Relevant bibliographies by topics / Implicit Runge-Kutta method

Academic literature on the topic 'Implicit Runge-Kutta method'

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Published: 10 March 2023

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Journal articles on the topic "Implicit Runge-Kutta method"

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Liu, M. Y., L. Zhang, and C. F. Zhang. "Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations." Mathematical Problems in Engineering 2019 (October 10, 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, and M. Bellal Hossain. "An Implicit Method for Numerical Solution of Singular and Stiff Initial Value Problems." Journal of Computational Engineering 2013 (September 26, 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 (October 13, 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, 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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IMAI, Yohsuke, Takayuki Aoki, and 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, and Bojan Orel. "Implicit Runge-Kutta method for molecular dynamics integration." Journal of Chemical Information and Modeling 33, no. 2 (March 1, 1993): 252–57. http://dx.doi.org/10.1021/ci00012a011.

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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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Gardner, David J., Jorge E. Guerra, François P. Hamon, Daniel R. Reynolds, Paul A. Ullrich, and Carol S. Woodward. "Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models." Geoscientific Model Development 11, no. 4 (April 17, 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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Huang, Juntao, and 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 (March 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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Cong, Y. H., and 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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Dissertations / Theses on the topic "Implicit Runge-Kutta method"

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Roberts, Steven Byram. "Multimethods for the Efficient Solution of Multiscale Differential Equations." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/104872.

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Mathematical models involving ordinary differential equations (ODEs) play a critical role in scientific and engineering applications. Advances in computing hardware and numerical methods have allowed these models to become larger and more sophisticated. Increasingly, problems can be described as multiphysics and multiscale as they combine several different physical processes with different characteristics. If just one part of an ODE is stiff, nonlinear, chaotic, or rapidly-evolving, this can force an expensive method or a small timestep to be used. A method which applies a discretization and timestep uniformly across a multiphysics problem poorly utilizes computational resources and can be prohibitively expensive. The focus of this dissertation is on "multimethods" which apply different methods to different partitions of an ODE. Well-designed multimethods can drastically reduce the computation costs by matching methods to the individual characteristics of each partition while making minimal concessions to stability and accuracy. However, they are not without their limitations. High order methods are difficult to derive and may suffer from order reduction. Also, the stability of multimethods is difficult to characterize and analyze. The goals of this work are to develop new, practical multimethods and to address these issues. First, new implicit multirate Runge–Kutta methods are analyzed with a special focus on stability. This is extended into implicit multirate infinitesimal methods. We introduce approaches for constructing implicit-explicit methods based on Runge–Kutta and general linear methods. Finally, some unique applications of multimethods are considered including using surrogate models to accelerate Runge–Kutta methods and eliminating order reduction on linear ODEs with time-dependent forcing.
Doctor of Philosophy
Almost all time-dependent physical phenomena can be effectively described via ordinary differential equations. This includes chemical reactions, the motion of a pendulum, the propagation of an electric signal through a circuit, and fluid dynamics. In general, it is not possible to find closed-form solutions to differential equations. Instead, time integration methods can be employed to numerically approximate the solution through an iterative procedure. Time integration methods are of great practical interest to scientific and engineering applications because computational modeling is often much cheaper and more flexible than constructing physical models for testing. Large-scale, complex systems frequently combine several coupled processes with vastly different characteristics. Consider a car where the tires spin at several hundred revolutions per minute, while the suspension has oscillatory dynamics that is orders of magnitude slower. The brake pads undergo periods of slow cooling, then sudden, rapid heating. When using a time integration scheme for such a simulation, the fastest dynamics require an expensive and small timestep that is applied globally across all aspects of the simulation. In turn, an unnecessarily large amount of work is done to resolve the slow dynamics. The goal of this dissertation is to explore new "multimethods" for solving differential equations where a single time integration method using a single, global timestep is inadequate. Multimethods combine together existing time integration schemes in a way that is better tailored to the properties of the problem while maintaining desirable accuracy and stability properties. This work seeks to overcome limitations on current multimethods, further the understanding of their stability, present new applications, and most importantly, develop methods with improved efficiency.
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Ijaz, Muhammad. "Implicit runge-kutta methods to simulate unsteady incompressible flows." Texas A&M University, 2007. http://hdl.handle.net/1969.1/85850.

