scholarly journals Stability of correction procedure via reconstruction with summation-by-parts operators for Burgers' equation using a polynomial chaos approach

2018 ◽  
Vol 52 (6) ◽  
pp. 2215-2245 ◽  
Author(s):  
Philipp Öffner ◽  
Jan Glaubitz ◽  
Hendrik Ranocha
Keyword(s):  
Polynomial Chaos ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Conditions ◽  
High Sensitivity ◽  
Correction Procedure ◽  
Summation By Parts ◽  
Numerical Fluxes ◽  
Entropy Stable ◽  
The Given

In this paper, we consider Burgers’ equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic system of deterministic equations, which can be solved by several numerical methods. Here, we apply the correction procedure via reconstruction (CPR) using summation-by-parts operators. We focus especially on stability, which is proven for CPR methods and the systems arising from the PC approach. Due to the usage of split-forms, the major challenge is to construct entropy stable numerical fluxes. For the first time, such numerical fluxes are constructed for all systems resulting from the PC approach for Burgers' equation. In numerical tests, we verify our results and show also the performance of the given ansatz using CPR methods. Moreover, one of the simulations, i.e. Burgers’ equation equipped with an initial shock, demonstrates quite fascinating observations. The behaviour of the numerical solutions from several methods (finite volume, finite difference, CPR) differ significantly from each other. Through careful investigations, we conclude that the reason for this is the high sensitivity of the system to varying dissipation. Furthermore, it should be stressed that the system is not strictly hyperbolic with genuinely nonlinear or linearly degenerate fields.

New Travelling Wave Solutions of Burgers Equation with Finite Transport Memory

2010 ◽  
Vol 65 (8-9) ◽  
pp. 633-640 ◽  
Author(s):  
Rathinasamy Sakthivel ◽  
Changbum Chun ◽  
Jonu Lee
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Diffusion Processes ◽  
Initial Conditions ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Wide Range ◽  
Finite Memory

The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation

Analytical and numerical solutions of generalized Burgers' equation via Buckingham's Pi-theorem

2005 ◽  
Vol 83 (10) ◽  
pp. 1035-1049
Author(s):  
I A Hassanien ◽  
A A Salama ◽  
H A Hosham
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Similarity Solutions ◽  
Initial Value ◽  
Generalized Burgers Equation ◽  
One Step ◽  
Pi Theorem ◽  
Scaling Invariant

A generalized dimensional analysis performed by using Buckingham's Pi-theorem for the generalized Burgers' equation is presented. The application of the Buckingham Pi-theorem is used to reduce the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with appropriate corresponding conditions. By using a scaling invariant we simplify the similarity solutions, which are discussed for a specific choice of boundary conditions, and yield analytical solutions, which are in closed form. Also, using extended one-step methods of order five we solve the final ordinary differential equations. This criterion for solvability involves converting the boundary value problem to an initial value problem. PACS Nos.: 02.60.Lj, 47.27.Jv

Oscillatory flow past a circular cylinder in a rotating frame

1997 ◽  
Vol 345 ◽  
pp. 101-131
Author(s):  
M. D. KUNKA ◽  
M. R. FOSTER
Keyword(s):  
Circular Cylinder ◽  
Steady Flow ◽  
Stagnation Point ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Temporal Integration ◽  
Oncoming Flow ◽  
Flow Past ◽  
Rear Stagnation Point

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals Numerical daemons of hydrological models are summoned by extreme precipitation

2021 ◽  
Author(s):  
Peter T. La Follette ◽  
Adriaan J. Teuling ◽  
Nans Addor ◽  
Martyn Clark ◽  
Koen Jansen ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Extreme Precipitation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Computational Cost ◽  
Numerical Error ◽  
Hydrological Models ◽  
Multiple Model ◽  
Numerical Techniques ◽  
Modeling Framework

Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on approximate numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation like that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation events. In this experiment, a large number of hydrographs is generated with the modular modeling framework FUSE, using eight numerical techniques across a variety of forcing datasets. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational expense and numerical error associated with each hydrograph were recorded. It was found that numerical error (root mean square error) usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both low numerical error and low computational cost. A basic literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be sub-optimal. We conclude that relatively large numerical errors might be common in current models, and because these will likely become larger as the climate changes, we advocate for the use of low cost, low error numerical methods.

