Relevant bibliographies by topics / Runge-Kutta 4th and 5th order

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Runge-Kutta 4th and 5th order'

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Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Runge-Kutta 4th and 5th order"

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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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Raza, J., Fateh Mebarek-Oudina, Paras Ram, and S. Sharma. "MHD Flow of Non-Newtonian Molybdenum Disulfide Nanofluid in a Converging/Diverging Channel with Rosseland Radiation." Defect and Diffusion Forum 401 (May 2020): 92–106. http://dx.doi.org/10.4028/www.scientific.net/ddf.401.92.

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The steady two-dimensional flow of an incompressible non-Newtonian Molybdenum Disulfide nanofluid in the presence of source or sink between two stretchable or shrinkable walls under the influence of thermal radiation is investigated numerically. A generalized transformation is applied to convert the constructed set of partial differential equations (PDEs) into the system of non-linear coupled ordinary differential equations (ODEs). The obtained system of ODEs are solved by using Runge-Kutta 4th and 5th order. The influence of physical parameters, shrinking/ stretching parameter, Casson parameter, Hartmann number, Reynolds number, solid volume fraction, opening angle of the channel and radiation parameter on the velocity and temperature distribution are observed for converging and diverging channels. It is noticed that thermal boundary layer thickness is diminished for increased thermal radiation resulting in gradual temperature fall. The results also reveal that velocity and temperature profile both are elevated on raising the stretching parameter and Hartmann number. A comparative analysis is made out to validate the present results.
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Makukula, Z. G., S. S. Motsa, and S. Shateyi. "Numerical Analysis for the Synthesis of Biodiesel Using Spectral Relaxation Method." Mathematical Problems in Engineering 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/601374.

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Biodiesel is an alternative diesel fuel chemically defined as the mono-alkyl esters of long chain fatty acids derived from vegetable oils or animal fat. It is becoming more attractive as an alternative fuel due to the depleting fossil fuel resources. A mathematical model for the synthesis of biodiesel from vegetable oils and animal fats is presented in this study. Numerical solutions of the model are found using a spectral relaxation method. The method, originally developed for boundary value problems, is an iterative scheme based on the Chebyshev spectral collocation method developed by decoupling systems of equations using Gauss-Seidel type of techniques. The effects of the reaction rate constants and initial concentrations of the reactants on the amount of the final product are being investigated. The accuracy of the numerical results is validated by comparison with known analytical results and numerical results obtained usingode45, an efficient explicit 4th and 5th order Runge-Kutta method used to integrate both linear and nonlinear differential equations.
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Zhao, Xin, Jian Li, Wansuo Duan, and Dongqian Xue. "Numerical Analysis of the Mixed 4th-Order Runge-Kutta Scheme of Conditional Nonlinear Optimal Perturbation Approach for the EI Niño-Southern Oscillation Model." Advances in Applied Mathematics and Mechanics 8, no. 6 (September 19, 2016): 1023–35. http://dx.doi.org/10.4208/aamm.2014.m786.

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AbstractIn this paper, we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation (CNOP) approach for the EI Niño-Southern Oscillation (ENSO) model. This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method. Convergence, the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved. Furthermore, optimal control problem is developed and the gradient of the cost function is determined.
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Side, Syafruddin, Maya Sari Wahyuni, and Arifuddin R. "Solusi Numerik Model Verhulst pada Estimasi Pertumbuhan Hasil Panen Padi dengan Metode Adam Bashforth-Moulton (ABM)." Journal of Mathematics, Computations, and Statistics 2, no. 1 (May 12, 2020): 91. http://dx.doi.org/10.35580/jmathcos.v2i1.12463.

