Статті в журналах: "Dissipative Scheme"

1

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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2

Burkhardt, Ulrike, and Erich Becker. "A Consistent Diffusion–Dissipation Parameterization in the ECHAM Climate Model." Monthly Weather Review 134, no. 4 (April 1, 2006): 1194–204. http://dx.doi.org/10.1175/mwr3112.1.

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Abstract The diffusion–dissipation parameterizations usually adopted in GCMs are not physically consistent. Horizontal momentum diffusion, applied in the form of a hyperdiffusion, does not conserve angular momentum and the associated dissipative heating is commonly ignored. Dissipative heating associated with vertical momentum diffusion is often included, but in a way that is inconsistent with the second law of thermodynamics. New, physically consistent, dissipative heating schemes due to horizontal diffusion (Becker) and vertical diffusion (Becker, and Boville and Bretherton) have been developed and tested. These schemes have now been implemented in 19- and 39-level versions of the ECHAM4 climate model. The new horizontal scheme requires the replacement of the hyperdiffusion with a ∇2 scheme. Dissipation due to horizontal momentum diffusion is found to have maximum values in the upper troposphere/lower stratosphere in midlatitudes and in the winter hemispheric sponge layer, resulting in a warming of the area around the tropopause and of the polar vortex in Northern Hemispheric winter. Dissipation associated with vertical momentum diffusion is largest in the boundary layer. The change in parameterization acts to strengthen the vertical diffusion and therefore the associated dissipative heating. Dissipation due to vertical momentum diffusion has an indirect effect on the upper-tropospheric/stratospheric temperature field in northern winter, which is to cool and strengthen the northern polar vortex. The warming in the area of the tropopause resulting from the change in both dissipation parameterizations is quite similar in both model versions, whereas the response in the temperature of the northern polar vortex depends on the model version.

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3

Chen, Xiaowei, Mingzhan Song, and Songhe Song. "A Fourth Order Energy Dissipative Scheme for a Traffic Flow Model." Mathematics 8, no. 8 (July 28, 2020): 1238. http://dx.doi.org/10.3390/math8081238.

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We propose, analyze and numerically validate a new energy dissipative scheme for the Ginzburg–Landau equation by using the invariant energy quadratization approach. First, the Ginzburg–Landau equation is transformed into an equivalent formulation which possesses the quadratic energy dissipation law. After the space-discretization of the Fourier pseudo-spectral method, the semi-discrete system is proved to be energy dissipative. Using diagonally implicit Runge–Kutta scheme, the semi-discrete system is integrated in the time direction. Then the presented full-discrete scheme preserves the energy dissipation, which is beneficial to the numerical stability in long-time simulations. Several numerical experiments are provided to illustrate the effectiveness of the proposed scheme and verify the theoretical analysis.

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4

Najafiyazdi, Mostafa, Luc Mongeau, and Siva Nadarajah. "Low-dissipation low-dispersion explicit Taylor-Galerkin schemes from the Runge-Kutta kernels." International Journal of Aeroacoustics 17, no. 1-2 (February 24, 2018): 88–113. http://dx.doi.org/10.1177/1475472x17743657.

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A multi-stage approach was adopted to investigate similarities and differences between the explicit Taylor-Galerkin and the explicit Runge-Kutta time integration schemes. It was found that the substitution of some, but not all, of second-order temporal derivatives in a Taylor-Galerkin scheme by additional stages makes it analogous to a Runge-Kutta scheme while preserving its original dissipative property for node-to-node oscillations. The substitution of all second-order temporal derivatives transforms Taylor-Galerkin schemes into Runge-Kutta schemes with zero attenuation at the grid cut-off. The application of this approach to an existing two-stage Taylor-Galerkin scheme yields a low-dissipation low-dispersion Taylor-Galerkin formulation. Two one-dimensional benchmarks were simulated to study the performance of this new scheme. The reverse process yields a general approach for transforming m-stage Runge-Kutta schemes into ( m−1)-stage Taylor-Galerkin schemes while preserving the same order of accuracy. The dissipation and dispersion properties for several new Taylor-Galerkin schemes were compared to those of their corresponding Runge-Kutta form.

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5

Zlotnik, Alexander, and Timofey Lomonosov. "VERIFICATION OF AN ENTROPY DISSIPATIVE QGD-SCHEME." Mathematical Modelling and Analysis 24, no. 2 (February 5, 2019): 179–94. http://dx.doi.org/10.3846/mma.2019.013.

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An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh L1-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for L2-dissipativity of the Cauchy problem for a linearized QGD-scheme.