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A numerical method (SIMPLE DIRK Method) for unsteady incompressible viscous flow simulation is presented. The proposed method can be used to achieve arbitrarily high order of accuracy in time-discretization which is otherwise limited to second order in majority of the currently used simulation techniques. A special class of implicit Runge-Kutta methods is used for time discretization in conjunction with finite volume based SIMPLE algorithm. The algorithm was tested by solving for velocity field in a lid-driven square cavity. In the test case calculations, power law scheme was used in spatial discretization and time discretization was performed using a second-order implicit Runge-Kutta method. Time evolution of velocity profile along the cavity centerline was obtained from the proposed method and compared with that obtained from a commercial computational fluid dynamics software program, FLUENT 6.2.16. Also, steady state solution from the present method was compared with the numerical solution of Ghia, Ghia, and Shin and that of Erturk, Corke, and Goökçöl. Good agreement of the solution of the proposed method with the solutions of FLUENT; Ghia, Ghia, and Shin; and Erturk, Corke, and Goökçöl establishes the feasibility of the proposed method.
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Biehn, Neil David. "Implicit Runge-Kutta Methods for Stiff and Constrained Optimal Control Problems." NCSU, 2001. http://www.lib.ncsu.edu/theses/available/etd-20010322-165913.

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The purpose of the research presented in this thesis is to better understand and improve direct transcription methods for stiff and state constrained optimal control problems. When some implicit Runge-Kutta methods are implemented as approximations to the dynamics of an optimal control problem, a loss of accuracy occurs when the dynamics are stiff or constrained. A new grid refinement strategy which exploits the variation of accuracy is discussed. In addition, the use of a residual function in place of classical error estimation techniques is proven to work well for stiff systems. Computational experience reveals the improvement in efficiency and reliability when the new strategies are incorporated as part of a direct transcription algorithm. For index three differential-algebraic equations, the solutions of some implicit Runge-Kutta methods may not converge. However, computational experience reveals apparent convergence for the same methods used when index three state inequality constraints become active. It is shown that the solution chatters along the constraint boundary allowing for better approximations. Moreover, the consistency of the nonlinear programming problem formed by a direct transcription algorithm using an implicit Runge-Kutta approximation is proven for state constraints of arbitrary index.

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Al-Harbi, Saleh M. "Implicit Runge-Kutta methods for the numerical solution of stiff ordinary differential equation." Thesis, University of Manchester, 1999. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.488322.

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The primary aim of this thesis is to calculate the numerical solution of a given stiff system of ordinary differential equations. We deal with the implementation of the implicit Runge-Kutta methods, in particular for Radau IIA order 5 which is now a competitive method for solving stiff initial value problems. New software based on Radau IIA, called IRKMR5 written in MATLAB has been developed for fixed order (order 5) with variable stepsizes, which is quite efficient when it is used to solve stiff problems. The code is organized in a modular form so that it facilitates both the understanding of it and its modification whenever needed. The new software is not only more functional than its Fortran 77 Radau IIA counterpart but also more robust and better documented. When implicit methods are used to solve nonlinear problems it is necessary to solve systems of nonlinear algebraic equations. New investigations for a modified Newton iteration are undertaken. This new strategy manages the iterative solutions of nonlinear equations in the ODEs solver. It also involves when to re-evaluate the Jacobian and the iteration matrix. The strategy also significantly reduces the number of function evaluations and linear solves. We subsequently consider the mathematical analysis of the nonlinear algebraic equations that arise from using s-stage fully implicit Runge-Kutta methods. Results for uniqueness of solutions and an error bound was established. The termination criterion in the iterative solution of the nonlinear equations is also studied as well as two types of termination criterion (displacement and the residual test). The residual test has been compared with the displacement test on some test examples and the results are tabulated.
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Wood, Dylan M. "Solving Unsteady Convection-Diffusion Problems in One and More Dimensions with Local Discontinuous Galerkin Methods and Implicit-Explicit Runge-Kutta Time Stepping." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1461181441.

