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Auteur : Grafiati
Publié le 10 mars 2023

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1

Liu, M. Y., L. Zhang et C. F. Zhang. « Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations ». Mathematical Problems in Engineering 2019 (10 octobre 2019) : 1–8. http://dx.doi.org/10.1155/2019/4850872.

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The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.
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Hasan, M. Kamrul, M. Suzan Ahamed, M. S. Alam et M. Bellal Hossain. « An Implicit Method for Numerical Solution of Singular and Stiff Initial Value Problems ». Journal of Computational Engineering 2013 (26 septembre 2013) : 1–5. http://dx.doi.org/10.1155/2013/720812.

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An implicit method has been presented for solving singular initial value problems. The method is simple and gives more accurate solution than the implicit Euler method as well as the second order implicit Runge-Kutta (RK2) (i.e., implicit midpoint rule) method for some particular singular problems. Diagonally implicit Runge-Kutta (DIRK) method is suitable for solving stiff problems. But, the derivation as well as utilization of this method is laborious. Sometimes it gives almost similar solution to the two-stage third order diagonally implicit Runge-Kutta (DIRK3) method and the five-stage fifth order diagonally implicit Runge-Kutta (DIRK5) method. The advantage of the present method is that it is used with less computational effort.
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Muhammad, Raihanatu. « THE ORDER AND ERROR CONSTANT OF A RUNGE-KUTTA TYPE METHOD FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM ». FUDMA JOURNAL OF SCIENCES 4, no 2 (13 octobre 2020) : 743–48. http://dx.doi.org/10.33003/fjs-2020-0402-256.

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Implicit Runge- Kutta methods are used for solving stiff problems which mostly arise in real life situations. Analysis of the order, error constant, consistency and convergence will help in determining an effective Runge- Kutta Method (RKM) to use. Due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation. In this paper, we examine in simpler details how to obtain the order, error constant, consistency and convergence of a Runge -Kutta Type method (RKTM) when the step number .
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Ahmad, S. Z., F. Ismail, N. Senu et 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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IMAI, Yohsuke, Takayuki Aoki et Tetsuya Kobara. « Implicit IDO scheme by using Runge-Kutta method ». Proceedings of The Computational Mechanics Conference 2003.16 (2003) : 151–52. http://dx.doi.org/10.1299/jsmecmd.2003.16.151.

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Janezic, Dusanka, et Bojan Orel. « Implicit Runge-Kutta method for molecular dynamics integration ». Journal of Chemical Information and Modeling 33, no 2 (1 mars 1993) : 252–57. http://dx.doi.org/10.1021/ci00012a011.

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Chauhan, Vijeyata, et 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 (1 avril 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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8

Gardner, David J., Jorge E. Guerra, François P. Hamon, Daniel R. Reynolds, Paul A. Ullrich et Carol S. Woodward. « Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models ». Geoscientific Model Development 11, no 4 (17 avril 2018) : 1497–515. http://dx.doi.org/10.5194/gmd-11-1497-2018.

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Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.
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9

Huang, Juntao, et Chi-Wang Shu. « A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model ». Mathematical Models and Methods in Applied Sciences 27, no 03 (mars 2017) : 549–79. http://dx.doi.org/10.1142/s0218202517500099.

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In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.
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10

Cong, Y. H., et C. X. Jiang. « Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order ». Scientific World Journal 2014 (2014) : 1–7. http://dx.doi.org/10.1155/2014/147801.

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The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.
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11

Nguyen Thu, Thuy. « Parallel iteration of two-step Runge-Kutta methods ». Journal of Science Natural Science 66, no 1 (mars 2021) : 12–24. http://dx.doi.org/10.18173/2354-1059.2021-0002.

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In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.
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12

Do, Nguyen B., Aldo A. Ferri et Olivier A. Bauchau. « Efficient Simulation of a Dynamic System with LuGre Friction ». Journal of Computational and Nonlinear Dynamics 2, no 4 (18 mars 2007) : 281–89. http://dx.doi.org/10.1115/1.2754304.