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Penelitian ini menerapkan metode Adam Bashforth-Moulton untuk menentukan solusi model Verhulst. Bentuk solusi yang diperoleh adalah estimasi hasil panen padi di Kabupaten Gowa dengan menggunakan persamaan berikut . Persamaan model Verhulst terlebih dahulu diselesaikan dengan metode Runge-Kutta orde-4 untuk mendapatkan solusi awal ; ; dan . Selanjutnya nilai awal disubstitusi pada persamaan Adam-Bashforth orde-4 untuk mendapatkan nilai prediksi, kemudian nilai prediksi yang diperoleh diperbaiki menggunakan persamaan korektor Adam Moulton orde-4. Pada iterasi ke-14 yaitu saat menunjukkan tahun diperoleh nilai prediktor dan nilai korektor sehingga estimasi hasil panen padi di Kabupaten Gowa pada tahun 2021 dengan menggunakan metode Adam Bashforth-Moulton saat adalah ton.Kata Kunci: Model Verhulst, Metode Runge-Kutta, Metode Adam Bashforth-Moulton This research applied Adam Bashforth-Moulton Method to determine the solution of Verhust Model. The form of the solution obtained is estimatation of rice harvest in Gowa Regency by using the following equation . Verhulst model equation firstly solved by using 4th order of Runge-Kutta method to get initial solutions of ; ; and . Furthermore, the initial values subtituted on the 4th order of Adam-Bashforth equation to get the prediction value, then the prediction value obtained was corrected using the corrector equation of 4th order of Adam Moulton. On the 14th iteration that is when shows the year of 2021 retrieved the predictor value of and corrector value of so estimation of rice harvets in Gowa Regency in 2021 by using Adam Bashforth-Moulton method when is ton.Keywords: Verhulst Model, Runge-Kutta Method, Adam Bashforth-Moulton
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Álvarez, Cristhian, Edwin Espinel, and Carlos J. Noriega. "Study of Different Alternatives for Dynamic Simulation of a Steam Generator Using MATLAB." Fluids 6, no. 5 (April 29, 2021): 175. http://dx.doi.org/10.3390/fluids6050175.

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This work presents the simulation of a steam generator or water-tube boiler through the implementation in MATLAB® for a proposed mathematical model. Mass and energy balances for the three main components of the boiler—the drum, the riser and down-comer tubes—are presented. Three alternative solutions to the ordinary differential equation (ODE) were studied, based on Runge–Kutta 4th order method, Heun’s method, and MATLAB function Ode45. The best results were obtained using MATLAB® function Ode45 based on the Runge–Kutta 4th Order Method. The error was less than 5% for the simulation of the steam pressure in the drum, the total volume of water in the boiler, and the mixture quality in relation to what was reported.
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Ashgi, Rizky. "Comparison of Numerical Simulation of Epidemiological Model between Euler Method with 4th Order Runge Kutta Method." International Journal of Global Operations Research 2, no. 1 (March 14, 2021): 37–44. http://dx.doi.org/10.47194/ijgor.v2i1.67.

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Coronavirus Disease 2019 has become global pandemic in the world. Since its appearance, many researchers in world try to understand the disease, including mathematics researchers. In mathematics, many approaches are developed to study the disease. One of them is to understand the spreading of the disease by constructing an epidemiology model. In this approach, a system of differential equations is formed to understand the spread of the disease from a population. This is achieved by using the SIR model to solve the system, two numerical methods are used, namely Euler Method and 4th order Runge-Kutta. In this paper, we study the performance and comparison of both methods in solving the model. The result in this paper that in the running process of solving it turns out that using the euler method is faster than using the 4th order Runge-Kutta method and the differences of solutions between the two methods are large.
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Randhir, D., S. Umashankar, D. Vijayakumar, and D. P. Kothari. "Comparative Analysis of Solution Methods to Power Electronic Interface Modeling for Renewable Energy Applications." Advanced Materials Research 768 (September 2013): 9–15. http://dx.doi.org/10.4028/www.scientific.net/amr.768.9.

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This paper presents the comparison between various numerical methods namely Euler's Method, Midpoint Method, 2nd order Runge Kutta Method and 4th order Runge Kutta Method with the analytical method to solve a power electronic system in both single phase and three phase configuration using decoupled methodology. The values of source current, load current and DC link voltage are obtained for each method using Matlab software and compared with each other. Also, the error in each numerical method with respect to analytical method is calculated and tabulated. These power electronic models could be an excellent research platform for testing the renewable energy systems without going for full scale or scaled down models.
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HANSEN, JAKOB, ALEXEI KHOKHLOV, and IGOR NOVIKOV. "PROPERTIES OF FOUR NUMERICAL SCHEMES APPLIED TO A NONLINEAR SCALAR WAVE EQUATION WITH A GR-TYPE NONLINEARITY." International Journal of Modern Physics D 13, no. 05 (May 2004): 961–82. http://dx.doi.org/10.1142/s021827180400502x.

