In this paper are presented some recent advances in multiscale splitting methods, based on additive and iterative schemes and applied to deterministic and stochastic differential equations.
Several interesting algorithmic aspects of these novel splitting schemes will be discussed. For example, why a decomposed, or split, system may be the key to many important applications in multiply scaled subjects, and why iterative splitting methods can be powerful and more appropriate for well-balanced coupled nonlinear problems. Both theoretical and practical aspects of the recent advances in splitting methods for multiphysics and multiscale applications will be discussed.
AbdulleA.EngquistW. E. B., and Vanden-EijndenE.. The heterogeneous multiscale method. Acta Numerica21, 1–87, 2012.
2.
BergerM.J. and ColellaP.. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82:64–84, 1989.
3.
BlancX.Le BrisC. and LionsP.-L.Atomistic to continuum limits for computational materials science. ESAIM, Mathematical Modelling and Numerical Analysis, 41(2):391–426, 2007.
4.
BrandtA.. Multilevel adaptive solutions to boundary value problems. Mathematics of Computation, 31:333–390, 1977.
5.
BrandtA.. Multiscale scientific computation: Review 2000. In: Multiscale and Multiresolution Methods, Springer-Verlag, Berlin, 2001, pp. 1–96.
ChinS.GeiserJ.. Multi-product operator splitting as a general method of solving autonomous and non-autonomous equations. IMA Journal of Numerical Analysis, accepted, June 2010.
8.
DullweberA.LeimkuhlerB., and McLachlanR.I.. Split-Hamiltonian methods for rigid body molecular dynamics. J. Chem. Phys.107(5):5840–5851, 1997.
9.
EngquistW. E. B.. Principles of Multiscale Modelling.Cambridge University Press, 2010.
10.
EngquistW. E. B.LiX.RenW., and Vanden-EijndenE.. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., 2(3):367–450, 2007.
11.
EngquistW. E. B. and HuangZ.. Heterogeneous multiscale method: A general methodology for multiscale modeling. Physical Review B67:092101, 2003.
12.
EngquistW. E. B.. The heterogenous multiscale methods. Commun. Math. Sci.1(1):87–132, 2003.
13.
FaragoI. and GeiserJ.. Iterative operator-splitting methods for linear problems. International Journal of Computa- tional Science and Engineering3(4):255–263, 2007.
14.
GanderM.J. and VanderwalleS.. Analysis of the Parareal time-parallel time-integration method. SIAM Journal of Scientific Computing, 29(2):556–578, 2007.
15.
GearC.W.HymanJ.M.KevrekididP.G.KevrekidisI.G.RunborgO., and TheodoropoulosC.. Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis. Comm. Math. Sciences, 1(4):715–762, 2003.
16.
GeiserJ.. Decomposition Methods for Partial Differential Equations: Theory and Applications in Multiphysics Problems.Chapman & Hall/CRC, 2009.
17.
GeiserJ.UckV.B. and ArabM.. Model of PE-CVD apparatus: Verification and Simulations.Mathematical Problems in Engineering, Hindawi Publishing Corp., New York, accepted, April 2010.
18.
GeiserJ.. An iterative splitting method via waveform relaxation. International Journal of Computer Mathematics, 88(17):3646–3665, 2011.
19.
GeiserJ.. Iterative Splitting Methods for Differential Equations.Chapman & Hall/CRC, 2011.
20.
GeiserJ.. Multiscale splitting for stochastic differential equations: Applications in particle collisions. Journal of Coupled Systems and Multiscale Dynamics, August 2013.
21.
GeiserJ. and GüttelS.. Coupling methods for heat transfer and heat flow: Operator splitting and the parareal algorithm. Journal of Mathematical Analysis and Applications, 388(2):873–887, 2012.
22.
GeiserJ.. Iterative splitting methods for multiscale problems. Proceedings paper, DCABES 2013 (2–4 September, 2013), London, accepted, June 2013.
23.
GeiserJ.. Multiscale modeling of PE-CVD apparatus: Simulations and approximations. Invited paper to special issue on “Multiscale Simulations in Soft Matter”, accepted, January 2013.