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Friction is a difficult phenomenon to model and simulate. One promising friction model is the LuGre model, which captures key frictional behavior from experiments and from other friction models. While displaying many modeling advantages, the LuGre model of friction can result in numerically stiff system dynamics. In particular, the LuGre friction model exhibits very slow dynamics during periods of sticking and very fast dynamics during periods of slip. This paper investigates the best simulation strategies for application to dynamic systems with LuGre friction. Several simulation strategies are applied including the explicit Runge–Kutta, implicit Trapezoidal, and implicit Radau-IIA schemes. It was found that both the Runge–Kutta and Radau-IIA methods performed well in simulating the system. The Runge–Kutta method had better accuracy, but the Radau-IIA method required less integration steps.
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13

Ghawadri, Nizam, Norazak Senu, Firas Adel Fawzi, Fudziah Ismail et Zarina Ibrahim. « Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation ». Algorithms 12, no 1 (29 décembre 2018) : 10. http://dx.doi.org/10.3390/a12010010.

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In this study, fifth-order and sixth-order diagonally implicit Runge–Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge–Kutta (RK) methods.
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14

BUTCHER, J. C. « PRACTICAL RUNGE–KUTTA METHODS FOR SCIENTIFIC COMPUTATION ». ANZIAM Journal 50, no 3 (janvier 2009) : 333–42. http://dx.doi.org/10.1017/s1446181109000030.

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AbstractImplicit Runge–Kutta methods have a special role in the numerical solution of stiff problems, such as those found by applying the method of lines to the partial differential equations arising in physical modelling. Of particular interest in this paper are the high-order methods based on Gaussian quadrature and the efficiently implementable singly implicit methods.
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15

Butcher, J. C., et D. J. L. Chen. « A new type of singly-implicit Runge–Kutta method ». Applied Numerical Mathematics 34, no 2-3 (juillet 2000) : 179–88. http://dx.doi.org/10.1016/s0168-9274(99)00126-9.

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16

Qiu, Ruofan, Rongqian Chen et Yancheng You. « An implicit-explicit finite-difference lattice Boltzmann subgrid method on nonuniform meshes ». International Journal of Modern Physics C 28, no 04 (avril 2017) : 1750045. http://dx.doi.org/10.1142/s0129183117500450.

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In this paper, an implicit-explicit finite-difference lattice Boltzmann method with subgrid model on nonuniform meshes is proposed. The implicit-explicit Runge–Kutta scheme, which has good convergence rate, is used for the time discretization and a mixed difference scheme, which combines the upwind scheme with the central scheme, is adopted for the space discretization. Meanwhile, the standard Smagorinsky subgrid model is incorporated into the finite-difference lattice Boltzmann scheme. The effects of implicit-explicit Runge–Kutta scheme and nonuniform meshes of present lattice Boltzmann method are discussed through simulations of a two-dimensional lid-driven cavity flow on nonuniform meshes. Moreover, the comparison simulations of the present method and multiple relaxation time lattice Boltzmann subgrid method are conducted qualitatively and quantitatively.
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17

Nataliya Bondarenko et Vasiliy Pechuk. « CONSTRUCTION OF EXPLICIT RUNGE-KUTTA METHODS FOR MODELING OF DYNAMIC SYSTEMS WITH DELAY ». APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no 99 (17 décembre 2020) : 16–27. http://dx.doi.org/10.32347/0131-579x.2020.99.16-27.