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6

Appadu, A. R. "Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes." Journal of Applied Mathematics 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/734374.

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Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted byhandk, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values ofkandh. Two optimisation techniques are then implemented to find the optimal values ofkwhenh=0.02for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

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7

Lin, F. B., and F. Sotiropoulos. "Assessment of Artificial Dissipation Models for Three-Dimensional Incompressible Flow Solutions." Journal of Fluids Engineering 119, no. 2 (June 1, 1997): 331–40. http://dx.doi.org/10.1115/1.2819138.

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Various approaches for constructing artificial dissipation terms for three-dimensional artificial compressibility algorithms are presented and evaluated. Two, second-order accurate, central-differencing schemes, with explicitly added scalar and matrix-valued fourth-difference artificial dissipation, respectively, and a third-order accurate flux-difference splitting upwind scheme are implemented in a multigrid time-stepping procedure and applied to calculate laminar flow through a strongly curved duct. Extensive grid-refinement studies are carried out to investigate the grid sensitivity of each discretization approach. The calculations indicate that even the finest mesh employed, consisting of over 700,000 grid nodes, is not sufficient to establish grid independent solutions. However, all three schemes appear to converge toward the same solution as the grid spacing approaches zero. The matrix-valued dissipation scheme introduces the least amount of artificial dissipation and should be expected to yield the most accurate solutions on a given mesh. The flux-difference splitting upwind scheme, on the other hand, is more dissipative and, thus, particularly sensitive to grid resolution, but exhibits the best overall convergence characteristics on grids with large aspect ratios.

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8

Zhang, Yang, Laiping Zhang, Xin He, and Xiaogang Deng. "An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on Hybrid Grids." Communications in Computational Physics 20, no. 2 (July 21, 2016): 459–85. http://dx.doi.org/10.4208/cicp.190915.240216a.

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AbstractA hybrid grid based second-order finite volume algorithm has been developed for Detached-Eddy Simulation (DES) of turbulent flows. To alleviate the effect caused by the numerical dissipation of the commonly used second order upwind schemes in implementing DES with unstructured computational fluid dynamics (CFD) algorithms, an improved second-order hybrid scheme is established through modifying the dissipation term of the standard Roe's flux-difference splitting scheme and the numerical dissipation of the scheme can be self-adapted according to the DES flow field information. By Fourier analysis, the dissipative and dispersive features of the new scheme are discussed. To validate the numerical method, DES formulations based on the two most popular background turbulence models, namely, the one equation Spalart-Allmaras (SA) turbulence model and the two equationk–ωShear Stress Transport model (SST), have been calibrated and tested with three typical numerical examples (decay of isotropic turbulence, NACA0021 airfoil at 60° incidence and 65° swept delta wing). Computational results indicate that the issue of numerical dissipation in implementing DES can be alleviated with the hybrid scheme, the resolution for turbulence structures is significantly improved and the corresponding solutions match the experimental data better. The results demonstrate the potentiality of the present DES solver for complex geometries.

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9

Lu, Changna, Qianqian Gao, Chen Fu, and Hongwei Yang. "Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh." Discrete Dynamics in Nature and Society 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/3427376.

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A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative termuxxplays a more important role thanuxxxxin dissipation.

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10

Mai-Duy, N., N. Phan-Thien, and T. Tran-Cong. "An improved dissipative particle dynamics scheme." Applied Mathematical Modelling 46 (June 2017): 602–17. http://dx.doi.org/10.1016/j.apm.2017.01.086.

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11

Bragin, M. D. "Upwind bicompact schemes for hyperbolic conservation laws." Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ 517, no. 1 (October 3, 2024): 50–56. http://dx.doi.org/10.31857/s2686954324030097.

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For the first time, upwind bicompact schemes of third order approximation in space are presented. A formula is obtained for the transition factor of an arbitrary fully discrete bicompact scheme with integration in time by a Runge–Kutta method. Stability and monotonicity of the first-order in time scheme are investigated, dissipative and dispersion properties of the third-order in time scheme are analyzed. Advantages of the new schemes relative to their centered counterparts are demonstrated.

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12

Appadu, A. R., and A. A. I. Peer. "Optimized Weighted Essentially Nonoscillatory Third-Order Schemes for Hyperbolic Conservation Laws." Journal of Applied Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/428681.

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We describe briefly how a third-order Weighted Essentially Nonoscillatory (WENO) scheme is derived by coupling a WENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative properties when used to approximate the 1D linear advection equation and use a technique of optimisation to find the optimal cfl number of the scheme. We carry out some numerical experiments dealing with wave propagation based on the 1D linear advection and 1D Burger’s equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lowerL1errors. Lastly, we test numerically the order of convergence of the WENO3 scheme.

