Average energy dissipation rates of explicit exponential Runge-Kutta methods for gradient flow problems

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EPTRK) methods for first-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the effcient sequential Runge-Kutta method dopri5.

This paper is aimed at analyzing the consistency and convergence of a derived explicit fourth-stage fourth-order Runge-Kutta method. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

A probe consisting of four X-wires (a total of eight hot wires operated in constant temperature mode) and two pairs of parallel cold wires (operated in constant current mode) is in principle capable of providing simultaneous data for the instantaneous energy and temperature dissipation rates. To evaluate the performance of this probe, measurements have been made in decaying turbulence downstream of a grid/screen combination. The directly measured mean values of the energy and temperature dissipation rates are compared with those obtained from the streamwise decay rates of the mean turbulent energy and temperature variance. The probe also yields all three fluctuating vorticity components; after applying spatial resolution corrections, their spectra are in close agreement with isotropic calculations over nearly all wavenumbers. Both the vorticity variance and the mean energy dissipation rate exhibit the same power-law decay rate. Transport equations for the mean energy and temperature dissipation rates are satisfied to within ±10%.

Vortex optical communication is a method to improve the spectral efficiency of underwater wireless optical communication. However, facing the complex marine environment, the turbulence effect in the offshore area is a big challenge to vortex optical communication. Oceanic turbulence will influence the transmission performance of the vortex beam. After turbulent transmission, crosstalk will inevitably occur between different topological modes. This paper studies the degree of crosstalk of Hankel–Bessel (HB) vortex light in different oceanic turbulences. By establishing a mathematical model of anisotropic oceanic turbulence, the main focus is on average temperature dissipation rate, turbulent energy dissipation rate, and temperature-salinity balance parameters. The influence of different turbulence parameters on the propagation of HB vortex light is discussed, and the degree of influence of the above parameters on the channel capacity of the orbital angular momentum HB (OAM-HB) optical communication system is studied. Simulation experiments show that anisotropic oceanic turbulence will cause energy migration in different topological models and cause significant decreases in channel capacity. We observed the capacity of HB vortex beams and compared the communication effects of Laguerre–Gaussian and HB vortex beams in anisotropic ocean turbulence. The communication performance of OAM-HB light in anisotropic turbulent is better than that in an isotropic one. The research results of the transmission characteristics of the vortex HB light underwater show that the OAM-HB underwater communication is worthy of further study.

Twoand three-dimensional turbulent wakes which develop under the influence of streamline curvature and rotation were calculated with three turbulence closure models. The first model was comprised of transport equations for the turbulent kinetic energy and the rate of energy dissipation. The second model was comprised of equations for the rate of turbulent kinetic energy dissipation and Reynolds stresses, but the effects of the convection and diffusion in the Reynolds stress transport equation were handled collectively. The third model utilizes equations of turbulent kinetic energy dissipation and Reynolds stresses in nearly exact form. All three of the models were modified for the effects of streamline curvature. The second and third models include the effects of rotation through the rotation-originated redistribution term in the transport equation of Reynolds stresses. The numerical results for the wakes of airfoils, cascades, and turbomachinery rotor blades demonstrate that the second and third models provide accurate predictions, but computer time and storage can be considerably saved with the second model.

In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

On the economization of explicit Runge-Kutta methods

The stability region of an explicit and consistentm-stage Runge-Kutta method cannot contain the closed disk with diameter [−2m, 0] as a proper subset.

[1] In the early 1980s, in situ observations were made of the spatial-temporal variability of the density field at three separate shelf edge sites off the Canadian east coast to improve our understanding of the dynamics of the generation, transformation, and dissipation of internal wave energy through an analysis of the potential energy budget in the water column. A description of the procedure for estimating the perturbation potential energy and its principal characteristics is given within a conceptual framework, where an internal tide is generated, propagates shoreward from the shelf edge, is transformed by nonlinear effects into shorter undulations such as solitary waves and internal bores, and eventually dissipates. On the basis of observations we find that the linear approximation of the potential energy is a major fraction of available potential energy, especially in large-amplitude internal waves. The potential energy in the generated internal tide is shown to covary with the squared current volume flux in the generating region. Turbulence and mixing are expected to occur as the short, energetic waves become unstable. From the observations, the rate of energy dissipation is smallest on the Grand Bank (0.007 W m−2 over a 30 km distance). In the Scotian Gulf the loss rate is 0.05 W m−2 over a 13 km distance. At The Gully site, where the energy concentrations are the largest, dissipation is estimated to be as large as 0.2 W m−2 over a 10 km distance.