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A system of differential equations with delay is considered, which is a mathematical model of many technical processes with a time delay. Numerical Runge-Kutta methods and the method of expansion onTaylorseries on time delay are most often used to model such systems.When the value of the delay is greater than the step of numerical integration, the numerical solution of these systems is not difficult. For this, interpolation of the prehistory of the model and numerical methods for conventional systems of differential equations are used. For example, explicit Runge-Kutta methods. Using the method of steps, a numerical solution is obtained for the required period of time. But, if the time interval is large enough in comparison with the delay, then the number of steps turns out to be large, which slows down the process of numerical integration and leads to the accumulation of errors.For small delays,Taylortime delay expansions are used and the further solution of the usual system of differential equations by a numerical method, for example, by the Runge-Kutta method. This approach has limitations on the amount of delay and is not applicable for many models.Thus, it is often necessary to apply a step of numerical integration that is larger than the delay value and continuous implicit Runge-Kutta methods, which leads to a complication of the numerical algorithm, since at each step of numerical integration it is necessary to solve systems of nonlinear equations.In this paper, based on the construction of Newton's and Taylor's polynomials, an algorithm has been developed that allows using explicit Runge-Kutta methods to solve systems with delay and the step size of numerical integration is more than the delay value. For systems with delay, an explicit Runge-Kutta method of the fifth order of approximation is constructed on the basis of the most frequently used explicit Runge-Kutta methods of the fifth order of approximation. These methods are convenient in programming, have a higher speed of calculation than implicit methods, and are applicable for steps of numerical integration that are large in comparison with the lag.
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Wu, Jie, Xianbin Du, Yijiang Ma et Peng Ren. « Research of Precise Time Integration Method and its Derived Formats on Helicopter Rotor Dynamics ». International Journal of Computational Methods 17, no 08 (9 juillet 2019) : 1950059. http://dx.doi.org/10.1142/s0219876219500592.

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The aeroelastic coupling dynamic equation of helicopter rotor is essentially a set of nonlinear and inhomogeneous equations with large rigidity, in which the inhomogeneous term is a function of blade motion and aerodynamic load. In this paper, the precise time integration method and its derived formats are introduced to solve the rotor blade dynamic equation, and the Duhamel integral item can be calculated by various numerical methods. In terms of computational accuracy and numerical stability, the precise Kutta method and high precision direct integration method (HPD method) are carefully selected to compare with classical Runge–Kutta method numerically. HPD method is used to solve the rotor blade dynamic equation, and the transient response of the rotor blade is examined by Newmark and implicit trapezoidal methods. Results indicate that HPD method dominates the classical Runge–Kutta method in step size independence, and gets close to implicit methods in numerical stability and accuracy for dynamic equation of helicopter rotor blade.
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19

Ahmad, N. A., N. Senu, Z. B. Ibrahim et M. Othman. « Stability Analysis of Diagonally Implicit Two Derivative Runge-Kutta methods for Solving Delay Differential Equations ». Malaysian Journal of Mathematical Sciences 16, no 2 (29 avril 2022) : 215–35. http://dx.doi.org/10.47836/mjms.16.2.04.

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The stability properties of fourth and fifth-order Diagonally Implicit Two Derivative Runge-Kutta method (DITDRK) combined with Lagrange interpolation when applied to the linear Delay Differential Equations (DDEs) are investigated. This type of stability is known as P-stability and Q-stability. Their stability regions for (λ,μ∈R) and (μ∈C,λ=0) are determined. The superiority of the DITDRK methods over other same order existing Diagonally Implicit Runge-Kutta (DIRK) methods when solving DDEs problems are clearly demonstrated by plotting the efficiency curves of the log of both maximum errors versus function evaluations and the CPU time taken to do the integration.
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20

Senu, Norazak, Mohamed Suleiman, Fudziah Ismail et Norihan Md Arifin. « New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs ». Discrete Dynamics in Nature and Society 2012 (2012) : 1–20. http://dx.doi.org/10.1155/2012/324989.

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New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.
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21

Senu, Norazak, Nur Amirah Ahmad, Zarina Bibi Ibrahim et Mohamed Othman. « Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS ». Sains Malaysiana 50, no 6 (30 juin 2021) : 1799–814. http://dx.doi.org/10.17576/jsm-2021-5006-25.

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A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.
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22

Williams, Russell, Kevin Burrage, Ian Cameron et Minnie Kerr. « A four-stage index 2 Diagonally Implicit Runge–Kutta method ». Applied Numerical Mathematics 40, no 3 (février 2002) : 415–32. http://dx.doi.org/10.1016/s0168-9274(01)00090-3.

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23

Xie, Dexuan. « An improved approximate Newton method for implicit Runge–Kutta formulas ». Journal of Computational and Applied Mathematics 235, no 17 (juillet 2011) : 5249–58. http://dx.doi.org/10.1016/j.cam.2011.05.027.

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24

Mahmoud, Sayed, et Xiaojun Chen. « A verified inexact implicit Runge–Kutta method for nonsmooth ODEs ». Numerical Algorithms 47, no 3 (19 février 2008) : 275–90. http://dx.doi.org/10.1007/s11075-008-9180-0.

