investigate the order of convergence. The numerical results show that our method is consistent with the theoretical order of convergence of the Monte-Carlo simulation. However, in terms of the Runge-Kutta method, we cannot accept the consistency of our method with the theoretical order of convergence without further research.
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Booth, Andrew S. "Collocation methods for a class of second order initial value problems with oscillatory solutions." Thesis, Durham University, 1993. http://etheses.dur.ac.uk/5664/.

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We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given.
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Malroy, Eric Thomas. "Solution of the ideal adiabatic stirling model with coupled first order differential equations by the Pasic method." Ohio : Ohio University, 1998. http://www.ohiolink.edu/etd/view.cgi?ohiou1176410606.

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Alhojilan, Yazid Yousef M. "Higher-order numerical scheme for solving stochastic differential equations." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/15973.

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We present a new pathwise approximation method for stochastic differential equations driven by Brownian motion which does not require simulation of the stochastic integrals. The method is developed to give Wasserstein bounds O(h3/2) and O(h2) which are better than the Euler and Milstein strong error rates O(√h) and O(h) respectively, where h is the step-size. It assumes nondegeneracy of the diffusion matrix. We have used the Taylor expansion but generate an approximation to the expansion as a whole rather than generating individual terms. We replace the iterated stochastic integrals in the method by random variables with the same moments conditional on the linear term. We use a version of perturbation method and a technique from optimal transport theory to find a coupling which gives a good approximation in Lp sense. This new method is a Runge-Kutta method or so-called derivative-free method. We have implemented this new method in MATLAB. The performance of the method has been studied for degenerate matrices. We have given the details of proof for order h3/2 and the outline of the proof for order h2.
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Jewell, Jeffrey Steven. "Higher-order Runge--Kutta type schemes based on the Method of Characteristics for hyperbolic equations with crossing characteristics." ScholarWorks @ UVM, 2019. https://scholarworks.uvm.edu/graddis/1028.

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The Method of Characteristics (MoC) is a well-known procedure used to find the numerical solution of systems of hyperbolic partial differential equations (PDEs). The main idea of the MoC is to integrate a system of ordinary differential equations (ODEs) along the characteristic curves admitted by the PDEs. In principle, this can be done by any appropriate numerical method for ODEs. In this thesis, we will examine the MoC applied to systems of hyperbolic PDEs with straight-line and crossing characteristics. So far, only first- and second-order accurate explicit MoC schemes for these types of systems have been reported. As such, the purpose of this thesis is to develop MoC schemes which are of an order greater than two. The order of the global truncation error of an MoC scheme goes hand-in-hand with the order of the ODE solver used. The MoC schemes which have already been developed use the first-order Simple Euler (SE) and second-order Modified Euler (ME) methods as the ODE solvers. The SE and ME methods belong to a larger family of numerical methods for ODEs known as the Runge--Kutta (RK) methods. First, we will attempt to develop third- and fourth-order MoC schemes by using the classical third- and fourth-order RK methods as the ODE solver. We will show that the resulting MoC schemes can be strongly unstable, meaning that the error in the numerical solution becomes unbounded rather quickly. We then turn our attention to the so-called pseudo-RK (pRK) methods for ODEs. The pRK methods are at the intersection of RK and multistep methods, and a variety of third- and fourth-order schemes can be constructed. We show that when certain pRK schemes are used in the MoC, at most a weak instability, or no instability at all, is present, and thus the resulting methods are suitable for long-time computations. Finally, we present some numerical results confirming that the MoC using third- and fourth-order pRK schemes have the desired accuracy.
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KUMAR, PRADEEP. "COVID-19 USING NUMERICAL METHOD." Thesis, 2021. http://dspace.dtu.ac.in:8080/jspui/handle/repository/20443.