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13

Zhou, Hanmei, Qishui Zhong, Shaoyu Hu, Jin Yang, Kaibo Shi, and Shouming Zhong. "Dissipative Discrete PID Load Frequency Control for Restructured Wind Power Systems via Non-Fragile Design Approach." Mathematics 11, no. 14 (July 24, 2023): 3252. http://dx.doi.org/10.3390/math11143252.

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This article proposes a discrete proportional-integral-derivative (PID) load frequency control (LFC) scheme to investigate the dissipative analysis issue of restructured wind power systems via a non-fragile design approach. Firstly, by taking the different power-sharing rates of governors into full consideration, a unified model is constructed for interconnected power systems containing multiple governors. Secondly, unlike existing LFC schemes, a non-fragile discrete PID control scheme is designed, which has the performance of tolerating control gain fluctuation and relieving the huge computational burden. Further, by constructing a discrete-type Lyapunov–Krasovskii functional, improved stability criteria with a strict dissipative performance index are established. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed control method.

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14

Kang, Lei, and Chun-Hian Lee. "An efficient low-dissipative WENO filter scheme." International Journal for Numerical Methods in Fluids 69, no. 2 (April 1, 2011): 273–93. http://dx.doi.org/10.1002/fld.2555.

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15

Aregba–Driollet, D., J. Breil, S. Brull, B. Dubroca, and E. Estibals. "Modelling and numerical approximation for the nonconservative bitemperature Euler model." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 4 (July 2018): 1353–83. http://dx.doi.org/10.1051/m2an/2017007.

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This paper is devoted to the study of the nonconservative bitemperature Euler system. We firstly introduce an underlying two species kinetic model coupled with the Poisson equation. The bitemperature Euler system is then established from this kinetic model according to an hydrodynamic limit. A dissipative entropy is proved to exist and a solution is defined to be admissible if it satisfies the related dissipation property. Next, four different numerical methods are presented. Firstly, the kinetic model gives rise to kinetic schemes for the fluid system. The second approach belongs to the family of the discrete BGK schemes introduced by Aregba–Driollet and Natalini. Finally, a quasi-linear relaxation approach and a Lagrange-remap scheme are considered.

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16

DEN OTTER, W. K., and J. H. R. CLARKE. "THE TEMPERATURE IN DISSIPATIVE PARTICLE DYNAMICS." International Journal of Modern Physics C 11, no. 06 (September 2000): 1179–93. http://dx.doi.org/10.1142/s0129183100001012.

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The two most popular algorithms for dissipative particle dynamics (DPD) are critically discussed. In earlier papers, the Groot–Warren algorithm with λ = 1/2 was recommended over the original Hoogerbrugge–Koelman scheme on the basis of a marked difference in their equilibrium temperatures. We show, however, that both schemes produce identical trajectories. Expressions for the temperatures of an ideal gas and a liquid as functions of the simulation parameters are presented. Our findings indicate that the current DPD algorithms do not possess a unique temperature because of the way in which the dissipative and random forces are included. The commonly used large time steps are beyond the stability limits of the conservative force field integrator.

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17

Chabassier, Juliette, Julien Diaz, and Sébastien Imperiale. "Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 3 (April 1, 2020): 845–78. http://dx.doi.org/10.1051/m2an/2019079.

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This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

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18

Wang, Zhenming, Jun Zhu, Chunwu Wang, and Ning Zhao. "Finite difference alternative unequal-sized weighted essentially non-oscillatory schemes for hyperbolic conservation laws." Physics of Fluids 34, no. 11 (November 2022): 116108. http://dx.doi.org/10.1063/5.0123597.

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In this paper, two unequal-sized weighted essentially non-oscillatory (US-WENO) schemes are proposed for solving hyperbolic conservation laws. First, an alternative US-WENO (AUS-WENO) scheme based directly on the values of conserved variables at the grid points is designed. This scheme can inherit all the advantages of the original US-WENO scheme [J. Zhu and J. Qiu, “A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws,” J. Comput. Phys. 318, 110–121 (2016).], such as the arbitrariness of the linear weights. Moreover, this presented AUS-WENO scheme enables any monotone fluxes applicable to this framework, whereas the original US-WENO scheme is only suitable for the more dissipative smooth flux splitting. Therefore, the method in this paper has a smaller L1 and [Formula: see text] numerical errors than the original scheme under the same conditions. Second, in order to further improve the computational efficiency of the above AUS-WENO scheme, a hybrid AUS-WENO scheme is proposed by combining a hybrid strategy. This strategy identifies the discontinuous regions directly based on the extreme points of the reconstruction polynomial corresponding to the five-point stencil, which brings the important advantage that it does not depend on the specific problem and does not contain any artificial adjustable parameters. Finally, the performance of the above two AUS-WENO schemes in terms of low dissipation, shock capture capability, discontinuity detection capability, and computational efficiency is verified by some benchmark one- and two-dimensional numerical examples.

