Artykuły w czasopismach: „Kutta condition”

1

Xu, Cheng. "Kutta Condition for sharp edge flows". Mechanics Research Communications 25, nr 4 (lipiec 1998): 415–20. http://dx.doi.org/10.1016/s0093-6413(98)00054-8.

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Crighton, D. G. "The Kutta Condition in Unsteady Flow". Annual Review of Fluid Mechanics 17, nr 1 (styczeń 1985): 411–45. http://dx.doi.org/10.1146/annurev.fl.17.010185.002211.

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Wang, Youjiang. "An easy-to-implement highly efficient algorithm for nonlinear Kutta condition in boundary element method". Physics of Fluids 34, nr 12 (grudzień 2022): 127111. http://dx.doi.org/10.1063/5.0131509.

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An easy-to-implement highly efficient algorithm for the nonlinear Kutta condition in the boundary element method is proposed. The main innovation is to pre-calculate an inverse matrix and use it to replace a solving system of equations with vector–scalar multiplication and matrix–vector multiplication. This allows calculating the Jacobian matrix in each nonlinear Kutta condition iteration with little computational effort, which is important for fast and robust convergence. The open-water characteristics of four different propellers are calculated with the linear and nonlinear Kutta conditions. The simulations show that the nonlinear Kutta condition results in more accurate open-water characteristics and more physically reasonable surface pressure distributions. In addition, the nonlinear Kutta condition takes no more than 3 extra seconds for an open-water simulation, and this extra time does not increase much with the number of simulation steps. The method proposed in this work is expected to improve the computational speed of the boundary element method while maintaining the same accuracy, or improve the accuracy with little extra computational time.

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4

Pierce, Allan D. "David Crighton and the unsteady Kutta condition". Journal of the Acoustical Society of America 109, nr 5 (maj 2001): 2469–70. http://dx.doi.org/10.1121/1.4744766.

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Gu, Wei, Ming Wang i Dongfang Li. "Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations". Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/304071.

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The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last.

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6

Poling, D. R., i D. P. Telionis. "The Trailing Edge of a Pitching Airfoil at High Reduced Frequencies". Journal of Fluids Engineering 109, nr 4 (1.12.1987): 410–14. http://dx.doi.org/10.1115/1.3242681.

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Trailing edge flows are visualized for a pitching airfoil. The validity of the quasi-steady and an extension to an unsteady Kutta condition, namely the Giesing-Maskell condition are examined. A new dynamic similarity parameter is proposed. Earlier work and the present results are re-evaluated in terms of this parameter. A range is identified in which no Kutta-type condition may apply.

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7

Zannetti, Luca, i Alexandre Gourjii. "Two-vortex equilibrium in the flow past a flat plate at incidence". Journal of Fluid Mechanics 755 (14.08.2014): 50–61. http://dx.doi.org/10.1017/jfm.2014.418.

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AbstractThe two-dimensional inviscid incompressible steady flow past an inclined flat plate is considered. A locus of asymmetric equilibrium configurations for vortex pairs is detected. It is shown that the flat geometry has peculiar properties compared to other geometries: (i) in order to satisfy the Kutta condition at both edges, which ensures flow regularity, the total circulation and the force acting on the plate must be zero; and (ii) the Kutta condition and the free vortex equilibrium conditions are not independent of each other. The non-existence of symmetric equilibrium configurations for an orthogonal plate is extended to more general asymmetric flows.

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8

Taha, Haithem, i Amir S. Rezaei. "Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem". Journal of Fluid Mechanics 868 (8.04.2019): 141–75. http://dx.doi.org/10.1017/jfm.2019.159.

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The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

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9

Mohebbi, Farzad, i Mathieu Sellier. "On the Kutta Condition in Potential Flow over Airfoil". Journal of Aerodynamics 2014 (1.04.2014): 1–10. http://dx.doi.org/10.1155/2014/676912.

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This paper proposes a novel method to implement the Kutta condition in irrotational, inviscid, incompressible flow (potential flow) over an airfoil. In contrast to common practice, this method is not based on the panel method. It is based on a finite difference scheme formulated on a boundary-fitted grid using an O-type elliptic grid generation technique. The proposed algorithm uses a novel and fast procedure to implement the Kutta condition by calculating the stream function over the airfoil surface through the derived expression for the airfoils with both finite trailing edge angle and cusped trailing edge. The results obtained show the excellent agreement with the results from analytical and panel methods thereby confirming the accuracy and correctness of the proposed method.