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We study stability, dispersion and dissipation properties of four numerical schemes (Itera-tive Crank–Nicolson, 3rd and 4th order Runge–Kutta and Courant–Fredrichs–Levy Nonlinear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar nonlinear wave equation with a type of nonlinearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant–Fredrichs–Levy Nonlinear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4th order Runge–Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.
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ANASTASSI, Z. A., and T. E. SIMOS. "SPECIAL OPTIMIZED RUNGE–KUTTA METHODS FOR IVPs WITH OSCILLATING SOLUTIONS." International Journal of Modern Physics C 15, no. 01 (January 2004): 1–15. http://dx.doi.org/10.1142/s0129183104006510.

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In this paper we present a family of explicit Runge–Kutta methods of 5th algebraic order, one of which has variable coefficients, for the efficient solution of problems with oscillating solutions. Emphasis is placed on the phase-lag property in order to show its importance with regards to problems with oscillating solutions. Basic theory of Runge–Kutta methods, phase-lag analysis and construction of the new methods are described. Numerical results obtained for known problems show the efficiency of the new methods when they are compared with known methods in the literature. Furthermore we note that the method with variable coefficients appears to have much higher accuracy, which gets close to double precision, when the product of the frequency with the step-length approaches certain values. These values are constant and independent of the problem solved and depend only on the method used and more specifically on the expressions used to achieve higher algebraic order.
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Dissertations / Theses on the topic "Runge-Kutta 4th and 5th order"

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Yaakub, Abdul Razak Bin. "Computer solution of non-linear integration formula for solving initial value problems." Thesis, Loughborough University, 1996. https://dspace.lboro.ac.uk/2134/25381.

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This thesis is concerned with the numerical solutions of initial value problems with ordinary differential equations and covers single step integration methods. focus is to study the numerical the various aspects of Specifically, its main methods of non-linear integration formula with a variety of means based on the Contraharmonic mean (C.M) (Evans and Yaakub [1995]), the Centroidal mean (C.M) (Yaakub and Evans [1995]) and the Root-Mean-Square (RMS) (Yaakub and Evans [1993]) for solving initial value problems. the applications of the second It includes a study of order C.M method for parallel implementation of extrapolation methods for ordinary differential equations with the ExDaTa schedule by Bahoshy [1992]. Another important topic presented in this thesis is that a fifth order five-stage explicit Runge Kutta method or weighted Runge Kutta formula [Evans and Yaakub [1996]) exists which is contrary to Butcher [1987] and the theorem in Lambert ([1991] ,pp 181). The thesis is organized as follows. An introduction to initial value problems in ordinary differential equations and parallel computers and software in Chapter 1, the basic preliminaries and fundamental concepts in mathematics, an algebraic manipulation package, e.g., Mathematica and basic parallel processing techniques are discussed in Chapter 2. Following in Chapter 3 is a survey of single step methods to solve ordinary differential equations. In this chapter, several single step methods including the Taylor series method, Runge Kutta method and a linear multistep method for non-stiff and stiff problems are also considered. Chapter 4 gives a new Runge Kutta formula for solving initial value problems using the Contraharmonic mean (C.M), the Centroidal mean (C.M) and the Root-MeanSquare (RMS). An error and stability analysis for these variety of means and numerical examples are also presented. Chapter 5 discusses the parallel implementation on the Sequent 8000 parallel computer of the Runge-Kutta contraharmonic mean (C.M) method with extrapolation procedures using explicit assignment scheduling Kutta RK(4, 4) method (EXDATA) strategies. A is introduced and the data task new Rungetheory and analysis of its properties are investigated and compared with the more popular RKF(4,5) method, are given in Chapter 6. Chapter 7 presents a new integration method with error control for the solution of a special class of second order ODEs. In Chapter 8, a new weighted Runge-Kutta fifth order method with 5 stages is introduced. By comparison with the currently recommended RK4 ( 5) Merson and RK5(6) Nystrom methods, the new method gives improved results. Chapter 9 proposes a new fifth order Runge-Kutta type method for solving oscillatory problems by the use of trigonometric polynomial interpolation which extends the earlier work of Gautschi [1961]. An analysis of the convergence and stability of the new method is given with comparison with the standard Runge-Kutta methods. Finally, Chapter 10 summarises and presents conclusions on the topics discussed throughout the thesis.
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Book chapters on the topic "Runge-Kutta 4th and 5th order"

