Dispersion and Gibbs phenomenon associated with difference approximations to initial boundary-value problems for hyperbolic equations☆
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On the dissipation at a shock wave in an elastic bar
2022, International Journal of Solids and StructuresCitation Excerpt :There is a rich literature on the dynamics of one-dimensional lattices. A few of these papers include: the celebrated Fermi–Pasta–Ulam–Tsingou (FPUT) problem where the authors investigated the transfer of energy between modes in a one-dimensional chain of particles, (Fermi et al., 1955); the closed form solution to a dynamic problem for a harmonic chain, Synge (1973) and Chin (1975); the motion of a Frenkel–Kontorova dislocation, e.g. Atkinson and Cabrera (1965); the dynamics of phase transitions, e.g. Slepyan et al. (2005), Kresse and Truskinovsky (2003), Truskinovsky and Vainchtein (2005), Puglisi and Truskinovsky (2000), Purohit and Bhattacharya (2003), Zhao and Purohit (2016); the dispersive evolution of pulses in a lattice, e.g. Giannoulis and Mielke (2006); the derivation by Aubry and Proville (2009) of Rankine–Hugoniot type jump conditions for a discrete damped nonlinear lattice; and so on. The rigorous transition from a discrete model to a continuous one is subtle, e.g. see Giannoulis et al. (2006).
Effect of the inter-fiber friction on fiber damage propagation and ballistic limit of 2-D woven fabrics under a fully confined boundary condition
2016, International Journal of Impact EngineeringSteady discrete shocks of 5th and 7th-order RBC schemes and shock profiles of their equivalent differential equations
2014, Journal of Computational PhysicsAn explicit time integration scheme for the analysis of wave propagations
2013, Computers and StructuresCitation Excerpt :Accurate finite element solutions of wave propagations are difficult to obtain. Numerical errors due to the spatial and time discretizations resulting in artificial period elongations and amplitude decays, seen as numerical dispersions and dissipations, often render finite element solutions of wave propagation problems to be quite inaccurate [1,6–10]. In particular, large errors in just the few highest frequency modes contained in the mesh shown as spurious oscillations can severely impair the accuracy of the solution.
Performance of an implicit time integration scheme in the analysis of wave propagations
2013, Computers and StructuresCitation Excerpt :Direct time integration is widely used in the finite element solutions of transient wave propagation problems, see e.g. [4,5]. However, accurate solutions are difficult to obtain because of numerical dispersion and dissipation, resulting from period elongations and amplitude decays [4–11]. Spurious oscillations, especially for high wave numbers, can severely ruin the accuracy of the solution.
A finite element method enriched for wave propagation problems
2012, Computers and StructuresCitation Excerpt :Therefore, for problems with short waves, very fine meshes are required to obtain reasonable solutions, so fine, that the numerical solution effort may be prohibitive. In the case of transient wave propagations, the solution may exhibit spurious oscillations, related to the Gibb’s phenomenon, and the numerical wave propagation velocity may be significantly different from the physical velocity, due to the numerical period elongation and amplitude decay [2,11] resulting in the dispersion and dissipation errors [11–23]. When a wave travels long distances, the errors become large and the numerical solution is very inaccurate.
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This work was performed under the auspices of the U.S. Atomic Energy Commission.