24.
HackbuschW.. Multi-Grid Methods and Applications.Springer-Verlag, Berlin, 1985.
25.
HairerE.LubichC., and WannerG.. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations.SCM, Springer-VerlagBerlin, No. 31, 2002.
26.
HansenE. and OstermannA.. Exponential splitting for unbounded operators. Mathematics of Computation, 78:1485–1496, 2009.
27.
HartenA.. Multiresolution Representation and Numerical Algorithms: A Brief Review. NASA Report, 1994.
28.
HoldenH.KarlsenK. H.LieK.-A., and RisebroN.H.. Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB programs. EMS Series of Lectures in Mathematics.European Mathematical Society, Zürich, 2010.
29.
HundsdorferW.H. and VerwerJ.G.. Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations, Springer-Verlag, Berlin, 2003.
30.
JahnkeT. and LubichC.. Error bounds for exponential operator splittings. BIT Numerical Mathematics, 40(4):735–745, 2000.
31.
KloedenP.E. and PlatenE.. The Numerical Solution of Stochastic Differential Equations.Springer-Verlag, Berlin, 1992.
32.
KevrekidisI.G. and SamaeyG.. Equation-free multiscale computation: Algorithms and applications. Annual Review in Physical Chemistry60:321–344, 2009.
33.
KarlsenK.H.LieK.-A.NatvigJ.R.NordhaugH.F., and DahleH.K.. Operator splitting methods for systems of convection–diffusion equations: Nonlinear error mechanisms and correction strategies. J. Comput. Phys., 173(2):636–663, 2001.
34.
LadicsT. and FaragoI.. Generalizations and error analysis of the iterative operator splitting method. Central European Journal of Mathematics, 11(8):1416–1428, 2013.
35.
LemonsD.S.WinskeD.DaughtonW., and AlbrightB.. Small-angle Coulomb collision model for particle-in-cell simulations. J. Comput. Phys., 228(5):1391–1403, 2009.
36.
LionsJ. L.MadayY.TurinciniG.. A “parareal” in time discretization of PDEs. C. R. Acad. Sci. Paris Sér. I Math., 332:661–668, 2001.
37.
MarchukG.I.Some applications of splitting-up methods to the solution of mathematical physics problems. Aplikace Matematiky, 13:103–132, 1968.
38.
MarchukG.I.. Splitting and alternating direction methods. In Handbook of Numerical Analysis, CiarletP. G.LionsJ. L. (eds), vol. 1. Elsevier, Amsterdam, 1990.
39.
McLachlanR.I. and QuispelG.R.W.. Splitting methods. Acta Numerica, 11:341–434, 2002.
40.
NevanlinnaO.. Remarks on Picard–Lindelöf Iteration, Part I. BIT, 29:328–346, 1989.
41.
NevanlinnaO.. Remarks on Picard–Lindelöf Iteration, Part II. BIT, 29:535–562, 1989.
42.
NevanlinnaO.. Linear Acceleration of Picard–Lindelöf. Numerische Mathematik, 57:147–156, 1990.
43.
NinomiyaS. and VictoirN., Weak approximation of stochastic differential equations and application to derivative pricing. Applied Mathematical Finance, 15(2):107–121, 2008.
44.
PavliotisG.A. and StuartA.M.. Multiscale Methods: Averaging and Homogenization.Springer-Verlag, Heidelberg, 2008.
45.
RokhlinV.. Rapid solution of integral equations of classic potential theory. J. Computational Physics, 60:187–207, 1985.
46.
ShengQ.. Solving linear partial differential equations by exponential splitting. IMA Journal of Numer. Analysis, 9:199–212, 1989.
47.
ShengQ.. Global error estimate for exponential splitting. IMA J. Numer. Anal., 14:27–56, 1993.
48.
StrangG.. On the construction and comparision of difference schemes. SIAM J. Numer. Anal., 5:506–517, 1968.
49.
ThalhammerM.. High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM Journal on Numerical Analysis, 46(4):2022–2038, 2008.
50.
TrotterH.F.. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4):545–551, 1959.