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10

Schneid, J. "A necessary condition forB-convergence of Runge-Kutta methods". BIT 30, nr 1 (marzec 1990): 166–70. http://dx.doi.org/10.1007/bf01932143.

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11

Mohebbi, Farzad, Ben Evans i Mathieu Sellier. "On the Kutta Condition in Compressible Flow over Isolated Airfoils". Fluids 4, nr 2 (1.06.2019): 102. http://dx.doi.org/10.3390/fluids4020102.

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This paper presents a novel and accurate method to implement the Kutta condition in solving subsonic (subcritical) inviscid isentropic compressible flow over isolated airfoils using the stream function equation. The proposed method relies on body-fitted grid generation and solving the stream function equation for compressible flows in computational domain using finite-difference method. An expression is derived for implementing the Kutta condition for the airfoils with both finite angles and cusped trailing edges. A comparison of the results obtained from the proposed numerical method and the results from experimental and other numerical methods reveals that they are in excellent agreement, which confirms the accuracy and correctness of the proposed method.

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12

Amiet, R. K. "Gust response for flat-plate airfoils and the Kutta condition". AIAA Journal 28, nr 10 (październik 1990): 1718–27. http://dx.doi.org/10.2514/3.10465.

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13

Devinant, Ph, A. Leroy i M. Mudry. "Unsteady 3D nonlinear Kutta-Joukovski condition for thin lifting surfaces". Computational Mechanics 24, nr 2 (25.08.1999): 138–47. http://dx.doi.org/10.1007/s004660050446.

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14

Xiangyang, Li, i Chen Wanqiang. "Rolling Bearing Fault Diagnosis Based on Physical Model and One-Class Support Vector Machine". ISRN Mechanical Engineering 2014 (14.04.2014): 1–4. http://dx.doi.org/10.1155/2014/160281.

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This paper aims at diagnosing the fault of rolling bearings and establishes the system of dynamics model with the consideration of rolling bearing with nonlinear bearing force, the radial clearance, and other nonlinear factors, using Runge-Kutla such as Hertzian elastic contactforce and internal radial clearance, which are solved by the Runge-Kutta method. Using simulated data of the normal state, a self-adaptive alarm method for bearing condition based on one-class support vector machine is proposed. Test samples were diagnosed with a recognition accuracy over 90%. The present method is further applied to the vibration monitoring of rolling bearings. The alarms under the actual abnormal condition meet the demand of bearings monitoring.

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15

Huang, He, i David Hui. "Large-amplitude vibration of imperfect angleply laminated rectangular plates with four boundary conditions". World Journal of Engineering 12, nr 5 (1.10.2015): 421–30. http://dx.doi.org/10.1260/1708-5284.12.5.421.

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This paper solves the modified-Duffing ordinary differential equation for the largeamplitude vibration problem of imperfect angle-ply laminated rectangular plate. Two inplane and two out-of-plane constraints are considered to form four boundary conditions. The initial condition is chosen to be initial vibration amplitude. To solve this angle-ply laminated rectangular plate vibration problem, Lindstedt’s perturbation technique and Runge-Kutta method are applied. The solution from both methods are plotted and compared for a validity check. Lindstedt’s perturbation technique is proved to be accurate for a sufficiently small vibration amplitude especially when imperfection exists. The results from Runge-Kutta method are plotted to form the typical backbone curves.

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16

Sarkar, Tanmay. "A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation". Computational Methods in Applied Mathematics 20, nr 1 (1.01.2020): 121–40. http://dx.doi.org/10.1515/cmam-2018-0032.

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AbstractWe perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened {4/3}-CFL condition ({\Delta t\sim h^{4/3}}). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in {L^{2}}-sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree {l\geq 1} under the standard CFL condition.

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17

SAJID, M., N. ALI, Z. ABBAS i T. JAVED. "STRETCHING FLOWS WITH GENERAL SLIP BOUNDARY CONDITION". International Journal of Modern Physics B 24, nr 30 (10.12.2010): 5939–47. http://dx.doi.org/10.1142/s0217979210055512.