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The SIR model is used to discuss the spread of the covid-19 epidemic in the Indian state of Maharashtra and its eventual end. Here we have examined about the spread of Coronavirus pandemic in extraordinary profundity utilizing Runge-kutta fourth-order method. The Runge kutta fourth-order method is a solving of the non-linear ordinary differential. We have used the data of covid-19 Outbreak of state Maharashtra on 29 April, 2021. The total population of Maharashtra is 122153000, according to this data. For the initial stage of experimental purposes, we used 113814181 susceptible cases, 4539553 infectious cases, and 3799266 recovered cases. The SIR model was used to analyse data from a wide range of infectious diseases. As a result, several scientists and researchers have thoroughly tested this model for infectious diseases. As a result of the research and simulation of this proposed covid-19 model using data on the number of covid-19 outbreak cases in state Maharashtra of India, show that the covid-19 epidemic infection cases rise for a period of time after the outbreak decreases, and then the covid-19 outbreak ends in Maharashtra cases. The model's findings also show that the Runge-kutta fourth-order method is used for forecast and avoid the covid-19 outbreak in India's Maharashtra state. Finally, we determine that the outbreak of the covid-19 epidemic in Maharashtra will peak on 11 May 2021, after which it will progress steadily and will likely end in the fourth week of October 2021.
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Meng-HanLi and 李孟翰. "A High-Order Runge-Kutta Discontinuous Galerkin Method for The Two-Dimensional Wave Equation." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/60562488311569777411.

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碩士
國立成功大學
數學系應用數學碩博士班
98
In this work, we develop a high-order Runge-Kutta Discontinuous Galerkin (RKDG) method to solve the two-dimensional wave equations. We use DG methods to discretize the equations with high order elements in space, and then we use the mth-order, m-stage strong stability preserving Runge-Kutta (SSP-RK) scheme to solve the resulting semi-discrete equations. To discretize the equaiotns in spaces, we use the quadrilateral elements and the Q^k-polynomials as basis functions. The scheme achieves full high-order convergence in time and space while keeping the time-step proportional to the spatial mesh-size. Numerical results are presented that confirm the expected convergence properties. When all the local spaces contain the polynomials of degree p,the numerical experiments show that the numerical solution converges with order p+1.
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Kotovshchikova, Marina. "On a third-order FVTD scheme for three-dimensional Maxwell's Equations." 2016. http://hdl.handle.net/1993/31035.

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This thesis considers the application of the type II third order WENO finite volume reconstruction for unstructured tetrahedral meshes proposed by Zhang and Shu in (CCP, 2009) and the third order multirate Runge-Kutta time-stepping to the solution of Maxwell's equations. The dependance of accuracy of the third order WENO scheme on the small parameter in the definition of non-linear weights is studied in detail for one-dimensional uniform meshes and numerical results confirming the theoretical analysis are presented for the linear advection equation. This analysis is found to be crucial in the design of the efficient three-dimensional WENO scheme, full details of which are presented. Several multirate Runge-Kutta (MRK) schemes which advance the solution with local time-steps assigned to different multirate groups are studied. Analysis of accuracy of three different MRK approaches for linear problems based on classic order-conditions is presented. The most flexible and efficient multirate schemes based on works by Tang and Warnecke (JCM, 2006) and Liu, Li and Hu (JCP, 2010) are implemented in three-dimensional finite volume time-domain (FVTD) method. The main characteristics of chosen MRK schemes are flexibility in defining the time-step ratios between multirate groups and consistency of the scheme. Various approaches to partition the three-dimensional computational domain into multirate groups to maximize the achievable speedup are discussed. Numerical experiments with three-dimensional electromagnetic problems are presented to validate the performance of the proposed FVTD method. Three-dimensional results agree with theoretical and numerical accuracy analysis performed for the one-dimensional case. The proposed implementation of multirate schemes demonstrates greater speedup than previously reported in literature.
February 2016
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Books on the topic "RUNGE-KUTTA FOURTH ORDER METHOD"

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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National Institute of Standards and Technology (U.S.), ed. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, 1997.

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Carpenter, Mark H. Fourth-order 2N-storage Runge-Kutta schemes. Hampton, Va: Langley Research Center, 1994.

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A, Kennedy Christopher, and Langley Research Center, eds. Fourth-order 2N Runge-Kutta schemes. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1994.

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A, Kennedy Christopher, and Langley Research Center, eds. Fourth-order 2N Runge-Kutta schemes. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1994.

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Book chapters on the topic "RUNGE-KUTTA FOURTH ORDER METHOD"

1

Liu, Chunfeng, Haiming Wu, Li Feng, and Aimin Yang. "Parallel Fourth-Order Runge-Kutta Method to Solve Differential Equations." In Information Computing and Applications, 192–99. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-25255-6_25.

