Search Results (8)

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results. Full article

In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems at inter-cells. In comparison with the Runge–Kutta DG (RKDG) methods, the proposed method needs less computer storage, thanks to the absence of intermediate stages. In brief, this current method is one-step, one-stage, and fully-discrete. Moreover, this method can easily obtain arbitrary high-order accuracy both in space and in time. Numerical results for one- and two-dimensional shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolution for discontinuous solutions. Full article

The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical schemes cannot directly maintain discrete entropy inequalities. To address this, we here construct an ES finite difference scheme for the nonlinear hyperbolic systems of conservation laws. The proposed scheme can not only maintain the discrete entropy inequality, but also enjoy high-order accuracy. Firstly, we construct the second-order accurate semi-discrete entropy conservative (EC) schemes and ensure that the schemes meet the entropy identity when an entropy pair is given. Then, the second-order EC schemes are used as a building block to achieve the high-order accurate semi-discrete EC schemes. Thirdly, we add a dissipation term to the above schemes to obtain the high-order ES schemes. The term is based on the Weighted Essentially Non-Oscillatory (WENO) reconstruction. Finally, we integrate the scheme using the third-order Runge–Kutta (RK) approach in time. In the end, plentiful one- and two-dimensional examples are implemented to validate the capability of the scheme. In summary, the current scheme has sharp discontinuity transitions and keeps the genuine high-order accuracy for smooth solutions. Compared to the standard WENO schemes, the current scheme can achieve higher resolution. Full article

The numerical solution of hyperbolic conservation laws requires algorithms with upwind characteristics. Conventional methods such as the numerical difference method can realize this characteristic by constructing special distributions of nodes. However, there are still no reports on how to construct algorithms with upwind characteristics through wavelet theory. To solve this problem, a system of high-order and stable wavelet collocation upwind schemes was successfully proposed by constructing interpolation wavelets with specific symmetry and smoothness. The effects of the characteristics of the scaling functions on the schemes were explored based on numerical tests and Fourier analysis. The numerical results revealed that the stability of the constructed scheme is affected by the smoothness order, N, and the asymmetry of the scaling function. The dissipation analysis suggested that schemes with N ∈ even have negative dissipation coefficients, leading to unstable behaviors. Only scaling functions with N ∈ odd and a bias magnitude of 1 can be used to construct stable upwind schemes due to the non-negative dissipation coefficients. Typical numerical examples verified the effectiveness of the proposed method, which is proved to have high accuracy and efficiency in solving high-speed flow problems with multi-scale smooth structures and discontinuities. Full article

We investigated the derivation of numerical methods for solving partial differential equations, focusing on those that preserve physical properties of Hamiltonian systems. The formulation of these properties via symplectic forms gives rise to multisymplectic variational schemes. By using analogy with the smooth case, we defined a discrete Lagrangian density through the use of exponential functions, and derived its Hamiltonian by Legendre transform. This led to a discrete Hamiltonian system, the symplectic forms of which obey the conservation laws. The integration schemes derived in this work were tested on hyperbolic-type PDEs, such as the linear wave equations and the non-linear seismic wave equations, and were assessed for their accuracy and the effectiveness by comparing them with those of standard multisymplectic ones. Our error analysis and the convergence plots show significant improvements over the standard schemes. Full article

Piping systems (e.g., storm sewers) that transition between free-surface flow and surcharged flow are challenging to model in one-dimensional (1D) networks as the continuity equation changes from hyperbolic to elliptic as the water surface reaches the pipe ceiling. Previous network models are known to have poor mass conservation or unpredictable convergence behavior at such transitions. To address this problem, a new algorithm is developed for simulating unsteady 1D flow in closed conduits with both free-surface and surcharged flow. The shallow-water (hydrostatic) approximation is used as the governing equations. The artificial compressibility (AC) method is implemented as a dual-time-stepping discretization for a finite-volume solver with timescale interpolation used for face reconstruction. A new formulation for the AC celerity parameter is proposed such that the AC celerity matches the equivalent gravity wave speed for the local hydraulic head—which has some similarities to the classic Preissmann Slot used to approximate pressurized flow in conduits. The new approach allows the AC celerity to be set locally by the flow (i.e., non-uniform in space) and removes it as a free parameter of the AC solution method. The derivation of the AC method provides for only a minor change in the form of the solution equations when a computational element switches from free-surface to surcharged. The new solver is tested for both unsteady free-surface (supercritical, subcritical) and surcharged flow transitions in a circular pipe and is implemented in an open-source Python code available under the name “PipeAC.” The results are compared to laboratory experiments that include rapid flow changes due to opening/closing of gates. Results show that the new algorithm is satisfactory for 1D representation of unsteady transition behavior with two caveats: (i) sufficient grid resolution must be applied, and (ii) the shallow-water equation approximations (hydrostatic, single-fluid) limit the accuracy of the solution with regards to the celerity of the turbulent unsteady bore that propagates upstream. This research might benefit any piping network model that must smoothly handle unsteady transitions from free surface to surcharged flow. Full article

In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented. Full article

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes. Full article