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General slip boundary condition is used to solve the viscous incompressible flows induced by a stretching sheet. These flow problems corresponds to the planar and axisymmetric stretching. A similarity solution is developed by shooting method using Runge–Kutta algorithm. The results are graphically displayed and discussed under the influence of slip parameter and critical shear rate. The comparison of stretching flow problem subject to Navier's boundary condition in the planar case is made with the available numerical results in the literature.

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18

Poling, D. R., i D. P. Telionis. "The response of airfoils to periodic disturbances - The unsteady Kutta condition". AIAA Journal 24, nr 2 (luty 1986): 193–99. http://dx.doi.org/10.2514/3.9244.

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19

Calvo, M., J. I. Montijano i L. Rández. "Constructive characterization of Runge–Kutta methods that satisfy the M-condition". SeMA Journal 74, nr 3 (19.05.2017): 345–59. http://dx.doi.org/10.1007/s40324-017-0126-0.

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20

Takagi, Shohei. "Visualization of potential-flow streamlines around an airfoil under the Kutta condition". Transaction of the Visualization Society of Japan 34, nr 9 (2014): 29–34. http://dx.doi.org/10.3154/tvsj.34.29.

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21

Lee, Hyun Geun. "Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation". Mathematics 8, nr 1 (19.12.2019): 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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22

Khamis, Doran, i Edward James Brambley. "Acoustic boundary conditions at an impedance lining in inviscid shear flow". Journal of Fluid Mechanics 796 (4.05.2016): 386–416. http://dx.doi.org/10.1017/jfm.2016.273.

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The accuracy of existing impedance boundary conditions is investigated, and new impedance boundary conditions are derived, for lined ducts with inviscid shear flow. The accuracy of the Ingard–Myers boundary condition is found to be poor. Matched asymptotic expansions are used to derive a boundary condition accurate to second order in the boundary layer thickness, which shows substantially increased accuracy for thin boundary layers when compared with both the Ingard–Myers boundary condition and its recent first-order correction. Closed-form approximate boundary conditions are also derived using a single Runge–Kutta step to solve an impedance Ricatti equation, leading to a boundary condition that performs reasonably even for thicker boundary layers. Surface modes and temporal stability are also investigated.

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23

Bose, Neil. "Explicit Kutta condition for an unsteady two-dimensional constant potential panel method". AIAA Journal 32, nr 5 (maj 1994): 1078–80. http://dx.doi.org/10.2514/3.12097.

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Ayton, Lorna J., J. R. Gill i N. Peake. "The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction". Journal of Sound and Vibration 378 (wrzesień 2016): 28–37. http://dx.doi.org/10.1016/j.jsv.2016.05.036.

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Hinder, Rainer, i Erhard Meister. "Regarding Some Problems of the Kutta — Joukovskii Condition in Lifting Surface Theory". Mathematische Nachrichten 184, nr 1 (1997): 191–228. http://dx.doi.org/10.1002/mana.19971840109.

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Coclici, C. A., i W. L. Wendland. "On the Treatment of the Kutta-Joukowski Condition in Transonic Flow Computations". ZAMM 79, nr 8 (sierpień 1999): 507–34. http://dx.doi.org/10.1002/(sici)1521-4001(199908)79:8<507::aid-zamm507>3.0.co;2-b.

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Pornsawad, Pornsarp, Elena Resmerita i Christine Böckmann. "Convergence Rate of Runge-Kutta-Type Regularization for Nonlinear Ill-Posed Problems under Logarithmic Source Condition". Mathematics 9, nr 9 (4.05.2021): 1042. http://dx.doi.org/10.3390/math9091042.

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We prove the logarithmic convergence rate of the families of usual and modified iterative Runge-Kutta methods for nonlinear ill-posed problems between Hilbert spaces under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori discrepancy principle.

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Mandal, Anandadeep, i Ruchi Sharma. "Explicit time discretization programming approach to risk modelling". GIS Business 13, nr 6 (26.12.2018): 29–35. http://dx.doi.org/10.26643/gis.v13i6.3263.

