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Immersed Interface CIP for One Dimensional Hyperbolic Equations

Published online by Cambridge University Press:  03 June 2015

Kazufumi Ito  and
Tomoya Takeuchi
Kazufumi Ito*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Tomoya Takeuchi*
Affiliation:
Collaborative Research Center for Innovative Mathematical Modelling, Institute of Industrial Science, University of Tokyo, Tokyo, Japan FIRST, Aihara Innovative Mathematical Modelling Project, JST, Kawaguchi, Japan
*
Corresponding author.Email:kito@math.ncsu.edu
Email:takeuchi@sat.t.u-tokyo.ac.jp
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Abstract

The immersed interface technique is incorporated into CIP method to solve one-dimensional hyperbolic equations with piecewise constant coefficients. The proposed method achieves the third order of accuracy in time and space in the vicinity of the interface where the coefficients have jump discontinuities, which is the same order of accuracy of the standard CIP scheme. Some numerical tests are given to verify the accuracy of the proposed method.

Keywords

65M1265M25CIP methodimmersed interface methodhyperbolic equations in discontinuous media.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 96 - 114
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Agarwal, R. P., Sharp Hermite interpolation error bounds for derivatives, Nonlinear Analysis, 17 (1991), 773–786.Google Scholar
[2]Aoki, T., Interpolated differential operator (IDO) scheme for solving partial differential equations, Computer Physics Communications, 102 (1997), 132–146.Google Scholar
[3]Barada, D., Fukuda, T., Itoh, M., and Yatagai, T., Cubic interpolated propagation scheme in numerical analysis of lightwave and optical force, Opt. Express, 14 (2006), 4151–4168.Google Scholar
[4]Evans, L. C., Partial differential equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.Google Scholar
[5]Imai, Y. and Aoki, T., A higher-order implicit IDO scheme and its CFD application to local mesh refinement method, Computational Mechanics, 38 (2006), 211–221.Google Scholar
[6]LeVeque, R. J. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019–1044.Google Scholar
[7]Li, Z. and Ito, K., The immersed interface method, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.Google Scholar
[8]Ogata, Y., Yabe, T., and Odagaki, K., An accurate numerical scheme for Maxwell equation with CIP-method of characteristics, Commun. Comput. Phys., 1 (2006), 311–335.Google Scholar
[9]Strikwerda, J. C., Finite difference schemes and partial differential equations, The Wadsworth, Pacific Grove, CA, 1989.Google Scholar
[10]Takewaki, H., Nishiguchi, A., and Yabe, T., Cubic interpolated pseudo-particle method (CIP) for solving hyperbolic-type equations, J. Comput. Phys., 61 (1985), 261–268.Google Scholar
[11]Utsumi, T., Kunugi, T., and Aoki, T., Stability and accuracy of the cubic interpolated propagation scheme, Computer Physics Communications, 101 (1997), 9–20.Google Scholar
[12]Yabe, T., A universal cubic interpolation solver for compressible and incompressible fluids, ShockWaves, 1 (1991), 187–195.Google Scholar
[13]Yabe, T. and Aoki, T., A universal solver for hyperbolic equations by cubic-polynomial interpolation. I. One-dimensional solver, Comput. Phys. Comm., 66 (1991), 219–232.CrossRefGoogle Scholar
[14]Yabe, T., Ishikawa, T., Wang, P. Y., Aoki, T., Kadota, Y., and Ikeda, F., A universal solver for hyperbolic equations by cubic-polynomial interpolation. II. Two- and three-dimensional solvers, Comput. Phys. Comm., 66 (1991), 233–242.Google Scholar
[15]Yabe, T., Mizoe, H., Takizawa, K., Moriki, H., Im, H. N., and Ogata, Y., Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme, J. Comput. Phys., 194 (2004), 57–77.CrossRefGoogle Scholar
[16]Yabe, T., Xiao, F., and Utsumi, T., The constrained interpolation profile method for multiphase analysis, J. Comput. Phys., 169 (2001), 556–593.Google Scholar
[17]Zhang, C. and LeVeque, R. J., The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion, 25 (1997), 237–263.Google Scholar