Tip clearance is the distance required between the blade tip and the pump body wall of the impeller in the multiphase pump, of which the resulting tip leakage vortex (TLV) causes unstable flow, leading to energy dissipation. However, only a few studies have been made on the energy dissipation caused by the TLV, and the internal mechanism of energy dissipation has not been revealed. In the present work, enstrophy dissipation theory is innovatively applied to quantitatively study the energy dissipation of the TLV in impeller, to provide guidelines for controlling energy dissipation associated with the TLV, and to optimize the design of the multiphase pump. The location, mode, and energy dissipation rate caused by the TLV are analyzed. The relationship between vorticity and the enstrophy dissipation rate is summarized, and the energy dissipation law is revealed during the inception, development, and dissipation of the TLV. The present analysis indicates that the vorticity is highest at the core of the TLV and gradually weakens along the radial direction with the vortex core at the center; the enstrophy dissipation, however, presented the opposite distribution law. The enstrophy dissipation rate changes as the spatialtemporal evolution of the TLV. The gas phase significantly deteriorates the flow pattern of the TLV, enhances the volume enstrophy dissipation rate, and reduces the wall enstrophy dissipation rate. The volume enstrophy dissipation power increases by 45.33% with an inlet gas void fraction of 10%, the wall enstrophy dissipation power decreases by 23.90%, and the total enstrophy dissipation power increases by 17.21%.

In a stepped spillway, the spillway face is provided with a series of steps from near the crest to the toe. This type of spillway is attractive for water resource management because it dissipates energy that is within the flow. When energy is dissipated by the steps, it reduces the energy reduction that otherwise must occur at the toe of the spillway. In the present study, nine physical stepped spillway models with slopes of 15, 25, 45° and with a number of steps varying from 5 to 50 have been constructed. The crest of the spillways have standard Ogee profiles and a constant slope extends from the crest to the spillway toe. Two different-sized flumes were used in the experiments (25 and 50 cm wide). Several physical models have been constructed in order to measure the energy dissipation. The purpose of this research is to investigate energy dissipation rate in spillways with different slopes. It was found that the two most important dimensionless factors influencing energy dissipation are the Froude number and q2/(gHdam3). The spillway slope and step number have less impact on the rate of energy dissipation. However, for a constant discharge over a stepped spillway, increasing spillway slope, and increasing the number of steps will increase the energy dissipation. Using multiple regressions, some useful relations for determination energy dissipating are obtained.

To address the energy dissipation requirements of hydraulic engineering projects with medium-low water heads and medium-high unit discharges, counterflow-type underflow energy dissipation can significantly enhance the energy dissipation efficiency through the head-on collision of flows from spillways on both sides. In this study, the spillway of the Lieshen Reservoir was used as the prototype. Since gravity dominates the flow in spillways, we established a 1:15 physical model based on the Froude similarity criterion, and conducted numerical simulations using the volume of fluid method coupled with the realizable k-ε turbulence model. Furthermore, the hydraulic characteristics of counterflow energy dissipation under different flow rates and stilling basin length conditions were analyzed. The results show that the counterflow energy dissipation rate first increases before decreasing with increasing stilling basin length, and the maximum energy dissipation rate can exceed 85%; however, the change in the stilling basin depth has a small impact on the energy dissipation rate, especially under relatively high flow rates; furthermore, an empirical formula for the conjugate depth after a hydraulic jump suitable for counterflow energy dissipation with Froude number in the range of 2.0 < Fr1 < 9.7 and stilling basin depth of 0.5–1.5 m is proposed, with the relative error between its predicted and simulated values being less than 6%. Based on the analysis of the water depth outer envelope curve at the outlet section of the stilling basin, it is suggested that the sidewall height be set to 0.6–0.8 times the conjugate depth after the hydraulic jump.

Energy, time, and channel evolution in catastrophically disturbed fluvial systems

Stability of some continuously imbedded Runge-Kutta methods of Sarafyan