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Santos, Ricardo Dias dos. "Uma formulação implícita para o método Smoothed Particle Hydrodynamics." Universidade do Estado do Rio de Janeiro, 2014. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=6751.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Em uma grande gama de problemas físicos, governados por equações diferenciais, muitas vezes é de interesse obter-se soluções para o regime transiente e, portanto, deve-se empregar técnicas de integração temporal. Uma primeira possibilidade seria a de aplicar-se métodos explícitos, devido à sua simplicidade e eficiência computacional. Entretanto, esses métodos frequentemente são somente condicionalmente estáveis e estão sujeitos a severas restrições na escolha do passo no tempo. Para problemas advectivos, governados por equações hiperbólicas, esta restrição é conhecida como a condição de Courant-Friedrichs-Lewy (CFL). Quando temse a necessidade de obter soluções numéricas para grandes períodos de tempo, ou quando o custo computacional a cada passo é elevado, esta condição torna-se um empecilho. A fim de contornar esta restrição, métodos implícitos, que são geralmente incondicionalmente estáveis, são utilizados. Neste trabalho, foram aplicadas algumas formulações implícitas para a integração temporal no método Smoothed Particle Hydrodynamics (SPH) de modo a possibilitar o uso de maiores incrementos de tempo e uma forte estabilidade no processo de marcha temporal. Devido ao alto custo computacional exigido pela busca das partículas a cada passo no tempo, esta implementação só será viável se forem aplicados algoritmos eficientes para o tipo de estrutura matricial considerada, tais como os métodos do subespaço de Krylov. Portanto, fez-se um estudo para a escolha apropriada dos métodos que mais se adequavam a este problema, sendo os escolhidos os métodos Bi-Conjugate Gradient (BiCG), o Bi-Conjugate Gradient Stabilized (BiCGSTAB) e o Quasi-Minimal Residual (QMR). Alguns problemas testes foram utilizados a fim de validar as soluções numéricas obtidas com a versão implícita do método SPH.
In a wide range of physical problems governed by differential equations, it is often of interest to obtain solutions for the unsteady state and therefore it must be employed temporal integration techniques. One possibility could be the use of an explicit methods due to its simplicity and computational efficiency. However, these methods are often only conditionally stable and are subject to severe restrictions for the time step choice. For advective problems governed by hyperbolic equations, this restriction is known as the Courant-Friedrichs-Lewy (CFL) condition. When there is the need to obtain numerical solutions for long periods of time, or when the computational cost for each time step is high, this condition becomes a handicap. In order to overcome this restriction implicit methods can be used, which are generally unconditionally stable. In this study, some implicit formulations for time integration are used in the Smoothed Particle Hydrodynamics (SPH) method to enable the use of larger time increments and obtain a strong stability in the time evolution process. Due to the high computational cost required by the particles tracking at each time step, the implementation will be feasible only if efficient algorithms were applied for this type of matrix structure such as Krylov subspace methods. Therefore, we carried out a study for the appropriate choice of methods best suited to this problem, and the methods chosen were the Bi-Conjugate Gradient (BiCG), the Bi-Conjugate Gradient Stabilized (BiCGSTAB) and the Quasi-Minimal Residual(QMR). Some test problems were used to validate the numerical solutions obtained with the implicit version of the SPH method.
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Scandurra, Leonardo. "Numerical Methods for All Mach Number flows for Gas Dynamics." Doctoral thesis, Università di Catania, 2017. http://hdl.handle.net/10761/4042.

