scholarly journals Comparative Review on Computational Performance of Multistep Schemes in Solving One-Dimensional Linear Wave Equation

CFD letters ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1-14
Author(s):  
Serdar Hayytov ◽  
Wah Yen Tey ◽  
Hooi Siang Kang ◽  
Mohammed W. Muhieldeen ◽  
Omid Afshar
Keyword(s):  
Wave Equation ◽  
Multistep Methods ◽  
Single Step ◽  
Second Order ◽  
Hyperbolic Model ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Computational Performance ◽  
Upwind Method

Among several numerical methods used to solve the hyperbolic model of the linear wave equation, single-step algorithms can be the more popular ones. However, these algorithms are time-consuming while incurring numerical inaccuracy. Thus, multistep methods can be a suitable option as it has a high order of accuracy. This study aims to investigate and compare the computational performance of these multistep schemes in solving hyperbolic model based on one-dimensional linear wave equation. The techniques studied in this paper comprise the two-step Lax-Wendroff method, MacCormack method, second-order upwind method, Rusanov-Burstein-Mirin method, Warming-Kutler-Lomax method, and fourth-order Runge-Kutta method. Finite difference method is applied in discretisation. Our simulation found that although higher-order multistep methods are more stable than single-step algorithm, they suffer numerical diffusion. The two-step Lax-Wendroff method outperforms other schemes, although it is relatively simple compared with the other three and four steps schemes. The second-order upwind method is attractive as well because it is executable even with a high Courant number.

Estimates for the controls of the wave equation with a potential

2018 ◽  
Vol 24 (1) ◽  
pp. 289-309 ◽  
Author(s):  
Sorin Micu ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Space Variable ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Boundary Controls ◽  
The One ◽  
Variable Potential ◽  
Uniformly Bounded

This article studies the L2-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential a = a(x). It is known these controls depend on a and their norms may increase exponentially with ||a||L∞. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal L2−norm controls are uniformly bounded with respect to the potential a, if the initial data have only sufficiently high eigenmodes.

scholarly journals On weak regularity requirements of the relaxation modulus in viscoelasticity

2019 ◽  
Vol 10 (1) ◽  
pp. 78-87
Author(s):  
Sandra Carillo ◽  
Michel Chipot ◽  
Vanda Valente ◽  
Giorgio Vergara Caffarelli
Keyword(s):  
Wave Equation ◽  
Existence And Uniqueness ◽  
Relaxation Function ◽  
Relaxation Modulus ◽  
Linear Wave ◽  
Differential Problem ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Continuous Relaxation ◽  
Limit Procedure

Abstract The existence and uniqueness of solution to a one-dimensional hyperbolic integro-differential problem arising in viscoelasticity is here considered. The kernel, in the linear viscoelasticity equation, represents the relaxation function which is characteristic of the considered material. Specifically, the case of a kernel, which does not satisfy the classical regularity requirements is analysed. This choice is suggested by applications according to the literature to model a wider variety of materials. A notable example of kernel, not satisfying the classical regularity requirements, is represented by a wedge continuous function. Indeed, the linear integro-differential viscoelasticity equation, characterised by a suitable wedge continuous relaxation function, is shown to give the classical linear wave equation via a limit procedure.

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

scholarly journals Godunov type scheme for the linear wave equation with Coriolis source term

2017 ◽  
Vol 58 ◽  
pp. 1-26 ◽  
Author(s):  
Emmanuel Audusse ◽  
Stéphane Dellacherie ◽  
Minh Hieu Do ◽  
Pascal Omnes ◽  
Yohan Penel
Keyword(s):  
Wave Equation ◽  
Source Term ◽  
Linear Wave ◽  
Linear Wave Equation

scholarly journals Radial solutions to the Cauchy problem for □1+3U = 0 as limits of exterior solutions

2016 ◽  
Vol 13 (04) ◽  
pp. 833-860
Author(s):  
Helge Kristian Jenssen ◽  
Charis Tsikkou
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Cauchy Data ◽  
Linear Wave ◽  
Radial Solutions ◽  
Linear Wave Equation ◽  
Model Situation ◽  
Neumann Solutions

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation [Formula: see text] with radial Cauchy data [Formula: see text], [Formula: see text]. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in [Formula: see text] whenever [Formula: see text]. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in [Formula: see text], we obtain [Formula: see text]-solutions via exterior Neumann solutions, and only [Formula: see text]-solutions via exterior Dirichlet solutions.

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