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25

Jator, S. N. « Implicit third derivative Runge-Kutta-Nyström method with trigonometric coefficients ». Numerical Algorithms 70, no 1 (15 novembre 2014) : 133–50. http://dx.doi.org/10.1007/s11075-014-9938-5.

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26

Attili, Basem S., Khalid Furati et Muhammed I. Syam. « An efficient implicit Runge–Kutta method for second order systems ». Applied Mathematics and Computation 178, no 2 (juillet 2006) : 229–38. http://dx.doi.org/10.1016/j.amc.2005.11.044.

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27

Baker, Donald L. « A Second-Order Diagonally Implicit Runge-Kutta Time-Stepping Method ». Ground Water 31, no 6 (novembre 1993) : 890–95. http://dx.doi.org/10.1111/j.1745-6584.1993.tb00861.x.

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Sommeijer, B. P. « A note on a diagonally implicit Runge-Kutta-Nyström method ». Journal of Computational and Applied Mathematics 19, no 3 (septembre 1987) : 395–99. http://dx.doi.org/10.1016/0377-0427(87)90208-1.

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Bruder, Jürgen. « Numerical results for a parallel linearly-implicit Runge-Kutta method ». Computing 59, no 2 (juin 1997) : 139–51. http://dx.doi.org/10.1007/bf02684476.

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30

Iavernaro, Felice, et Francesca Mazzia. « A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods ». Mathematics 9, no 10 (13 mai 2021) : 1103. http://dx.doi.org/10.3390/math9101103.

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The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.
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31

Rahim, Y. F., et M. E. H. Hafidzuddin. « THREE POINTS BLOCK EMBEDDED DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING ODES ». Advances in Mathematics : Scientific Journal 10, no 11 (23 novembre 2021) : 3449–60. http://dx.doi.org/10.37418/amsj.10.11.6.

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Block Embedded Diagonally Implicit Runge-Kutta (BEDIRK4(3)) me- thod derived using Butcher analysis and equi-distribution of error approach is outperformed standard Runge-Kutta (RK) formulae. BEDIRK4(3) method produces approximation to the solution of initial value problem (IVP) at a block of three points simultaneously. The standard one step RK3(2) method is used to approximate the solution at the first point of the block. At the second points the solution is approximated using RK4(2) method which is generated by the previous research. The same approach is used to obtain the solution at the third point. The code for this method was built and the algorithm developed is suitable for solving stiff system. The efficiency of the method is supported by some numerical results.
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Honda, Ryuma, Hiroki Suzuki et Shinsuke Mochizuki. « Impact of difference between explicit and implicit second-order time integration schemes on isotropic/anisotropic steady incompressible turbulence field ». Journal of Physics : Conference Series 2090, no 1 (1 novembre 2021) : 012145. http://dx.doi.org/10.1088/1742-6596/2090/1/012145.

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Abstract This study presents the impact of the difference between the implicit and explicit time integration methods on a steady turbulent flow field. In contrast to the explicit time integration method, the implicit time integration method may produce significant kinetic energy conservation error because the widely used spatial difference method for discretizing the governing equations is explicit with respect to time. In this study, the second-order Crank-Nicolson method is used as the implicit time integration method, and the fourth-order Runge-Kutta, second-order Runge-Kutta and second-order Adams-Bashforth methods are used as explicit time integration methods. In the present study, both isotropic and anisotropic steady turbulent fields are analyzed with two values of the Reynolds number. The turbulent kinetic energy in the steady turbulent field is hardly affected by the kinetic energy conservation error. The rms values of static pressure fluctuation are significantly sensitive to the kinetic energy conservation error. These results are examined by varying the time increment value. These results are also discussed by visualizing the large scale turbulent vortex structure.
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Ullrich, Paul, et Christiane Jablonowski. « Operator-Split Runge–Kutta–Rosenbrock Methods for Nonhydrostatic Atmospheric Models ». Monthly Weather Review 140, no 4 (avril 2012) : 1257–84. http://dx.doi.org/10.1175/mwr-d-10-05073.1.