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An original numerical method to solve the all-Mach number flow for the Euler equations of gas dynamics on staggered grid is presented in this thesis. The system is discretized to second order in space on staggered grid, in a fashion similar to the Nessyahu-Tadmor central scheme for 1D model and Jang-Tadmor central scheme for 2D model, thus simplifying the flux computation. This approach turns out to be extremely simple, since it requires no equation splitting. We consider the isentropic case and the general case. For simplicity we assume a gamma-law gas in both cases. Both approaches are based on IMEX strategy, in which some term is treated explicitly, while other terms are treated implicitly, thus avoiding the classical CFL restriction due to acoustic waves. - In Isentropic Euler Case: The core if the implicit term contains a non-linear elliptic equation for the pressure, which has to be treated by a fully implicit technique. Because of the non-linearity, it is necessary to adopt an iterative method to compute the pressure. In our numerical experiments Newton's method worked with few iterations. - General Euler Case: In this case the implicit term is treated in a semi-implicit fashion, thus avoiding the use of Newton's iterations. In both cases the schemes are implemented to second order accuracy in time. Suitably well-prepared initial conditions are considered, which depend on the Mach number. In one space dimension we obtain the same profiles found in the literature for the isentropic case and for the general Euler system for all Mach numbers. In particular, the schemes have been shown to be AP, in the sense that they become a consistent discretizzation of the incompressible Euler equation as the Mach number approaches zero. Numerical evidence of such AP property is provided on a two dimensional test case. The last chapter deals with the piston problem in Lagrangian coordinates treated by a semi-implicit scheme. The implicit treatment of the boundary conditions is originally developed in the thesis. It is shown that for very low Mach number the scheme is able to recover the adiabatic solution with very large CFL numbers. For moderate Mach numbers, or in presence of an initial acoustic wave, loss of accuracy is observerd if the CFL is too large. This drawback can be cured by using a suitable time step control, which will be subject of future investigation. Current work is also related on the development of higher order accurate schemes for 1D and 2D problems.
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AbuAlSaud, Moataz. "Simulation of 2-D Compressible Flows on a Moving Curvilinear Mesh with an Implicit-Explicit Runge-Kutta Method." Thesis, 2012. http://hdl.handle.net/10754/244571.

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The purpose of this thesis is to solve unsteady two-dimensional compressible Navier-Stokes equations for a moving mesh using implicit explicit (IMEX) Runge- Kutta scheme. The moving mesh is implemented in the equations using Arbitrary Lagrangian Eulerian (ALE) formulation. The inviscid part of the equation is explicitly solved using second-order Godunov method, whereas the viscous part is calculated implicitly. We simulate subsonic compressible flow over static NACA-0012 airfoil at different angle of attacks. Finally, the moving mesh is examined via oscillating the airfoil between angle of attack = 0 and = 20 harmonically. It is observed that the numerical solution matches the experimental and numerical results in the literature to within 20%.
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Roskovec, Filip. "Numerické řešení nelineárních problémů konvekce-difuze pomocí adaptivních metod." Master's thesis, 2014. http://www.nusl.cz/ntk/nusl-340765.

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This thesis is concerned with analysis and implementation of Time discontinuous Galerkin method. Important part of it is constructing of algorithm for solving nonlinear convection-diffusion equations, which combines Discontinuous Galerkin method in space (DGFEM) with Time discontinuous Galerkin method (TDG). Nonlinearity of the problem is overcome by damped Newton-like method. This approach provides easy adaptivity manipulation as well as high order approximation with respect to both space and time variables. The second part of the thesis is focused on Time discontinuous Galerkin method, applied to ordinary differential equations. It is shown that the solution of Time discontinuous Galerkin equals the solution obtained by Radau IIA implicit Runge-Kutta method in the roots of right Radau Quadrature. By virtue of this relation, error estimates of the order higher by one than the standard order can be obtained in these points. Furthermore, almost two times higher order can be achieved in the endpoints of the intervals of time discretization. Finally, the thesis deals with the phenomenon of stiffness, which may dramatically decrease the order of the applied method. The theoretical results are verified by numerical experiments. Powered by TCPDF (www.tcpdf.org)
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FRASCA, CACCIA GIANLUCA. "A new efficient implementation for HBVMs and their application to the semilinear wave equation." Doctoral thesis, 2015. http://hdl.handle.net/2158/992629.