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19

Zhao, Jianli, Qingjie Hu, Xinxin Xie та Chao Zhang. "Local Discontinuous Galerkin Methods for the Two-component μ-Camassa-Holm Equations". Journal of Physics: Conference Series 2890, № 1 (1 листопада 2024): 012018. http://dx.doi.org/10.1088/1742-6596/2890/1/012018.

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Abstract In this paper, we have developed two local discontinuous Galerkin (LDG) methods for the two-component μ-Camassa-Holm equations: a conservative scheme and a dissipative scheme. Exploiting the bi-Hamiltonian structure of the two-component μ-Camassa-Holm system, we introduce two significant Hamiltonian invariants and demonstrate that both schemes preserve discrete versions of these invariants. Additionally, we provide and prove a priori error estimates for both LDG schemes. Numerical experiments are conducted to validate the accuracy and effectiveness of the proposed methods.

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20

Mamaev, M., L. C. G. Govia, and A. A. Clerk. "Dissipative stabilization of entangled cat states using a driven Bose-Hubbard dimer." Quantum 2 (March 27, 2018): 58. http://dx.doi.org/10.22331/q-2018-03-27-58.

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We analyze a modified Bose-Hubbard model, where two cavities having on-site Kerr interactions are subject to two-photon driving and correlated dissipation. We derive an exact solution for the steady state of this interacting driven-dissipative system, and use it show that the system permits the preparation and stabilization of pure entangled non-Gaussian states, so-called entangled cat states. Unlike previous proposals for dissipative stabilization of such states, our approach requires only a linear coupling to a single engineered reservoir (as opposed to nonlinear couplings to two or more reservoirs). Our scheme is within the reach of state-of-the-art experiments in circuit QED.

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21

Shokin, Yurii, Ireneusz Winnicki, Janusz Jasinski, and Slawomir Pietrek. "High order modified differential equation of the Beam–Warming method, II. The dissipative features." Russian Journal of Numerical Analysis and Mathematical Modelling 35, no. 3 (June 25, 2020): 175–85. http://dx.doi.org/10.1515/rnam-2020-0014.

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AbstractThis paper is a continuation of [38]. The analysis of the modified partial differential equation (MDE) of the constant-wind-speed linear advection equation explicit difference scheme up to the eighth-order derivatives is presented. In this paper the authors focus on the dissipative features of the Beam–Warming scheme. The modified partial differential equation is presented in the so-called Π-form of the first differential approximation. The most important part of this form includes the coefficients μ (p) at the space derivatives. Analysis of these coefficients provides indications of the nature of the dissipative errors. A fragment of the stencil for determining the modified differential equation for the Beam–Warming scheme is included. The derived and presented coefficients μ (p) as well as the analysis of the dissipative features of this scheme on the basis of these coefficients have not been published so far.

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22

Hicks, F. E., and P. M. Steffler. "Characteristic Dissipative Galerkin Scheme for Open‐Channel Flow." Journal of Hydraulic Engineering 118, no. 2 (February 1992): 337–52. http://dx.doi.org/10.1061/(asce)0733-9429(1992)118:2(337).

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23

Li, Ruo, and Wei Zhong. "Improvement of the WENO-NIP Scheme for Hyperbolic Conservation Laws." Axioms 11, no. 5 (April 20, 2022): 190. http://dx.doi.org/10.3390/axioms11050190.

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The WENO-NIP scheme was obtained by developing a class of L1-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, the WENO-NIP scheme produces severe spurious oscillations when solving 1D linear advection problems with discontinuities at long output times, and it is also very oscillatory near discontinuities for 1D Riemann problems. In this paper, we find that the spectral property of WENO-NIP exhibits the negative dissipation characteristic, and this is the reason why WENO-NIP is unstable near discontinuities. Using this knowledge, we develop a way of improving the WENO-NIP scheme by introducing an additional term to eliminate the negative dissipation interval. The proposed scheme, denoted as WENO-NIP+, maintains the same convergence property, as well as the same low-dissipation property, as the corresponding WENO-NIP scheme. Numerical examples confirm that the proposed scheme is much more stable near discontinuities for 1D linear advection problems with large output times and 1D Riemann problems than the WENO-NIP scheme. Furthermore, the new scheme is far less dissipative in the region with high-frequency waves. In addition, the improved WENO-NIP+ scheme can remove or at least greatly decrease the post-shock oscillations that are commonly produced by the WENO-NIP scheme when simulating 2D Euler equations with strong shocks.

