High-order essentially non-oscillatory scheme for viscoelasticity with fading memory
Authors:
Chi-Wang Shu and Yanni Zeng
Journal:
Quart. Appl. Math. 55 (1997), 459-484
MSC:
Primary 73V15; Secondary 35Q72, 65M06, 73F15
DOI:
https://doi.org/10.1090/qam/1466143
MathSciNet review:
MR1466143
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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 $\delta$-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.
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J. Casper, C.-W. Shu, and H. Atkins, A comparison of two formulations for high-order accurate essentially non-oscillatory schemes, AIAA J. 32, 1970–1977 (1994)
C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal. 91, 193–205 (1986)
C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4, 219–278 (1979)
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A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys. 71, 231–303 (1987)
W. J. Hrusa and J. A. Nohel, The Cauchy problem in one-dimensional nonlinear viscoelasticity, J. Differential Equations 59, 388–412 (1985)
S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh 106A, 169–194 (1987)
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, 611–613 (1964)
R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser-Verlag, Basel, 1990
T.-P. Liu, Nonlinear waves for viscoelasticity with fading memory, J. Differential Equations 76, 26–46 (1988)
T.-P. Liu and Y. Zeng, Large time behavior of solutions of general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., vol. 125, no. 599, 1997
R. C. MacCamy, A model for one-dimensional nonlinear viscoelasticity, Quart. Appl. Math. 35, 21–33 (1977)
P. Markowich and M. Renardy, Lax-Wendroff methods for hyperbolic history value problems, SIAM J. Numer. Anal. 21, 24–51 (1984). Errata in 22, 204 (1985)
J. A. Nohel and M. Renardy, Development of singularities in nonlinear viscoelasticity, in Amorphous Polymers and Non-Newtonian Fluids, C. Dafermos, J. L. Ericksen, and D. Kinderlehrer, eds., Springer, New York, 1987, pp. 139–152
J. A. Nohel, R. C. Rogers, and A. E. Tzavaras, Weak solutions for a nonlinear system in viscoelasticity, Comm. Partial Differential Equations 13, 97–127 (1988)
C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77, 439–471 (1988)
C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys. 83, 32–78 (1989)
C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput. 5, 127–149 (1990)
C.-W. Shu, T. A. Zang, G. Erlebacher, D. Whitaker, and S. Osher, High order ENO schemes applied to two and three dimensional compressible flow, Appl. Numer. Math. 9, 45–71 (1992)
O. Staffans, On a nonlinear hyperbolic Volterra equation, SIAM J. Math. Anal. 11, 793–812 (1980)
Y. Zeng, Convergence to diffusion waves of solutions to nonlinear viscoelastic model with fading memory, Comm. Math. Phys. 146, 585–609 (1992)
Y. Zeng, $L^{1}$ asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math. 47, 1053–1082 (1994)
Y. Zeng, $L^{p}$ asymptotic behavior of solutions to hyperbolic-parabolic systems of conservation laws, Arch. Math. 66, 310–319 (1996)
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Article copyright:
© Copyright 1997
American Mathematical Society