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In this paper we formulate an explicit time discretization model for modeling risk by establishing an initial value problem as a function of time. The model is proved stable and the scaled-stability regions can encapsulated the volatile macroeconomic condition pertaining to financial risk. The model is extended to multistage schemes where we test for convergence under higher-order difference equations. Further, for addressing advection problems we have used Runge-Kutta method to propose a multistep model and have shown its stability patterns against general and absolute stability conditions. The paper also provides second-order and forth-order algorithm for computational programming of the models in practice. We conclude by stating that explicit time discretization models are stable and adequate for changing business environment. Keywords: Explicit time discretization; Runge-Kutta Method; algorithms; computational programming; risk modeling.

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29

Guo, Y. P. "Sound generation by a supersonic aerofoil cutting through a steady jet flow". Journal of Fluid Mechanics 216 (lipiec 1990): 193–212. http://dx.doi.org/10.1017/s0022112090000398.

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This paper examines the sound generation process when a supersonic aerofoil cuts through a steady jet flow. It is shown that the principal sound is generated by the leading edge of the aerofoil when it interacts with the streaming jet. To the leading order in terms of the jet velocity, no trailing-edge sound is generated. This is not the result of the cancellation of a trailing-edge sound by that from vortex shedding through the imposition of the Kutta condition. Instead, the null acoustic radiation from the trailing edge is entirely because, to the leading order, there is no interaction between the trailing edge and the jet. The effect of the trailing edge is to diffract sound waves generated by the leading edge. It is shown that the diffracted field (as well as the incident field) is regular at the trailing edge and the issue of satisfying the Kutta condition does not arise during the diffraction process. Thus, there is no extra vortex shedding from the trailing edge owing to its interaction with the flow, apart from those resulting from the discontinuity across the aerofoil, generated by the flow-leading edge interaction. This is in sharp contrast to the case of subsonic aerofoils where the removal of the singularity in the diffracted field at the trailing edge through the imposition of the Kutta condition results in vortex shedding from the sharp edge and energy exchange between the sound field and the vortical wake.

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30

Xü, Hongbo. "Potential flow solution for a yawed surface-piercing plate". Journal of Fluid Mechanics 226 (maj 1991): 291–317. http://dx.doi.org/10.1017/s0022112091002392.

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This paper presents the results of an analytical investigation of the steady translation of a vertical surface-piercing plate at a small angle of attack. This problem is the antisymmetric equivalent of the symmetric thin-ship problem solved by Michell. The linearized boundary-value problem is transformed into an integral equation of the first kind by the method of Green functions. The Kelvin–Havelock Green function is used to satisfy the linearized free-surface boundary condition and radiation condition. A pressure Kutta condition is imposed at the trailing edge. Effective algorithms are developed to evaluate the hypersingular kernel without recourse to numerical integration. The resulting integral equation is solved by a collocation method with a refined scheme of discretization. After establishing the convergence of the present algorithm, computations are carried out for a surface-piercing rectangular plate of aspect ratio 0.5. The integrated lateral-force and yaw-moment coefficients show good agreement with experimental data. Other parameters of the flow such as pressure distributions, drag, strength of leading-edge singularity and free-surface profiles on the plate are also presented. The incompatibility between the pressure Kutta condition and the linearized free-surface condition does not affect the global solution.

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Rohan, Dewanto Harwin, i Nur Hayati. "Persamaan Lorenz untuk Keamanan Nomor Serial Sistem Operasi Window7". Jurnal Ilmiah FIFO 10, nr 2 (1.03.2019): 1. http://dx.doi.org/10.22441/fifo.2018.v10i2.001.

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Serial number of operating system windows 7 needs to be safeguarded, so can’t be used by the others. Security of the data can use by modern cryptography such as Vernam Cipher methods and classic cryptography such as Caesar Cipher methods. The security level both of this method depends on the keywords used and it will difficult to crack if the random key is used more and more. To get a random key, we can take from chaos of Lorenz equations as key-generator for encryption and description. Before utilizing chaos in the Lorenz equations, we have to find the maximum t (time) for the inverse problem solution to fit with the forward problem solution. We can use Runge-Kutta method in the Lorenz equations for forward problem solution and inverse problem solution. The solution of integral that obtained by the Runge-Kutta method can be searched by Trapezoidal method. The result of Runge-Kutta solution and Trapezoidal will be used as key-generator for encryption and description. In the simulations performed, the best orde in Runge-Kutta method is 4 and t max is 2. The encryption key is used as the initial condition of Lorenz equation, then the result is integrable by the Trapezoidal method. The result of orde 4 from Runge-Kutta method and Trapezoidal method used as a key-generator. Application of Lorenz equation as key-generator for encryption and decryption, may change the cryptography algorithms of symmetric to be asymmetric.