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24

CHRISTOV, C. I. "DISSIPATIVE QUASI-PARTICLES: THE GENERALIZED WAVE EQUATION APPROACH." International Journal of Bifurcation and Chaos 12, no. 11 (November 2002): 2435–44. http://dx.doi.org/10.1142/s0218127402005959.

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Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

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25

Liu, Kai. "A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs." Journal of Computational Mathematics 35, no. 6 (June 2017): 780–800. http://dx.doi.org/10.4208/jcm.1612-m2016-0604.

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26

Wang, Jingqun, Jing Li, and Lixin Tian. "Global well-posedness for the two-component Camassa–Holm equation with fractional dissipation." Asian-European Journal of Mathematics 12, no. 04 (July 2, 2019): 1950051. http://dx.doi.org/10.1142/s1793557119500517.

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In this paper, we investigate the two-component Camassa–Holm equation with fractional dissipation [Formula: see text] where [Formula: see text] denotes the fractional dissipative operator which is defined by the Fourier transform [Formula: see text]. We prove the local-posedness of this equation via the Littlewood–Paley theory and the suitable iterative scheme. Furthermore, under appropriate discussions, we give the global well-posedness of the above equation.

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27

Feireisl, Eduard, Mária Lukáčová-Medvid’ová, Hana Mizerová, and Bangwei She. "Convergence of a finite volume scheme for the compressible Navier–Stokes system." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 6 (November 2019): 1957–79. http://dx.doi.org/10.1051/m2an/2019043.

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We study convergence of a finite volume scheme for the compressible (barotropic) Navier–Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.

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28

Chen, Wei, and Song Ping Wu. "Perfectly Matched Layer as an Absorbing Boundary Condition for Computational Aero-Acoustic." Advanced Materials Research 726-731 (August 2013): 3153–58. http://dx.doi.org/10.4028/www.scientific.net/amr.726-731.3153.

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Advances in Computational Aeroacoustics (CAA) depend critically on the availability of accurate, nondispersive, least dissipative computation algorithm as well as high quality numerical boundary treatments. This paper focuses on the Perfectly Matched Layer (PML) for external boundaries in CAA. To achieve low dissipation and dispersion errors, Dispersion-Relation-Preserving (DRP) Schemes are used for spatial discretization of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equations for time discretization. Four cases are given to illustrate the 2D PML equations for the linearized/nonlinear Euler equations in Cartesian coordinates and Cylindrical coordinates. The results show that the PML is effective as absorbing boundary condition. Those are basis for PML in actual computations of acoustic problems.

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29

Wu, Conghai, Sujuan Yang, and Ning Zhao. "A Fifth-Order Low-Dissipative Conservative Upwind Compact Scheme Using Centered Stencil." Advances in Applied Mathematics and Mechanics 6, no. 06 (December 2014): 830–48. http://dx.doi.org/10.4208/aamm.2013.m-s3.

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AbstractIn this paper, a conservative fifth-order upwind compact scheme using centered stencil is introduced. This scheme uses asymmetric coefficients to achieve the upwind property since the stencil is symmetric. Theoretical analysis shows that the proposed scheme is low-dissipative and has a relatively large stability range. To maintain the convergence rate of the whole spatial discretization, a proper non-periodic boundary scheme is also proposed. A detailed analysis shows that the spatial discretization implemented with the boundary scheme proposed by Pirozzoli [J. Comput. Phys., 178 (2001), pp. 81–117] is approximately fourth-order. Furthermore, a hybrid methodology, coupling the compact scheme with WENO scheme, is adopted for problems with discontinuities. Numerical results demonstrate the effectiveness of the proposed scheme.

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30

Nguyen, Binh Huy, and Giang Song Le. "Comparative study of numerical schemes for strong shock simulation using the Euler equations." Science and Technology Development Journal 18, no. 1 (March 31, 2015): 73–88. http://dx.doi.org/10.32508/stdj.v18i1.943.