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"Numerical Integration: Runge–Kutta 4th Order Method." In Distillation, 137–38. Elsevier, 2016. http://dx.doi.org/10.1016/b978-1-78548-177-2.50012-2.

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"Numerical Integration 4th-order Runge–Kutta Method." In Adsorption-Dryers for Divided Solids, 261–62. Elsevier, 2016. http://dx.doi.org/10.1016/b978-1-78548-179-6.50010-5.

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"Numerical Integration 4th-order Runge–Kutta Method." In Liquid-Solid Separators, 247–48. Elsevier, 2016. http://dx.doi.org/10.1016/b978-1-78548-182-6.50012-9.

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Conference papers on the topic "Runge-Kutta 4th and 5th order"

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Laurent, Alexandre, David Lenoir, Louis Jezequel, and Bruno Mevel. "Predictive Dynamic Model of a Deep Groove Ball Bearing With a Flexible Outer Ring." In STLE/ASME 2008 International Joint Tribology Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/ijtc2008-71296.

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An explicit dynamic model of a deep groove ball bearing under a radial load is proposed. All components are treated as rigid bodies whereas the bearing outer ring flexibility is taken into account using fixed interface component mode synthesis (CMS). The classical lubricated Hertzian contact theory is used to calculate elastic deflections and non-linear contact forces. The dynamic loading of the outer ring interface nodes is ensured using C2-continuous rational cubic splines. A Runge-Kutta-Felhberg 4th/5th order integration scheme is used to solve the dynamic equilibrium of all components. Time and frequency domain analyses are then carried out to investigate the dynamic behaviour of the ball bearing. The accuracy of these works is validated by comparison with the results of an analytical model and a model based on finite elements proposed in prior researchs.
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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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Berland, Julien, Christophe Bogey, and Christophe Bailly. "Optimized Explicit Schemes: Matching and Boundary Schemes, and 4th-order Runge-Kutta Algorithm." In 10th AIAA/CEAS Aeroacoustics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2004. http://dx.doi.org/10.2514/6.2004-2814.

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Hadi, Miftachul, Malcolm Anderson, and Andri Husein. "Using 4th order Runge-Kutta method for solving a twisted Skyrme string equation." In THE 4TH INTERNATIONAL CONFERENCE ON THEORETICAL AND APPLIED PHYSICS (ICTAP) 2014. AIP Publishing LLC, 2016. http://dx.doi.org/10.1063/1.4943700.

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Shah, Harsh. "Parallel Techniques for Navier-Stokes Solver based on 4th Order Modified Runge-Kutta Scheme with TVD." In 22nd AIAA Computational Fluid Dynamics Conference. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2015. http://dx.doi.org/10.2514/6.2015-3054.

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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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Nonomura, Taku, and Kozo Fujii. "Computational Analysis of Characteristics and Mach Number Effects on Noise Emission From Ideally Expanded Highly Supersonic Free-Jet." In ASME/JSME 2007 5th Joint Fluids Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/fedsm2007-37539.

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In this study, aero-acoustic noise from super-sonic jet plume is computationally investigated. Three-dimensional Navier-Stokes equations are solved with seventh order weighted compact non-linear scheme and total validation diminishing Runge-Kutta time integration scheme. At first, the noise from Mach 2.0 ideally expanded super-sonic jet is computed and validated with the past experimental study. Then the noises from various Mach number (2.0–3.5) ideally expanded jet plumes are computed. Noise source positions, directivity and convective Mach numbers are discussed.
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Elissa, M. G., and S. P. Rooke. "Axially Lumped Versus Axially Distributed Modeling of Liquid-to-Gas Cross-Flow Heat Exchanger Transients." In ASME 1994 International Computers in Engineering Conference and Exhibition and the ASME 1994 8th Annual Database Symposium collocated with the ASME 1994 Design Technical Conferences. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/cie1994-0463.

