Abstract
In this paper we solve linear parabolic problems using the three stage noble algorithms. First, the time discretization is approximated using the Laplace transformation method, which is both parallel in time (and can be in space, too) and extremely high order convergent. Second, higher-order compact schemes of order four and six are used for the the spatial discretization. Finally, the discretized linear algebraic systems are solved using multigrid to show the actual convergence rate for numerical examples, which are compared to other numerical solution methods.
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Anantha Krishnaiah U., Manohar R., Stephenson J.W.: Fourth-order finite difference methods for three-dimensional general linear elliptic problems with variable coefficients. Numer. Methods Partial Differ. Equ. 3(3), 229–240 (1987)
Anantha Krishnaiah U., Manohar R., Stephenson J.W.: High-order methods for elliptic equations with variable coefficients. Numer Methods Partial Differ. Equ. 3(3), 219–227 (1987)
Bromwich T.J.I.: Normal coordinates in dynamical systems. Proc. Lond Math. Soc. 15(Ser. 2), 401–448 (1916)
Cohen A.M.: Numerical Methods for Laplace Transform Inversion. Springer, New York (2007)
Crump K.S.: Numerical inversion of Laplace transforms using a Fourier series approximation. J. ACM 23(1), 89–96 (1976)
Douglas, C.C.: Multi–grid algorithms for elliptic boundary–value problems. PhD thesis, Yale University (1982)
Douglas C.C.: Multi–grid algorithms with applications to elliptic boundary–value problems. SIAM J. Numer. Anal. 21, 236–254 (1984)
Douglas C.C.: Madpack: a family of abstract multigrid or multilevel solvers. Comput. Appl. Math. 14, 3–20 (1995)
Douglas C.C., Douglas J. Jr: A unified convergence theory for abstract multigrid or multilevel algorithms, serial and parallel. SIAM J. Numer. Anal. 30, 136–158 (1993)
Douglas C.C., Douglas J. Jr, Fyfe D.E.: A multigrid unified theory for non-nested grids and/or quadrature. E W J Numer. Math. 2, 285–294 (1994)
Faber V., Manteuffel T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21, 352–362 (1984)
Freund, R.W.: Solution of shifted linear systems by quasi-minimal residual iterations. In: Numerical Linear Algebra (Kent, OH, 1992), de Gruyter, Berlin, pp. 101–121 (1993)
Freund R.W., Nachtigal N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60(3), 315–339 (1991)
Gander M.J., Vandewalle S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29, 556–578 (2007)
Gavrilyuk I.P., Hackbusch W., Khoromskij B.N.: \({\fancyscript{H}}\) -matrix approximation for the operator exponential with applications. Numer. Math. 92, 83–111 (2002)
Gupta M.M., Manohar R.P., Stephenson J.W.: A single cell high order scheme for the convection-diffusion equation with variable coefficients. Intern. J. Numer. Methods Fluids 4(7), 641–651 (1984)
Gupta M.M., Manohar R.P., Stephenson J.W.: High-order difference schemes for two-dimensional elliptic equations. Numer. Methods Partial Differ. Equ. 1(1), 71–80 (1985)
Karaa S., Zhang J.: High order adi method for solving unsteady convection-diffusion problems. J. Comput. Phys. 198, 1–9 (2004)
Kwon, Y., Manohar, R., Stephenson, J.W.: Single cell fourth order methods for the biharmonic equation. In: Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), vol. 34, pp. 475–482 (1982)
Lee, H.: The laplace transformation method and its applications in financial mathematics. PhD thesis, Interdisciplinary Program in Computational Science and Technology, Seoul National University, Seoul (2009)
Lee J., Sheen D.: An accurate numerical inversion of Laplace transforms based on the location of their poles. Comput. Math. Appl. 48(10–11), 1415–1423 (2004)
Lee J., Sheen D.: A parallel method for backward parabolic problems based on the Laplace transformation. SIAM J. Numer. Anal. 44, 1466–1486 (2006)
Lions J.L., Maday Y., Turinici G.: A parareal in time discretization of PDE’s. C R Acad. Sci. Paris Ser. I Math. 332, 661–668 (2001)
López-Fernández M., Palencia C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004)
López-Fernández M., Palencia C., Schädle A.: A spectral order method for inverting sectorial laplace transforms. SIAM J. Numer. Anal. 44(3), 1332–1350 (2006)
Murli A., Rizzardi M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Softw. 16, 158–168 (1990)
Sheen D., Sloan I.H., Thomée V.: A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature. Math. Comp. 69(229), 177–195 (2000)
Sheen D., Sloan I.H., Thomée V.: A parallel method for time-discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal. 23(2), 269–299 (2003)
Singer I., Turkel E.: High order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163, 343–358 (1998)
Singer I., Turkel E.: Sixth order accurate finite difference schemes for the Helmholtz equation. J. Comput. Accoustics 14, 339–351 (2006)
Talbot A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Applics 23, 97–120 (1979)
Thomée V.: A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature. Int. J. Numer. Anal. Model 2, 121–139 (2005)
Weeks W.T.: Numerical inversion of Laplace transforms using Laguerre functions. J. ACM 13(3), 419–429 (1966)
Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76(259), 1341–1356 (electronic) (2007)
Widder D.V.: The Laplace transform. Princeton University Press, Princeton (1941)
Young, D.M., Dauwalder, J.H.: Discrete representation of partial differential operators. In: Error in Digital Computation, Wiley New York, vol. 2, pp. 181–207 (1965)
Zhang J.: An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation. Comm. Numer. Methods Eng. 14(3), 209–218 (1998)
Zhang J., Sun H., Zhao J.J.: High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 191(41-42), 4661–4674 (2002)
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Communicated by: C. W. Oosterlee and A. Borzi.
The research by Prof. Douglas is based on work supported in part by NSF grants CNS-1018072 and CNS-1018079 and Award No. KUS-C1-016-04, made by the King Abdullah University of Science and Technology (KAUST). The research by Prof. Sheen was partially supported by NRF-2008-C00043 and NRF-2009-0080533, 0450-20090014. The research by H. Lee was partially supported by Seoul R & D Program WR080951.
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Douglas, C., Kim, I., Lee, H. et al. Higher-order schemes for the Laplace transformation method for parabolic problems. Comput. Visual Sci. 14, 39 (2011). https://doi.org/10.1007/s00791-011-0156-6
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DOI: https://doi.org/10.1007/s00791-011-0156-6