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A numerical study of extremely strong shocks was presented. Various types of numerical schemes with first-order accuracy and higherorder accuracy with adaptive stencils were implemented to solve the one and twodimensional Euler equations based on the explicit finite difference method, including Roe’s first-order upwind, Steger-Warming Flux Vector splitting (FVS), Sweby’s flux-limited and Essentially Non-oscillatory (ENO) scheme. The result comparisons were carried out to discuss which scheme is the most suitable for strong shock problem. The dissipative nature of the firstorder scheme can be easily seen from the numerical solutions. High order ENO scheme had the best resolution for the case having weak discontinuity, but it over- predicted the shock wave location for the case of strong discontinuity.

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31

Lemaire, Vincent. "An adaptive scheme for the approximation of dissipative systems." Stochastic Processes and their Applications 117, no. 10 (October 2007): 1491–518. http://dx.doi.org/10.1016/j.spa.2007.02.004.

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32

Serrano, M., G. De Fabritiis, P. Español, and P. V. Coveney. "A stochastic Trotter integration scheme for dissipative particle dynamics." Mathematics and Computers in Simulation 72, no. 2-6 (September 2006): 190–94. http://dx.doi.org/10.1016/j.matcom.2006.05.019.

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33

Chanteur, G. "A Modified Fornberg-Whitham Scheme for Dissipative KdV Equations." Physica Scripta 33, no. 3 (March 1, 1986): 233–39. http://dx.doi.org/10.1088/0031-8949/33/3/010.

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34

Litvinov, S., M. Ellero, X. Y. Hu, and N. A. Adams. "A splitting scheme for highly dissipative smoothed particle dynamics." Journal of Computational Physics 229, no. 15 (August 2010): 5457–64. http://dx.doi.org/10.1016/j.jcp.2010.03.040.

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35

Zhang, Yu, Wenhui Pei, Qi Zhang, and Baosen Ma. "A Generalized Hamilton Robust Control Scheme of Trajectory Tracking for Intelligent Vehicles." Sensors 23, no. 15 (August 5, 2023): 6975. http://dx.doi.org/10.3390/s23156975.

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To ensure the accuracy and stability of intelligent-vehicle-trajectory tracking, a robust trajectory-tracking control strategy based on generalized Hamilton theory is proposed. Firstly, a dynamic Hamilton dissipative controller (DHDC) and trajectory-tracking Hamilton dissipative controller (TTHDC) were designed based on the established vehicle-dynamics control system and trajectory-tracking control system using the orthogonal decomposition method and control-switching method. Next, the feedback-dissipative Hamilton realizations of the two systems were obtained separately to ensure the convergence of the system. Secondly, based on the dissipative Hamilton system designed by TTHDC, a generalized Hamilton robust controller (GHRC) was designed. Finally, the co-simulation of Carsim and MATLAB/Simulink was used to verify the effectiveness of the three control algorithms. The simulation results show that DHDC and TTHDC can achieve self-stabilizing control of vehicles and enable certain control effects for the trajectory tracking of vehicles. The GHRC solves the problems of low tracking accuracy and poor stability of DHDC and TTHDC. Compared with the sliding mode controller (SMC) and linear quadratic regulator (LQR) controller, the GHRC can reduce the lateral error by 84.44% and the root mean square error (RMSE) by 83.92%, which effectively improves the accuracy and robustness of vehicle-trajectory tracking.

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36

Fülöp, Tamás, Róbert Kovács, Mátyás Szücs, and Mohammad Fawaier. "Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids." Entropy 22, no. 2 (January 28, 2020): 155. http://dx.doi.org/10.3390/e22020155.

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On the example of the Poynting–Thomson–Zener rheological model for solids, which exhibits both dissipation and wave propagation, with nonlinear dispersion relation, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error, and dispersive error are analyzed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest—elastic—level of modeling.

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37

Germanos, R. A. C., and L. F. De Souza. "ANALYSIS OF DISPERSION ERRORS IN ACOUSTIC WAVE SIMULATIONS." Revista de Engenharia Térmica 5, no. 1 (July 31, 2006): 62. http://dx.doi.org/10.5380/reterm.v5i1.61663.

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The governing equations of the acoustic problem are the compressible Euler equations. The discretization of these equations has to ensure that the acoustic waves are transported with non-dispersive and non-dissipative characteristics. In the present study numerical simulations of a standing acoustic wave are performed. Four different space discretization schemes are tested, namely, a second order finite-differences, a fourth order finitedifferences, a fourth order finite-differences compact scheme and a sixth order finite-differences compact scheme. The time integration is done with a fourth order Runge-Kutta scheme. The results obtained are compared with linearized analytical solutions. The influence of the dispersion on the simulation of a standing wave is analyzed. The results confirm that high order accuracy schemes can be more efficient for simulation of acoustic waves, especially the waves with high frequency.