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In this thesis we have provided a detailed description of the low-rank Runge-Kutta family of Hamiltonian Boundary Value Methods (HBVMs) for the numerical solution of Hamiltonian problems. In particular, we have studied in detail their main property: the conservation of polynomial Hamiltonians, which results into a practical conservation for generic suitably regular Hamiltonians. This property turns out to play a fundamental role in some problems where the error on the Hamiltonian, usually obtained even when using a symplectic method, would be not negligible to the point of affecting the dynamics of the numerical solution. The research developed in this thesis has addressed two main topics. The first one is a new procedure, based on a particular splitting of the matrix defining the method, which turns out to be more effective of the well-known blended-implementation, as well as of a classical fixed-point iteration when the problem at hand is stiff. This procedure has been applied also to second order problems with separable Hamiltonian function, resulting in a cheaper computational cost. The second topic addressed is the application of HBVMs for the full discretization of a method of lines approach to numerically solve Hamiltonian PDEs. In particular, we have considered the semilinear wave equation coupled with either periodic, Dirichlet or Neumann boundary conditions, and the application of a (practically) energy conserving HBVM method to the semi-discrete problem obtained by means of a second order finite-difference approximation in space. When the problem is coupled with periodic boundary conditions we have also considered the case of higher-order finite-difference spatial discretizations and the case when a Fourier-Galerkin method is used for the spatial semi-discretization. The proposed methods are able to provide a numerical solution such that the energy (which can be conserved or not, depending on the assigned boundary conditions) practically satisfies its prescribed variation in time. A few numerical tests for the sine-Gordon equation have given evidence that, for some problems, there is an effective advantage in using an energy-conserving method for the time integration, with respect to the use of a symplectic one. Moreover, even though HBVMs are implicit method, their computational cost for the considered problem turns out to be competitive even with respect to that of explicit solvers of the same order, which, furthermore, may suffer from stepsize restrictions due to stability reasons, whereas HBVMs are A-stable.
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Books on the topic "Implicit Runge-Kutta method"

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Keeling, Stephen L. On implicit Runge-Kutta methods for parallel computations. Hampton, Va: ICASE, 1987.

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Center, Langley Research, ed. On implicit Runge-Kutta methods for parallel computations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1987.

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Center, Langley Research, ed. On implicit Runge-Kutta methods for parallel computations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1987.

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Loon, M. van. Time-step enlargement for Runge-Kutta integration algorithms by implicit smoothing. Amsterdam: National Aerospace Laboratory, 1991.

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Davidson, Lars. Implementation of a semi-implicit k-e turbulence model into an explicit Runge-Kutta Navier-Stokes code. Toulouse: CERFACS, 1990.

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United States. National Aeronautics and Space Administration., ed. Flow simulations about steady-complex and unsteady moving configurations using structured-overlapped and unstructured grids: Abstract. [Washington, D.C: National Aeronautics and Space Administration, 1995.

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Nguyen, Hung. Interpolation and error control schemes for algebraic differential equations using continuous implicit Runge-Kutta methods. Toronto: University of Toronto, Dept. of Computer Science, 1995.

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Book chapters on the topic "Implicit Runge-Kutta method"

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Hecke, T., M. Daele, G. Berghe, and H. Meyer. "P-stable mono-implicit Runge-Kutta-Nyström modifications of the Numerov method." In Lecture Notes in Computer Science, 536–45. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/3-540-62598-4_135.

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Bouhamidi, A., and K. Jbilou. "A Fast Block Krylov Implicit Runge–Kutta Method for Solving Large-Scale Ordinary Differential Equations." In Optimization, Simulation, and Control, 319–30. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5131-0_20.

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Hairer, Ernst, and Gerhard Wanner. "Runge–Kutta Methods, Explicit, Implicit." In Encyclopedia of Applied and Computational Mathematics, 1282–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_144.

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Hairer, Ernst, and Gerhard Wanner. "Construction of Implicit Runge-Kutta Methods." In Springer Series in Computational Mathematics, 71–90. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/978-3-642-05221-7_5.

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Hairer, Ernst, and Gerhard Wanner. "Implementation of Implicit Runge-Kutta Methods." In Springer Series in Computational Mathematics, 118–30. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/978-3-642-05221-7_8.