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32

Reddy, Swathi, i Anindya Deb. "An Improved Finite Element Formulation for Potential Flow Problems Using a Kutta Condition". SAE International Journal of Aerospace 15, nr 1 (11.01.2022): 99–117. http://dx.doi.org/10.4271/01-15-01-0007.

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Davi, Giuseppe, Rosario M. A. Marretta i Alberto Milazzo. "Explicit Kutta Condition for Unsteady Two-Dimensional High-Order Potential Boundary Element Method". AIAA Journal 35, nr 6 (czerwiec 1997): 1080–81. http://dx.doi.org/10.2514/2.197.

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OHGAMI, Yoshifumi, i Teruaki AKAMATSU. "Numerical studying of unsteady Kutta condition by a vortex method with diffusion velocity." Transactions of the Japan Society of Mechanical Engineers Series B 55, nr 510 (1989): 298–305. http://dx.doi.org/10.1299/kikaib.55.298.

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Davi, Giuseppe, Rosario M. A. Maretta i Alberto Milazzo. "Explicit Kutta condition for unsteady two-dimensional high-order potential boundary element method". AIAA Journal 35 (styczeń 1997): 1080–81. http://dx.doi.org/10.2514/3.13629.

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36

Frederiks, W., H. C. J. Hilberink i J. A. Sparenberg. "On the Kutta condition for the flow along a semi-infinite elastic plate". Journal of Engineering Mathematics 20, nr 1 (marzec 1986): 27–50. http://dx.doi.org/10.1007/bf00039321.

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Nageswara Rao, B., B. P. Shastry i G. Venkateswara Rao. "Large deflections of a cantilever beam subjected to a tip concentrated rotational load". Aeronautical Journal 90, nr 897 (wrzesień 1986): 262–66. http://dx.doi.org/10.1017/s0001924000015840.

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SummaryLarge deflection analysis of a cantilever beam under a tip concentrated rotational load governed by a second order non-linear differential equation is solved using a fourth-order Runge-Kutta integration scheme. Initially the two point boundary value problem is converted to an initial value problem by estimating one of the two required initial conditions in an iterative process, so as to satisfy the other boundary condition. The details of load deflection curves are presented.

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38

Zhang, Xu-jiu, Yong-sheng Zhu, Ke Yan i You-yun Zhang. "A Front Tracking Method Based on Runge-Kutta Discontinuous Galerkin Methods". International Journal of Online Engineering (iJOE) 12, nr 12 (25.12.2016): 67. http://dx.doi.org/10.3991/ijoe.v12i12.6453.

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In this paper, a high-resolution front tracking method was presented for interface tracking simulation with Runge-Kutta discontinuous Galerkin methods. An interface treating method of the discontinuous methods is presented. This method don’t construct the ghost fluid and the flow information on both sides next to the interface is used to solve the interfacial status. The limiter adopted the combination of the shock detection and monotonicity-preserving limiter and level set method is used for tracking the interface. Result shown that the front tracking of the high-order accurate Runge-Kutta discontinuous Galerkin method exhibits very good agreement with exact solution in the interface condition that contain strong shock.

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39

Usuki, Tsuneo. "A theory for the finite displacement of a thin-walled Bernoulli–Euler beam and lateral post-buckling analysis". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, nr 2094 (5.03.2008): 1543–70. http://dx.doi.org/10.1098/rspa.2007.0256.