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38

Li, Yan, Xingli Li, and Jiasen Jin. "Dissipation-Induced Information Scrambling in a Collision Model." Entropy 24, no. 3 (February 27, 2022): 345. http://dx.doi.org/10.3390/e24030345.

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In this paper, we present a collision model to stroboscopically simulate the dynamics of information in dissipative systems. In particular, an all-optical scheme is proposed to investigate the information scrambling of bosonic systems with Gaussian environmental states. Varying the states of environments, in the presence of dissipation, transient tripartite mutual information of system modes may show negative value, signaling the appearance of information scrambling. We also find that dynamical indivisibility based non-Markovianity plays dual roles in affecting the dynamics of information.

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39

Jiang, Yi, Meiliang Mao, Xiaogang Deng, and Huayong Liu. "Extending Seventh-Order Dissipative Compact Scheme Satisfying Geometric Conservation Law to Large Eddy Simulation on Curvilinear Grids." Advances in Applied Mathematics and Mechanics 7, no. 4 (May 29, 2015): 407–29. http://dx.doi.org/10.4208/aamm.2013.m404.

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AbstractSeventh-order hybrid cell-edge and cell-node dissipative compact scheme (HDCS-E8T7) is extended to a new implicit large eddy simulation named HILES on stretched and curvilinear meshes. Although the conception of HILES is similar to that of monotone integrated LES (MILES), i.e., truncation error of the discretization scheme itself is employed to model the effects of unresolved scales, HDCS-E8T7 is a new high-order finite difference scheme, which can eliminate the surface conservation law (SCL) errors and has inherent dissipation. The capability of HILES is tested by solving several benchmark cases. In the case of flow past a circular cylinder, the solutions of HILES fulfilling the SCL have good agreement with the corresponding experiment data, however, the flowfield is gradually contaminated when the SCL error is enlarged. With the help of fulling the SCL, ability of HILES for handling complex geometry has been enhanced. The numerical solutions of flow over delta wing demonstrate the potential of HILES in simulating turbulent flow on complex configuration.

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40

Aursand, Peder, and Johanna Ridder. "The Role of Inertia and Dissipation in the Dynamics of the Director for a Nematic Liquid Crystal Coupled with an Electric Field." Communications in Computational Physics 18, no. 1 (July 2015): 147–66. http://dx.doi.org/10.4208/cicp.220414.231214a.

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AbstractWe consider the dynamics of the director in a nematic liquid crystal when under the influence of an applied electric field. Using an energy variational approach we derive a dynamic model for the director including both dissipative and inertial forces.A numerical scheme for the model is proposed by extending a scheme for a related variational wave equation. Numerical experiments are performed studying the realignment of the director field when applying a voltage difference over the liquid crystal cell. In particular, we study how the relative strength of dissipative versus inertial forces influence the time scales of the transition between the initial configuration and the electrostatic equilibrium state.

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41

Yee, Helen C., Bjorn Sjögreen, and Abdellah Hadjadj. "Comparative Study of Three High Order Schemes for LES of Temporally Evolving Mixing Layers." Communications in Computational Physics 12, no. 5 (November 2012): 1603–22. http://dx.doi.org/10.4208/cicp.261111.130412a.

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AbstractThree high order shock-capturing schemes are compared for large eddy simulations (LES) of temporally evolving mixing layers for different convective Mach numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime. The considered high order schemes are fifth-order WENO (WENO5), seventh-order WENO (WENO7) and the associated eighth-order central spatial base scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter step (WENO7fi). This high order nonlinear filter method of Yee & Sjögreen is designed for accurate and efficient simulations of shock-free compressible turbulence, turbulence with shocklets and turbulence with strong shocks with minimum tuning of scheme parameters. The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al., and published direct numerical simulations (DNS) work of Rogers & Moser and Pantano & Sarkar, whereas results by WENO5 and WENO7 compare poorly with experimental data and DNS computations.

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42

Chen, Chang-Yong, Shao-Hua Li, and Mang Feng. "Scheme for entangled-mesoscopic-state generation in weakly dissipative cavities." Journal of Physics B: Atomic, Molecular and Optical Physics 40, no. 14 (July 6, 2007): 2961–67. http://dx.doi.org/10.1088/0953-4075/40/14/013.

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43

Qin, Jiaxian, Yaming Chen, and Xiaogang Deng. "Stabilized seventh-order dissipative compact scheme using simultaneous approximation terms." Applied Mathematics and Mechanics 40, no. 6 (April 23, 2019): 823–36. http://dx.doi.org/10.1007/s10483-019-2483-7.