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This paper presents a new approach for discretizing the nonhydrostatic Euler equations in Cartesian geometry using an operator-split time-stepping strategy and unstaggered upwind finite-volume model formulation. Following the method of lines, a spatial discretization of the governing equations leads to a set of coupled nonlinear ordinary differential equations. In general, explicit time-stepping methods cannot be applied directly to these equations because the large aspect ratio between the horizontal and vertical grid spacing leads to a stringent restriction on the time step to maintain numerical stability. Instead, an A-stable linearly implicit Rosenbrock method for evolving the vertical components of the equations coupled to a traditional explicit Runge–Kutta formula in the horizontal is proposed. Up to third-order temporal accuracy is achieved by carefully interleaving the explicit and linearly implicit steps. The time step for the resulting Runge–Kutta–Rosenbrock–type semi-implicit method is then restricted only by the grid spacing and wave speed in the horizontal. The high-order finite-volume model is tested against a series of atmospheric flow problems to verify accuracy and consistency. The results of these tests reveal that this method is accurate, stable, and applicable to a wide range of atmospheric flows and scales.
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34

Gorgey, Annie, et Nor Azian Aini Mat. « Efficiency of Runge-Kutta Methods in Solving Simple Harmonic Oscillators ». MATEMATIKA 34, no 1 (28 mai 2018) : 1–12. http://dx.doi.org/10.11113/matematika.v34.n1.1039.

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Symmetric methods such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR) and 2-stage Gauss method are beneficial in solving Hamiltonian problems since they are also symplectic. Symplectic methods have advantages over non-symplectic methods in the long term integration of Hamiltonian problems. The study is to show the efficiency of IMR, ITR and the 2-stage Gauss method in solving simple harmonic oscillators (SHO). This study is done theoretically and numerically on the simple harmonic oscillator problem. The theoretical analysis and numerical results on SHO problem showed that the magnitude of the global error for a symmetric or symplectic method with stepsize h is linearly dependent on time t. This gives the linear error growth when a symmetric or symplectic method is applied to the simple harmonic oscillator problem. Passive and active extrapolations have been implemented to improve the accuracy of the numerical solutions. Passive extrapolation is observed to show quadratic error growth after a very short period of time. On the other hand, active extrapolation is observed to show linear error growth for a much longer period of time.
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35

Hằng, Phạm Thị Thu. « PREDICTOR-CORRECTOR TECHNIQUE FOR IMPLEMENTING AN SIXTH ORDER IMPLICIT RUNGE-KUTTA METHOD ». TNU Journal of Science and Technology 226, no 15 (30 novembre 2021) : 60–67. http://dx.doi.org/10.34238/tnu-jst.5230.

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Bài báo này đưa ra hướng tiếp cận mới đối với bài toán xây dựng trình thực thi cho một lớp các phương pháp Runge-Kutta dạng ẩn. Phương pháp Runge-Kutta dạng ẩn được nghiên cứu cụ thể ở đây được phát triển dựa trên các đa thức Gauss-Legendre, phương pháp xuất hiện đầu tiên trong bài báo của J. C. Butcher (2009). Sự cải tiến mà hướng tiếp cận mới mang lại là rất hữu ích. Điều này có được do những lợi thế của phương pháp một bước dạng ẩn chỉ có ba bước, đặc biệt phù hợp với các bài toán stiff, với khối lượng tính toán nhỏ mà độ chính xác cao của một phương pháp bậc sáu, một bậc tương đối cao của sự hội tụ mà vẫn đảm bảo điều kiện bền vững. Chứng minh cho sự hội tụ của phương pháp này được ra. Hướng tiếp cận này cũng có thể áp dụng cho một phương pháp Runge-Kutta dạng ẩn khác được đưa ra với bậc thấp hơn được xây dựng dựa trên các đa thức Gauss-Legendre. Sự kết hợp giữa hướng tiếp cận mới và phương pháp sai phân dạng khối Off-step bậc sáu có thể mang đến sự hợp lý trong việc xấp xỉ các bài toán stiff. Phương pháp này cũng được nghiên cứu trong bài báo. Sau cùng, các so sánh thực nghiệm đưa ra nhằm minh họa cho sự ưu việt của hướng tiếp cận đạt được.
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36

Sharifi, Mohammad, Ali Abdi, Michal Braś et Gholamreza Hojjati. « HIGH ORDER SECOND DERIVATIVE DIAGONALLY IMPLICIT MULTISTAGE INTEGRATION METHODS FOR ODES ». Mathematical Modelling and Analysis 28, no 1 (19 janvier 2023) : 53–70. http://dx.doi.org/10.3846/mma.2023.16102.