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The state vector equation for lateral buckling in finite displacement theory is formulated using only the hypothesis of the Bernoulli–Euler beam. By using an appropriate orthogonalization of the warping functions, the normally complicated calculation has been processed systematically using only matrix notation. As a numerical analysis, the lateral buckling load on the cantilever receiving a concentrated end load on the upper flange was calculated using the coefficient matrix of the first-order increment; the post-buckling behaviour was investigated with increasing load. Since the state vector equation is a higher order nonlinear equation, the original coefficient matrix was fixed with an arbitrary initial value and the solution was provided by the Runge–Kutta transfer matrix method. Subsequent calculations were pursued in the same way with the solution obtained via Runge–Kutta methods as a new initial value and then shifted to the next load condition. This theory and analysis method does not employ an assumed displacement function, such as the Ritz's method; it is therefore useful for the finite displacement analysis of a beam with arbitrary boundary conditions and intermediate support conditions.

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Liu, Hailiang, i Hairui Wen. "Error estimates of the third order runge-kutta alternating evolution discontinuous galerkin method for convection-diffusion problems". ESAIM: Mathematical Modelling and Numerical Analysis 52, nr 5 (wrzesień 2018): 1709–32. http://dx.doi.org/10.1051/m2an/2018020.

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In this paper, we present the stability analysis and error estimates for the alternating evolution discontinuous Galerkin (AEDG) method with third order explicit Runge-Kutta temporal discretization for linear convection-diffusion equations. The scheme is shown stable under a CFL-like stability condition c0τ ≤ ε ≤ c1h2. Here ε is the method parameter, and h is the maximum spatial grid size. We further obtain the optimal L2 error of order O(τ3 + hk+1). Key tools include two approximation finite element spaces to distinguish overlapping polynomials, coupled global projections, and energy estimates of errors. For completeness, the stability analysis and error estimates for second order explicit Runge-Kutta temporal discretization is included in the appendix.

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41

Li, Gao Hui, Hong Jun Jiang, Jian Zhang i Sheng Chen. "The Explicit Solution to Calculate Maximum Upsurge in Water Chamber Surge Tank under Superposed Condition". Applied Mechanics and Materials 427-429 (wrzesień 2013): 392–95. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.392.

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The explicit solution to calculate maximum upsurge in the water chamber surge tank when load rejection at the worst instant followed by load acceptance has been derived. Comparison is made between explicit solution and the numerical solution by means of the fourth order Runge-Kutta method, and the results show that the derived formula is more convenient for calculation and the precision is high. Therefore, the formula proposed in this study is of great value in practical engineering.

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42

Auerbach, David. "Experiments on attached flow about sharp edges and its relevance to the Kutta condition". Physics of Fluids A: Fluid Dynamics 4, nr 8 (sierpień 1992): 1848–50. http://dx.doi.org/10.1063/1.858406.

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Tang, Y. F. "The necessary condition for a Runge-Kutta scheme to be symplectic for Hamiltonian systems". Computers & Mathematics with Applications 26, nr 1 (lipiec 1993): 13–20. http://dx.doi.org/10.1016/0898-1221(93)90082-7.

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Nazarov, Sergueï A. "About singularities at angular points of a trailing edge under the Joukowskii-Kutta Condition". Mathematical Methods in the Applied Sciences 21, nr 10 (10.07.1998): 939–67. http://dx.doi.org/10.1002/(sici)1099-1476(19980710)21:10<939::aid-mma978>3.0.co;2-k.

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Rienstra, S. W. "A note on the Kutta condition in Glauert's solution of the thin aerofoil problem". Journal of Engineering Mathematics 26, nr 1 (luty 1992): 61–69. http://dx.doi.org/10.1007/bf00043226.

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Pornsawad, Pornsarp, Nantawan Sapsakul i Christine Böckmann. "A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems". Mathematics 7, nr 5 (10.05.2019): 419. http://dx.doi.org/10.3390/math7050419.

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In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥ F ( x δ ( T ) ) − y δ ∥ = τ δ + for some δ + > δ , and an appropriate source condition. We yield the optimal rate of convergence.

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YING, Z. G., Y. Q. Ni i Z. H. CHEN. "A SYMPLECTIC ALGORITHM FOR THE STABILITY ANALYSIS OF NONLINEAR PARAMETRIC EXCITED SYSTEMS". International Journal of Structural Stability and Dynamics 09, nr 03 (wrzesień 2009): 561–84. http://dx.doi.org/10.1142/s0219455409003156.