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Abstract Thermal transients of single-row finned-tube serpentine liquid-to-gas cross-flow heat exchangers utilizing single phase fluids are examined numerically in the temperature-time domain. Two modeling approaches are examined with the aim of contrasting levels of modeling sophistication for this specific heat exchanger application. One model is axially lumped (cases with more than one lump are examined), and the second model is axially distributed. The lumped model is solved using a 4th order Runge-Kutta integration scheme, and the distributed model is solved using an explicit upwind differencing scheme. Single lump representation is found to be undesirable, while multi-lump representation provides adequate comparison with the distributed model. Computation times become an issue for the lumped model if multiple lumps are solved using the 4th order Runge-Kutta scheme.
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Gonzalez-Buelga, Alicia, David Wagg, Simon Neild, and Oreste S. Bursi. "A Comparison of Runge Kutta and Novel L-Stable Methods for Real-Time Integration Methods for Dynamic Substructuring." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-15574.

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In this paper we compare the performance of Runge-Kutta and novel L-stable real-time (LSRT) integration algorithms for real-time dynamic substructuring testing. Substructuring is a hybrid numerical-experimental testing method which can be used to test critical components in a system experimentally while the remainder of the system is numerically modelled. The physical substructure and the numerical model must interact in real time in order to replicate the behavior of the whole (or emulated) system. The systems chosen for our study are mass-spring-dampers, which have well known dynamics and therefore we can benchmark the performance of the hybrid testing techniques and in particular the numerical integration part of the algorithm. The coupling between the numerical part and experimental part is provided by an electrically driven actuator and a load cell. The real-time control algorithm provides bi-directional coupling and delay compensation which couples together the two parts of the overall system. In this paper we consider the behavior of novel L-stable real-time (LSRT) integration algorithms, which are based on Rosenbrock's method. The new algorithms have considerable advantages over 4th order Runge-Kutta in that they are unconditionally stable, real-time compatible and less computationally intensive. They also offer the possibility of damping out unwanted high frequencies and integrating stiff problems. The paper presents comparisons between 4th order Runge-Kutta and the LSRT integration algorithms using three experimental configurations which demonstrate these properties.
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Escobar, Jose, Ismail Celik, and Donald Ferguson. "Development of a Log-Time Integration Method for Reactive Flows." In ASME 2012 Fluids Engineering Division Summer Meeting collocated with the ASME 2012 Heat Transfer Summer Conference and the ASME 2012 10th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/fedsm2012-72090.

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In reactive flow simulations integration of the stiff species transport equations consumes most of the computational time. Another important aspect of combustion simulation is the need to simulate at least tens of species in order to accurately predict emissions and the related combustion dynamics. Small time scales and systems with tens of species lead to very high computational costs. Classic integration methods such as Euler method are restricted by the smallest characteristic time scale, and explicit Runge-Kutta methods require intermediate predictor corrector steps which make the problem computationally expensive. On the other hand, implicit methods are also computationally expensive due the calculation of the Jacobian. This work presents a strategy to significantly reduce computational time for integration of species transport equations using a new explicit integration scheme called Log-Time Integration Method (LTIM). LTIM is fairly robust and can compete with methods such as the 5th order Runge-Kutta method. Results showed that LTIM applied to the solution of a zero dimensional reactive system which consists of 4 chemical species obtains the solution around 4 times faster than 5th order Runge-Kutta method. LTIM was also applied to the solution of a one dimensional methane-air flame. The chemical reactions were modeled using a reduced chemical mechanism ARM9 that consists of 9 chemical species and 5 global reactions. The solution was carried out for 9 species transport equations along with the energy equation. Governing equations were decoupled into flow and chemical parts and were solved separately using a split formulation. Thermodynamic properties were obtained using NASA format polynomials and transport properties using kinetic-theory formulation. It is shown that the new one dimensional flame code is able to calculate the adiabatic flame temperature of the system and corresponding flame speed for the methane-air flame thus validating its robustness and accuracy.
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