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Zhang, Baoji, and Lupeng Fu. "Study on the Analysis Method of Ship Surf-Riding/Broaching Based on Maneuvering Equations." In Proceeding of 2021 International Conference on Wireless Communications, Networking and Applications, 569–75. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-2456-9_58.

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AbstractIn order to understand the mechanism of the surf-riding/broaching profoundly, the four- degree- of-freedom(4DOF) maneuvering equation (surge, sway, yaw and roll) is simplified to a one- degree-of-freedom (1DOF) equation, and the fourth-order Runge-Kutta method is used to integrate a 1DOF surge equation in the time domain to analyze the two motion states of the ship during the surging and surf-riding. The critical Froude number is calculated using the Melnikov method. Taking a fishing boat as an example, the ship’s surf-riding/broaching phenomenon is simulated under the condition of wavelength-to-ship-length ratio and wave steepness, 1 and 1/10 respectively, providing technical support for the formulation of the second generation intact stability criteria.
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Abadi, Maryam Asghari Hemmat, and Bing Yuan Cao. "Solving First Order Fuzzy Initial Value Problem by Fourth Order Runge-Kutta Method Based on Different Means." In Advances in Intelligent Systems and Computing, 356–69. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-66514-6_36.

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Ben Amma, B., Said Melliani, and L. S. Chadli. "A Fourth Order Runge-Kutta Gill Method for the Numerical Solution of Intuitionistic Fuzzy Differential Equations." In Recent Advances in Intuitionistic Fuzzy Logic Systems, 55–68. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-02155-9_5.

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Griffiths, David F., and Desmond J. Higham. "Runge–Kutta Method—I: Order Conditions." In Numerical Methods for Ordinary Differential Equations, 123–34. London: Springer London, 2010. http://dx.doi.org/10.1007/978-0-85729-148-6_9.

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Rabiei, Faranak, Fudziah Ismail, Norihan Arifin, and Saeid Emadi. "Third Order Accelerated Runge-Kutta Nyström Method for Solving Second-Order Ordinary Differential Equations." In Informatics Engineering and Information Science, 204–9. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-25462-8_17.

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Eremin, Alexey S., Nikolai A. Kovrizhnykh, and Igor V. Olemskoy. "Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations." In Computational Science and Its Applications – ICCSA 2019, 89–102. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-24289-3_8.

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Sundnes, Joakim. "Stable Solvers for Stiff ODE Systems." In Solving Ordinary Differential Equations in Python, 35–60. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-46768-4_3.

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AbstractIn the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.
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Ben Amma, Bouchra, Said Melliani, and S. Chadli. "The Numerical Solution of Intuitionistic Fuzzy Differential Equations by the Third Order Runge-Kutta Nyström Method." In Intuitionistic and Type-2 Fuzzy Logic Enhancements in Neural and Optimization Algorithms: Theory and Applications, 119–32. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35445-9_11.

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Sawhney, Himanshu, Kedar S. Pakhare, Rameshchandra P. Shimpi, P. J. Guruprasad, and Yogesh M. Desai. "Single Variable New First-Order Shear Deformation Plate Theory: Numerical Solutions of Lévy-Type Plates Using Fourth-Order Runge-Kutta Technique." In Recent Advances in Computational Mechanics and Simulations, 477–85. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-8315-5_40.

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Conference papers on the topic "RUNGE-KUTTA FOURTH ORDER METHOD"

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You, Xiong, Xinmeng Yao, and Xin Shu. "An Optimized Fourth Order Runge-Kutta Method." In 2010 Third International Conference on Information and Computing Science (ICIC). IEEE, 2010. http://dx.doi.org/10.1109/icic.2010.195.

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Nurhakim, Abdurrahman, Nanang Ismail, Hendri Maja Saputra, and Saepul Uyun. "Modified Fourth-Order Runge-Kutta Method Based on Trapezoid Approach." In 2018 4th International Conference on Wireless and Telematics (ICWT). IEEE, 2018. http://dx.doi.org/10.1109/icwt.2018.8527811.

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Hussain, Kasim, Fudziah Ismail, Norazak Senu, and Faranak Rabiei. "Optimized fourth-order Runge-Kutta method for solving oscillatory problems." In INNOVATIONS THROUGH MATHEMATICAL AND STATISTICAL RESEARCH: Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS2016). Author(s), 2016. http://dx.doi.org/10.1063/1.4952512.