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44

Danca, Marius-F., Wallace K. S. Tang, and Guanrong Chen. "A switching scheme for synthesizing attractors of dissipative chaotic systems." Applied Mathematics and Computation 201, no. 1-2 (July 2008): 650–67. http://dx.doi.org/10.1016/j.amc.2008.01.003.

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45

Gajewski, H., and K. Gärtner. "A dissipative discretization scheme for a nonlocal phase segregation model." ZAMM 85, no. 11 (November 2, 2005): 815–22. http://dx.doi.org/10.1002/zamm.200510233.

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46

Ortega, Roberto, Geraldine Farías, Marcela Cruchaga, Matías Rivero, Mariano Vázquez, Eva Casoni, and Guillaume Houzeaux. "Modeling the damped dynamic behavior of a flexible pendulum." Journal of Strain Analysis for Engineering Design 54, no. 2 (February 2019): 116–29. http://dx.doi.org/10.1177/0309324719832735.

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The focus of this work is on the computational modeling of a pendulum made of a hyperelastic material and the corresponding experimental validation with the aim of contributing to the study of a material commonly used in seismic absorber devices. From the proposed dynamics experiment, the motion of the pendulum is recorded using a high-speed camera. The evolution of the pendulum’s positions is recovered using a capturing motion technique by tracking markers. The simulation of the problem is developed in the framework of a parallel multi-physics code. Particular emphasis is placed on the analysis of the Newmark integration scheme and the use of Rayleigh damping model. In particular, the time step size effect is analyzed. A strong time step size dependency is obtained for dissipative time integration schemes, while the Rayleigh damping formulation without time integration dissipation shows time step–independent results when convergence is achieved.

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47

Frost, Miroslav, and Jan Valdman. "Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys." Mathematics 10, no. 23 (November 23, 2022): 4412. http://dx.doi.org/10.3390/math10234412.

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The incremental energy minimization principle provides a compact variational formulation for evolutionary boundary problems based on constitutive models of rate-independent dissipative solids. In this work, we develop and implement a versatile computational tool for the resolution of these problems via the finite element method (FEM). The implementation is coded in the MATLAB programming language and benefits from vector operations, allowing all local energy contributions to be evaluated over all degrees of freedom at once. The monolithic solution scheme combined with gradient-based optimization methods is applied to the inherently nonlinear, non-smooth convex minimization problem. An advanced constitutive model for shape memory alloys, which features a strongly coupled rate-independent dissipation function and several constraints on internal variables, is implemented as a benchmark example. Numerical simulations demonstrate the capabilities of the computational tool, which is suited for the rapid development and testing of advanced constitutive laws of rate-independent dissipative solids.

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48

Materassi, Massimo, and Emanuele Tassi. "Algebrizing friction: a brief look at the Metriplectic Formalism." Intellectual Archive 1, no. 3 (July 28, 2012): 45–52. http://dx.doi.org/10.32370/ia_2012_07_3.

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The formulation of Action Principles in Physics, and the introduction of the Hamiltonian framework, reduced dynamics to bracket algebra of observables. Such a framework has great potentialities, to understand the role of symmetries, or to give rise to the quantization rule of modern microscopic Physics. Conservative systems are easily algebrized via the Hamiltonian dynamics: a conserved observable H generates the variation of any quantity f via the Poisson bracket {f,H}. Recently, dissipative dynamical systems have been algebrized in the scheme presented here, referred to as metriplectic framework: the dynamics of an isolated system with dissipation is regarded as the sum of a Hamiltonian component, generated by H via a Poisson bracket algebra; plus dissipation terms, produced by a certain quantity S via a new symmetric bracket. This S is in involution with any other observable and is interpreted as the entropy of those degrees of freedom statistically encoded in friction. In the present paper, the metriplectic framework is shown for two original “textbook” examples. Then, dissipative Magneto-Hydrodynamics (MHD), a theory of major use in many space physics and nuclear fusion applications, is reformulated in metriplectic terms.

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49

Allahverdiev, B. P., and Ahmet Canoǧlu. "Spectral analysis of dissipative Schrödinger operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 6 (1997): 1113–21. http://dx.doi.org/10.1017/s0308210500026962.

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Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

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50

Kim, Valentine Aleksandrovich, and Roman Ivanovich Parovik. "Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation." Fractal and Fractional 6, no. 5 (May 19, 2022): 274. http://dx.doi.org/10.3390/fractalfract6050274.

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In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid.

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