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A symplectic numerical approach to the stability analysis of nonlinear parametric excited systems with multi-degree-of-freedom is developed and the stability of a nonlinear vibration system depends on the dynamic behavior of its perturbation. The nonlinear parameter-varying differential equations for the perturbation motion are expressed in the form of Hamiltonian equations with time-varying Hamiltonian, and are converted further into the classical Hamiltonian equations with extended time-invariant Hamiltonian by augmenting the state variables. The solution to the augmented Hamiltonian equations has the symplectic structure in terms of the symplectic transformation. Then the difference equations of the symplectic Runge-Kutta algorithm with a sufficient condition are constructed, which is proved to preserve the intrinsic symplectic structure of the original solution. In particular, the symplectic Gauss-Runge-Kutta algorithm with stage 2 and order 4 is proposed and applied to the stability analysis of a nonlinear system. Unstable regions based on the nonlinear periodic-parameter perturbation equation are obtained by using the symplectic Gauss-Runge-Kutta algorithm, analytical solution method and non-symplectic conventional Runge-Kutta algorithm to verify the higher accuracy of the proposed algorithm. Unstable regions based on the nonlinear perturbation are given to illustrate the improvement over those based on the linear perturbation. The developed symplectic approach to the stability analysis can preserve the symplectic structure of the original system and is applicable to nonlinear parametric excited systems with multi-degree-of-freedom.

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Rana, Sohel, Jeevan Kanesan, Ahmed Reza i Harikrishnan Ramiah. "Tickhonov based well-condition asymptotic waveform evaluation for dual-phase-lag heat conduction". Thermal Science 20, nr 6 (2016): 1891–902. http://dx.doi.org/10.2298/tsci140410104r.

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The Tickhonov based well condition asymptotic waveform evaluation (TWCAWE) is presented here to study the non-Fourier heat conduction problems with various boundary conditions. In this paper, a novel TWCAWE method is proposed to overwhelm ill-conditioning of the asymptotic waveform evaluation (AWE) technique for thermal analysis and also presented for time-reliant problems. The TWCAWE method is capable to evade the instability of AWE and also efficaciously approximates the initial high frequency and delay similar as well-established numerical method, such as Runge-Kutta (R-K). Furthermore, TWCAWE method is found 1.2 times faster than the AWE and also 4 times faster than the traditional R-K method.

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Aghakhani, M., M. Suhatril, M. Mohammadhassani, M. Daie i A. Toghroli. "A Simple Modification of Homotopy Perturbation Method for the Solution of Blasius Equation in Semi-Infinite Domains". Mathematical Problems in Engineering 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/671527.

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A simple modification of the homotopy perturbation method is proposed for the solution of the Blasius equation with two different boundary conditions. Padé approximate is used to deal with the boundary condition at infinity. The results obtained from the analytical method are compared to Howarth’s numerical solution and fifth order Runge-Kutta Fehlberg method indicating a very good agreement. The proposed method is a simple and reliable modification of homotopy perturbation method, which does not require the existence of a small parameter, linearization of the equation, or computation of Adomian’s polynomials.

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La Mantia, M., i P. Dabnichki. "Note on the Physical Basis of the Kutta Condition in Unsteady Two-Dimensional Panel Methods". Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/708541.

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Force generation in avian and aquatic species is of considerable interest for possible engineering applications. The aim of this work is to highlight the theoretical and physical foundations of a new formulation of the unsteady Kutta condition, which postulates a finite pressure difference at the trailing edge of the foil. The condition, necessary to obtain a unique solution and derived from the unsteady Bernoulli equation, implies that the energy supplied for the wing motion generates trailing-edge vortices and their overall effect, which depends on the motion initial parameters, is a jet of fluid that propels the wing. The postulated pressure difference (the value of which should be experimentally obtained) models the trailing-edge velocity difference that generates the thrust-producing jet. Although the average thrust values computed by the proposed method are comparable to those calculated by assuming null pressure difference at the trailing edge, the latter (commonly used) approach is less physically meaningful than the present one, as there is a singularity at the foil trailing edge. Additionally, in biological applications, that is, for autonomous flapping, the differences ought to be more significant, as the corresponding energy requirements should be substantially altered, compared to the studied oscillatory motions.

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