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Strehmel, Karl, and Rüdiger Weiner. "Linear-implizite Runge-Kutta-Methoden." In Teubner-Texte zur Mathematik, 120–88. Wiesbaden: Vieweg+Teubner Verlag, 1992. http://dx.doi.org/10.1007/978-3-663-10673-9_4.

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Strehmel, Karl, and Rüdiger Weiner. "Partitionierte linear-implizite Runge-Kutta-Methoden." In Teubner-Texte zur Mathematik, 189–236. Wiesbaden: Vieweg+Teubner Verlag, 1992. http://dx.doi.org/10.1007/978-3-663-10673-9_5.

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Trobec, Roman, Bojan Orel, and Boštjan Slivnik. "Coarse-grain parallelisation of multi-implicit Runge-Kutta methods." In Lecture Notes in Computer Science, 498–504. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/3-540-62598-4_131.

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Lindblad, E., D. M. Valiev, B. Müller, J. Rantakokko, P. Lütstedt, and M. A. Liberman. "Implicit-explicit Runge-Kutta methods for stiff combustion problems." In Shock Waves, 299–304. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-85168-4_47.

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Cordero-Carrión, Isabel, and Pablo Cerdá-Durán. "Partially Implicit Runge-Kutta Methods for Wave-Like Equations." In Advances in Differential Equations and Applications, 267–78. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-06953-1_26.

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Conference papers on the topic "Implicit Runge-Kutta method"

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Ma, Can, Xinrong Su, Jinlan Gou, and Xin Yuan. "Runge-Kutta/Implicit Scheme for the Solution of Time Spectral Method." In ASME Turbo Expo 2014: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/gt2014-26474.

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This paper investigates the Runge-Kutta implicit scheme applied to the solution of the time spectral method for periodic unsteady flow simulation. Several explicit and implicit time integration schemes including the Runge-Kutta scheme, Block-Jacobi SSOR (symmetric successive over relaxation)scheme and Block-Jacobi Runge-Kutta/Implicit scheme are implemented into an in-house code and applied to the time marching solution of the time spectral method. The time integration is coupled with Full Approximation Storage (FAS) type multi-grid method for convergence acceleration. The in-house code is based on the finite volume method and solves the RANS (Reynolds Averaged Navier-Stokes) equations on multi-block structured mesh. For spatial discretization the 3rd/5th order WENO (weighted essentially nonoscillatory) upwind scheme is used for reconstruction and the convective flux is computed with Roe approximate Riemann solver. The widely used one-equation Spalart-Allmaras turbulence model is used in the simulations. The time integration schemes for the solution of the time spectral method are tested with two different compressor cascades with periodically oscillating inlet boundary conditions. The first case is a low speed compressor stator with inlet flow angle varying with time. The second case is a high speed compressor rotor with inlet boundary condition profile to simulation the influence of upstream wakes. The results show that for moderate frequencies and wave mode numbers, the Block-Jacobi Runge-Kutta/Implicit scheme shows favorable convergence behavior compared to the other schemes. However, for extremely high frequencies and wave mode numbers such as in the simulation of high rotating speed compressors, the advantage of the Block-Jacobi Runge-Kutta/Implicit scheme over the explicit Runge-Kutta scheme is totally lost.
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Kalogiratou, Zacharoula, Theodore Monovasilis, and T. E. Simos. "A sixth order symmetric and symplectic diagonally implicit Runge-Kutta method." In INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING 2014 (ICCMSE 2014). AIP Publishing LLC, 2014. http://dx.doi.org/10.1063/1.4897862.

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Zhang, Zhizhu, and Yun Cai. "A Numerical Solution to the Point Kinetic Equations Using Diagonally Implicit Runge Kutta Method." In 2016 24th International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/icone24-60011.

