High-order essentially non-oscillatory scheme for viscoelasticity with fading memory

Abstract

Abstract.In this paper we describe the application of high-order essentially nonoscillatory (ENO) finite difference schemes to the viscoelastic model with fading memory. ENO schemes can capture shocks as well as various smooth structures in the solution to a high-order accuracy without spurious numerical oscillations.We first verify the stability and resolution of the scheme. We apply the scheme to a nonlinear problem with a known smooth solution and check the order of accuracy. Then we apply the scheme to a linear problem with initial discontinuities.Discontinuity locations and strengths in the solutions of such problems can be found explicitly by making use of a pointwise estimate obtained in this paper for the Green's function of the equations, which contains two Dirac ^-functions decaying exponentially. We check the resolution of the discontinuities by the scheme. After verifying that the scheme is indeed high-order accurate, produces sharp, non-oscillatory shocks with the correct location and strength, we then proceed in applying it to the nonlinear case with discontinuous or smooth initial conditions, and study the local properties (in time) as well as the long time behavior of the solutions. We conclude that the ENO scheme is a robust, accurate numerical tool to supplement theoretical analysis to study such equations with memory terms. It should also provide an efficient and reliable practical tool when such equations must be solved numerically in applications.

Weighted Essentially Non-oscillatory Schemes on Triangular Meshes

“…ENO (essentially non-oscillatory) schemes (Harten et al [16], Shu and Osher [28,29]) have been successfully applied to solve hyperbolic conservation laws and other convection dominated problems, for example in simulating shock turbulence interactions (Shu and Osher [29], Shu et al [30], and Adams and Shariff [2]), in the direct simulation of compressible turbulence (Shu et al [30], Walsteijn [35], and Ladeinde et al [20]), in solving the relativistic hydrodynamics equations (Dolezal and Wong [8]), in shock vortex interactions and other gas dynamics problems (Casper and Atkins [6] and Erlebacher et al [10]), in incompressible flow calculations (E and Shu [9] and Harabetian et al [13]), in solving the viscoelasticity equations with fading memory (Shu and Zeng [31]), in semiconductor device simulation (Fatemi et al [11] and Jerome and Shu [17,18]), and in image processing and level set methods (Osher and Sethian [24], Sethian [26], and Siddiqi et al [32]). The original ENO paper by Harten et al [16] was for a one-dimensional finite volume formulation.…”

Thermal Non-Equilibrium Flows in Three Space Dimensions

“…This is based on the expansions as the Fourier variable ξ approaches to zero and to infinity, respectively, obtained in Section 3. As in the case of one space dimension [18,21,25,26], the long time behavior of G(x, t) depends on the expansions ofĜ(ξ, t) for small ξ , while the local behavior depends on those for large ξ . Naturally, we divide the integral in the inverse transform into three parts: over |ξ | < ε, |ξ | > R and ε < |ξ | < R for some small constant ε and some large constant R.…”

High-Order Methods for Computational Physics

“…This problem was studied extensively in Woodward and Colella [92] and later by many others. This problem was studied extensively in Woodward and Colella [92] and later by many others.…”

“…For the rectangle based triangulation, we use a rectangular computational domain [0,4] x [0,1], as in [92]. The reflecting wall lies at the bottom of the computational domain for ~ $ x $ 4.…”

“…Unlike in [92] and many other papers, we do not modify our scheme near the corner in any way. Unlike in [92] and many other papers, we do not modify our scheme near the corner in any way.…”