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Kalogiratou, Z., Th Monovasilis, and T. E. Simos. "A fourth order modified trigonometrically fitted symplectic Runge-Kutta-Nyström method." In 11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013. AIP, 2013. http://dx.doi.org/10.1063/1.4825719.

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Abel Mejía Marcacuzco, Jesús, and Edwin Pino Vargas. "Computation of Gradually Varied Flow by Fourth Order Runge-Kutta Method (SRK)." In 38th IAHR World Congress. The International Association for Hydro-Environment Engineering and Research (IAHR), 2019. http://dx.doi.org/10.3850/38wc092019-0999.

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Wing, Moo Kwong, Norazak Senu, Fudziah Ismail, and Mohamed Suleiman. "A fourth order phase-fitted Runge-Kutta-Nyström method for oscillatory problems." In PROCEEDINGS OF THE 20TH NATIONAL SYMPOSIUM ON MATHEMATICAL SCIENCES: Research in Mathematical Sciences: A Catalyst for Creativity and Innovation. AIP, 2013. http://dx.doi.org/10.1063/1.4801141.

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Rabiei, Faranak, and Fudziah Ismail. "Fourth order 4-stages improved Runge-Kutta method with minimized error norm." In STATISTICS AND OPERATIONAL RESEARCH INTERNATIONAL CONFERENCE (SORIC 2013). AIP Publishing LLC, 2014. http://dx.doi.org/10.1063/1.4894341.

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Tan, Jiabo. "3-Order Symplectic Runge-Kutta Method Based on Radau-Right Quadrature Formula." In 2012 Fourth International Conference on Computational and Information Sciences (ICCIS). IEEE, 2012. http://dx.doi.org/10.1109/iccis.2012.10.

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Pu Gaojun, Liu Zhongbo, Fang Kezhao, and Kang Haigui. "Modified Boussinesq-Type Water Wave Model Based on Fourth-Order Runge-Kutta Method." In 2013 Fourth International Conference on Digital Manufacturing & Automation (ICDMA). IEEE, 2013. http://dx.doi.org/10.1109/icdma.2013.189.

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Rabiei, Faranak, Fudziah Ismail, and Saeid Emadi. "Solving Fuzzy Differential Equation Using Fourth Order 4-stages Improved Runge-Kutta Method." In Annual International Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2014). GSTF, 2014. http://dx.doi.org/10.5176/2251-1911_cmcgs14.18.

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Reports on the topic "RUNGE-KUTTA FOURTH ORDER METHOD"

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Tang, Hai C. Parallelizing a fourth-order Runge-Kutta method. Gaithersburg, MD: National Institute of Standards and Technology, 1997. http://dx.doi.org/10.6028/nist.ir.6031.

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Trahan, Corey, Jing-Ru Cheng, and Amanda Hines. ERDC-PT : a multidimensional particle tracking model. Engineer Research and Development Center (U.S.), January 2023. http://dx.doi.org/10.21079/11681/48057.

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This report describes the technical engine details of the particle- and species-tracking software ERDC-PT. The development of ERDC-PT leveraged a legacy ERDC tracking model, “PT123,” developed by a civil works basic research project titled “Efficient Resolution of Complex Transport Phenomena Using Eulerian-Lagrangian Techniques” and in part by the System-Wide Water Resources Program. Given hydrodynamic velocities, ERDC-PT can track thousands of massless particles on 2D and 3D unstructured or converted structured meshes through distributed processing. At the time of this report, ERDC-PT supports triangular elements in 2D and tetrahedral elements in 3D. First-, second-, and fourth-order Runge-Kutta time integration methods are included in ERDC-PT to solve the ordinary differential equations describing the motion of particles. An element-by-element tracking algorithm is used for efficient particle tracking over the mesh. ERDC-PT tracks particles along the closed and free surface boundaries by velocity projection and stops tracking when a particle encounters the open boundary. In addition to passive particles, ERDC-PT can transport behavioral species, such as oyster larvae. This report is the first report of the series describing the technical details of the tracking engine. It details the governing equation and numerical approaching associated with ERDC-PT Version 1.0 contents.
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