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It would take a long time to solve the point kinetics equations by using full implicit Runge-Kutta (FIRK) for the strong stiffness. Diagonally implicit Runge-Kutta (DIRK) is a useful tool like FIRK to solve the stiff differential equations, while it could greatly reduce the computation compared to FIRK. By embedded low-order Runge-Kutta, DIRK is implemented with the time step adaptation technique, which improves the computation efficiency of DIRK. Through four typical cases with step, ramp sinusoidal and zig-zag reactivity insertions, it shows that the results obtained by DIRK are in perfect agreement with other available results and DIRK with adaptive time step technique has more efficiency than DIRK with the fixed time step.
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Kalogiratou, Z., Th Monovasilis, T. E. Simos, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "A Diagonally Implicit Symplectic Runge-Kutta Method with Minimum Phase-lag." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3637001.

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Do, Nguyen B., Aldo A. Ferri, and Olivier Bauchau. "Efficient Simulation of a Dynamic System With LuGre Friction." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85339.

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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 laws. 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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SENU, N., M. SULEIMAN, F. ISMAIL, and M. OTHMAN. "A SINGLY DIAGONALLY IMPLICIT RUNGE-KUTTA-NYSTRÖM METHOD WITH DISPERSION OF HIGH ORDER." In Special Edition of the International MultiConference of Engineers and Computer Scientists 2011. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814390019_0009.

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Liu, P. F., H. Wei, B. Li, and B. Zhou. "Transient stability constrained optimal power flow using 2-stage diagonally implicit Runge-Kutta method." In 2013 IEEE PES Asia-Pacific Power and Energy Engineering Conference (APPEEC). IEEE, 2013. http://dx.doi.org/10.1109/appeec.2013.6837193.

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Wing, Moo Kwong, Norazak Senu, Mohamed Suleiman, and Fudziah Ismail. "A five-stage singly diagonally implicit Runge-Kutta-Nyström method with reduced phase-lag." In INTERNATIONAL CONFERENCE ON FUNDAMENTAL AND APPLIED SCIENCES 2012: (ICFAS2012). AIP, 2012. http://dx.doi.org/10.1063/1.4757486.

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Franco, Michael, Per-Olof Persson, Will Pazner, and Matthew J. Zahr. "An Adjoint Method using Fully Implicit Runge-Kutta Schemes for Optimization of Flow Problems." In AIAA Scitech 2019 Forum. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2019. http://dx.doi.org/10.2514/6.2019-0351.

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Zhang, Yining, Haochun Zhang, Yang Su, and Guangbo Zhao. "A Comparative Study of 10 Different Methods on Numerical Solving of Point Reactor Neutron Kinetics Equations." In 2017 25th International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/icone25-67275.

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Point reactor neutron kinetics equations describe the time dependent neutron density variation in a nuclear reactor core. These equations are widely applied to nuclear system numerical simulation and nuclear power plant operational control. This paper analyses the characteristics of 10 different basic or normal methods to solve the point reactor neutron kinetics equations. These methods are: explicit and implicit Euler method, explicit and implicit four order Runge-Kutta method, Taylor polynomial method, power series method, decoupling method, end point floating method, Hermite method, Gear method. Three different types of step reactivity values are introduced respectively at initial time when point reactor neutron kinetics equations are calculated using different methods, which are positive reactivity, negative reactivity and higher positive reactivity. The calculation results show that (i) minor relative error can be gain after three types of step reactivity are introduced, when explicit or implicit four order Runge-Kutta method, Taylor polynomial method, power series method, end point floating method or Hermite method is taken. These methods which are mentioned above are appropriate for solving point reactor neutron kinetics equations. (ii) the relative error of decoupling method is large, under the calculation condition of this paper. When a higher reactivity is introduced, the calculation of decoupling method cannot be convergence. (iii) after three types of step reactivity are introduced respectively, the relative error of implicit Euler method is higher than any other method except decoupling method. The third highest is Gear method. (iv) when the higher reactivity is introduced, the relative error of explicit and implicit Euler method are almost coincident, and higher than any other methods obviously. (v) 4 methods are suitable for solution on these given conditions, which are implicit Runge-Kutta method, Taylor polynomial method, power series method and end point floating method, considering both the accuracy and stiffness.
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