Effects of geometric similarity ratio on the disturbance amplitude of shock-wave front in viscosity measurement

2021 ◽  
pp. 2150330
Author(s):  
Kai Yang ◽  
Quan-Yu Xu ◽  
Xiao Wu ◽  
Xiao-Juan Ma
Keyword(s):  
Shock Wave ◽  
Phase Shift ◽  
Wave Front ◽  
Shock Wave Front ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Viscosity Measurement ◽  
Geometric Similarity ◽  
Similarity Ratio ◽  
Simulation Results

Geometric similarity ratio is one of the important factors that affects the disturbance amplitude of shock-wave front in viscosity measurement. In this paper, the Euler difference scheme of two-dimensional (2D) equations of viscous fluid mechanics is used to simulate the disturbance amplitude damping curves under different geometric similarity ratios, and the corresponding numerical solutions are shown. The samples of aluminum shocked to 80 GPa are taken as an example. The simulation results show that the initial conditions, material viscosity, wavelength, and sample geometric similarity ratio affect the evolution of the shock front sine wave disturbance. For flyer-impact flow field, the phase shift increases from 0 to a certain value with the viscosity coefficient for sample with wavelength [Formula: see text] mm and geometric similarity ratio [Formula: see text], 0.1. So, the geometric similarity method can be used to measure the viscosity of material. But it is found that the phase shift is sensitive to the geometric similarity ratio, which should be considered in Zaidel’s equation. So, some flyer-impact experiments will be carried out to determine the simulation results, and find the quantity relation of phase shift and viscosity of material in the future investigation.

Image Encryption Algorithm Based on a Novel 4D Chaotic System

2021 ◽  
Vol 15 (4) ◽  
pp. 118-131
Author(s):  
Sadiq A. Mehdi
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
High Sensitivity ◽  
Encryption Algorithm ◽  
Pseudorandom Sequence ◽  
Key Space ◽  
Xor Operation ◽  
Change Intensity ◽  
And Performance ◽  
Image Cryptosystem

In this paper, a novel four-dimensional chaotic system has been created, which has characteristics such as high sensitivity to the initial conditions and parameters. It also has two a positive Lyapunov exponents. This means the system is hyper chaotic. In addition, a new algorithm was suggested based on which they constructed an image cryptosystem. In the permutation stage, the pixel positions are scrambled via a chaotic sequence sorting. In the substitution stage, pixel values are mixed with a pseudorandom sequence generated from the 4D chaotic system using XOR operation. A simulation has been conducted to evaluate the algorithm, using the standardized tests such as information entropy, histogram, number of pixel change rate, unified average change intensity, and key space. Experimental results and performance analyses demonstrate that the proposed encryption algorithm achieves high security and efficiency.

Chaos of interband cascade lasers in the mid-infrared regime

2020 ◽  
Author(s):  
Yu Deng ◽  
Zhuo-Fei Fan ◽  
Shiyuan Zhao ◽  
Frédéric Grillot ◽  
Cheng Wang
Keyword(s):  
Semiconductor Lasers ◽  
Near Infrared ◽  
Optical Feedback ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Electrical Power ◽  
High Sensitivity ◽  
Light Sources ◽  
Mid Infrared ◽  
Operation Conditions

Abstract Chaos in nonlinear dynamical systems is featured with irregular appearance and with high sensitivity to initial conditions. Near-infrared semiconductor lasers subject to optical feedback from an external reflector are popular chaotic light sources, which have enabled multiple applications. Here, we report the fully-developed chaos in a mid-infrared interband cascade laser with external optical feedback. The chaos leads to significant electrical power enhancement over a frequency span of 500 MHz. In addition, the laser also exhibits periodic oscillations or low-frequency fluctuations before producing chaos, depending on the operation conditions. This work paves the way for extending chaos investigations from the near-infrared regime to the mid-infrared regime, which can stimulate potential applications in this spectral range.

scholarly journals On the evolution of solutions of Burgers equation on the positive quarter-plane

2019 ◽  
Vol 52 (1) ◽  
pp. 237-248
Author(s):  
Esen Hanaç
Keyword(s):  
Boundary Condition ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Condition ◽  
Quarter Plane ◽  
Initial Boundary Value ◽  
Good Agreement

AbstractIn this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

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