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Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.
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37

Van Hecke, T., M. Van Daele, G. Vanden Berghe et H. De Meyer. « A mono-implicit Runge-Kutta-Nyström modification of the Numerov method ». Journal of Computational and Applied Mathematics 78, no 1 (février 1997) : 161–77. http://dx.doi.org/10.1016/s0377-0427(96)00139-2.

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38

Janezic, Dusanka, et Roman Trobec. « Parallelization of an Implicit Runge-Kutta Method for Molecular Dynamics Integration ». Journal of Chemical Information and Modeling 34, no 3 (1 mai 1994) : 641–46. http://dx.doi.org/10.1021/ci00019a025.

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39

Cîrulis, T., et O. Lietuvietis. « DEGENERATE MATRIX METHOD FOR SOLVING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS ». Mathematical Modelling and Analysis 3, no 1 (15 décembre 1998) : 45–56. http://dx.doi.org/10.3846/13926292.1998.9637085.

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Degenerate matrix method for numerical solving nonlinear systems of ordinary differential equations is considered. The method is based on an application of special degenerate matrix and usual iteration procedure. The method, which is connected with an implicit Runge‐Kutta method, can be simply realized on computers. An estimation for the error of the method is given.
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40

ABDULLE, ASSYR, et GILLES VILMART. « COUPLING HETEROGENEOUS MULTISCALE FEM WITH RUNGE–KUTTA METHODS FOR PARABOLIC HOMOGENIZATION PROBLEMS : A FULLY DISCRETE SPACETIME ANALYSIS ». Mathematical Models and Methods in Applied Sciences 22, no 06 (26 avril 2012) : 1250002. http://dx.doi.org/10.1142/s0218202512500029.

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Numerical methods for parabolic homogenization problems combining finite element methods (FEMs) in space with Runge–Kutta methods in time are proposed. The space discretization is based on the coupling of macro and micro finite element methods following the framework of the Heterogeneous Multiscale Method (HMM). We present a fully discrete analysis in both space and time. Our analysis relies on new (optimal) error bounds in the norms L2(H1), [Formula: see text], and [Formula: see text] for the fully discrete analysis in space. These bounds can then be used to derive fully discrete spacetime error estimates for a variety of Runge–Kutta methods, including implicit methods (e.g. Radau methods) and explicit stabilized method (e.g. Chebyshev methods). Numerical experiments confirm our theoretical convergence rates and illustrate the performance of the methods.
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41

Li, Qin, et Xu Yang. « Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation ». Communications in Computational Physics 15, no 4 (avril 2014) : 996–1011. http://dx.doi.org/10.4208/cicp.010113.160813s.

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AbstractThis paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi,SIAM Numer. Anal., 49 (2011), pp. 2057-2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied on, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.
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42

Oje, Eric Augustine, et Aggrey Eric Majuk. « SEMI-IMPLICIT RATIONAL RUNGE-KUTTA METHOD OF SOLVING SECOND ORDER DIFFERENTIAL EQUATION ». International Journal of Engineering Applied Sciences and Technology 04, no 09 (30 janvier 2020) : 104–15. http://dx.doi.org/10.33564/ijeast.2020.v04i09.011.

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43

Imoni, S. O., F. O. Otunta et T. R. Ramamohan. « Embedded implicit Runge–Kutta Nyström method for solving second-order differential equations ». International Journal of Computer Mathematics 83, no 11 (novembre 2006) : 777–84. http://dx.doi.org/10.1080/00207160601084505.

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44

Ozawa, Kazufumi. « A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method ». Japan Journal of Industrial and Applied Mathematics 22, no 3 (octobre 2005) : 403–27. http://dx.doi.org/10.1007/bf03167492.

