Abstract
Different splitting methods have been playing an important role in computations of numerical solutions of partial differential equations. Modern numerical strategies including mesh adaptations, linear and nonlinear transformations are also utilized together with splitting algorithms in applications. This survey concerns two cornerstones of the splitting methods, that is, the Alternating Direction Implicit (ADI) and Local One-Dimensional (LOD) methods, as well as their applications together with an eikonal mapping for solving highly oscillatory paraxial Helmholtz equations in slowly varying envelope approximations of active laser beams. The resulted finite difference scheme is not only oscillation-free, but also asymptotically stable. This ensures the high efficiency and applicability in optical wave applications.
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References
Chin, S.A.: A fundamental theorem on the structure of symplectic integrators. Phys. Lett. A 354, 373–376 (2006)
Descartes, R.: Discourse on the Méthod (trans: Lafleur, L.J., 1637). The Liberal Arts Press, New York (1960)
Descombes, S., Thalhammer, M.: An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT 50, 729–749 (2010)
D’Yakonov, E.G.: Difference schemes with splitting operator for multi-dimensional nonstationary problems. Zh. Vychisl. Mat. i Mat. Fiz. 2, 549–568 (1962)
Douglas Jr., J., Rachford Jr., H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
Frey, P., George, P.-L.: Mesh Generation, 2nd edn. Wiley-ISTE, New York (2008)
Hausdorff, F.: Die symbolische Exponentialformel in der Gruppentheorie. Ber Verh Saechs Akad Wiss Leipzig 58, 19–48 (1906)
Hundsdorfer, W.H., Verwer, J.G.: Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Math. Comput. 53, 81–101 (1989)
Iserles, A.: A First Course in the Numerical Analysis of Differential Equations, 2nd edn. Cambridge University Press, London (2011)
Jahnke, T., Lubich, C.: Error bounds for exponential operator splitting. BIT 40, 735–744 (2000)
Marchuk, G.I.: Some applicatons of splitting-up methods to the solution of problems in mathematical physics. Aplikace Matematiky 1, 103–132 (1968)
McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341–434 (2002)
Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–46 (2003)
Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
Sheng, Q.: Solving linear partial differential equations by exponential splitting. IMA J. Numer. Anal. 9, 199–212 (1989)
Sheng, Q.: Global error estimate for exponential splitting. IMA J. Numer. Anal. 14, 27–56 (1993)
Sheng, Q.: The ADI Methods. Encyclopedia of Applied and Computational Mathematics. Springer Verlag GmbH, Heidelberg (2015)
Sheng, Q.: ADI, LOD and modern decomposition methods for certain multiphysics applications. J. Algorithms Comput. Technol. 9, 105–120 (2015)
Sheng, Q., Sun, H.: On the stability of an oscillation-free ADI method for highly oscillatory wave equations. Commun. Comput. Phys. 12, 1275–1292 (2012)
Sheng, Q., Sun, H.: Exponential splitting for \(n\)-dimensional paraxial Helmholtz equation with high wavenumbers. Comput. Math. Appl. 68, 1341–1354 (2014)
Suzuki, M.: General theory of fractal path integrals with applications to manybody theories and statistical physics. J. Math. Phys. 32, 400–407 (1991)
Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)
Yanenko, N.N.: The Method of Fractional Steps; the Solution of Problems of Mathematical Physics in Several Variables. Springer, Berlin (1971)
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Sheng, Q. (2016). The Legacy of ADI and LOD Methods and an Operator Splitting Algorithm for Solving Highly Oscillatory Wave Problems. In: Singh, V., Srivastava, H., Venturino, E., Resch, M., Gupta, V. (eds) Modern Mathematical Methods and High Performance Computing in Science and Technology. Springer Proceedings in Mathematics & Statistics, vol 171. Springer, Singapore. https://doi.org/10.1007/978-981-10-1454-3_18
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