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45

Liao, Wenyuan, et Yulian Yan. « Singly diagonally implicit runge-kutta method for time-dependent reaction-diffusion equation ». Numerical Methods for Partial Differential Equations 27, no 6 (26 avril 2010) : 1423–41. http://dx.doi.org/10.1002/num.20589.

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46

Li, Liang, et Songping Wu. « A Hybrid Time Integration Scheme for the Discontinuous Galerkin Discretizations of Convection-Dominated Problems ». Energies 13, no 8 (11 avril 2020) : 1870. http://dx.doi.org/10.3390/en13081870.

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Discontinuous Galerkin (DG) method is a popular high-order accurate method for solving unsteady convection-dominated problems. After spatially discretizing the problem with the DG method, a time integration scheme is necessary for evolving the result. Owing to the stability-based restriction, the time step for an explicit scheme is limited by the smallest element size within the mesh, making the calculation inefficient. In this paper, a hybrid scheme comprising a three-stage, third-order accurate, and strong stability preserving Runge–Kutta (SSP-RK3) scheme and the three-stage, third-order accurate, L-stable, and diagonally implicit Runge–Kutta (LDIRK3) scheme is proposed. By dealing with the coarse and the refined elements with the explicit and implicit schemes, respectively, the time step for the hybrid scheme is free from the limitation of the smallest element size, making the simulation much more efficient. Numerical tests and comparison studies were made to show the performance of the hybrid scheme.
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47

Lee, Hyun Geun. « Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation ». Mathematics 8, no 1 (19 décembre 2019) : 11. http://dx.doi.org/10.3390/math8010011.

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Strong-stability-preserving (SSP) implicit–explicit (IMEX) Runge–Kutta (RK) methods for the Cahn–Hilliard (CH) equation with a polynomial double-well free energy density were presented in a previous work, specifically H. Song’s “Energy SSP-IMEX Runge–Kutta Methods for the Cahn–Hilliard Equation” (2016). A linear convex splitting of the energy for the CH equation with an extra stabilizing term was used and the IMEX technique was combined with the SSP methods. And unconditional strong energy stability was proved only for the first-order methods. Here, we use a nonlinear convex splitting of the energy to remove the condition for the convexity of split energies and give a stability condition for the coefficients of the second-order method to preserve the discrete energy dissipation law. Along with a rigorous proof, numerical experiments are presented to demonstrate the accuracy and unconditional strong energy stability of the second-order method.
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48

Van Lent, Jan, et Stefan Vandewalle. « Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs ». SIAM Journal on Scientific Computing 27, no 1 (janvier 2005) : 67–92. http://dx.doi.org/10.1137/030601144.

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49

Tong, T. O., M. C. Kekana, M. Y. Shatalov et S. P. Moshokoa. « Accuracy Tests on Built-In Algorithms Applied to the Lorenz System ». Journal of Computational and Theoretical Nanoscience 16, no 10 (1 octobre 2019) : 4064–71. http://dx.doi.org/10.1166/jctn.2019.8486.

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This work investigate, An idea of checking accuracy of algorithms from mathematical black box by means of residual functions. Lorenz system is used as case study as the chaotic system does not have analytical solution. The numerical procedures examined include BDF, Adams method and Implicit Runge Kutta methods. The interval of numerical results is t ∈ [0; 10].
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50

Yu, Jui-Ling. « Adaptive Optimal -Stage Runge-Kutta Methods for Solving Reaction-Diffusion-Chemotaxis Systems ». Journal of Applied Mathematics 2011 (2011) : 1–25. http://dx.doi.org/10.1155/2011/389207.

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We present a class of numerical methods for the reaction-diffusion-chemotaxis system which is significant for biological and chemistry pattern formation problems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. Along with the implementation of the method of lines, implicit or semi-implicit schemes are typical time stepping solvers to reduce the effect on time step constrains due to the stability condition. However, these two schemes are usually difficult to employ. In this paper, we propose an adaptive optimal time stepping strategy for the explicit -stage Runge-Kutta method to solve reaction-diffusion-chemotaxis systems. Instead of relying on empirical approaches to control the time step size, variable time step sizes are given explicitly. Yet, theorems about stability and convergence of the algorithm are provided in analyzing robustness and efficiency. Numerical experiment results on a testing problem and a real application problem are shown.
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