Defect-based local error estimators for high-order splitting methods involving three linear operators

Prior work on high-order exponential operator splitting methods is extended to evolution equations defined by three linear operators. A posteriori local error estimators are constructed via a suitable integral representation of the local error involving the defect associated with the splitting solution and quadrature approximation via Hermite interpolation. In order to prove asymptotical correctness, a multiple integral representation involving iterated defects is deduced by repeated application of the variation-of-constant formula. The error analysis within the framework of abstract evolution equations provides the basis for concrete applications. Numerical examples for initial-boundary value problems of Schrödinger and of parabolic type confirm the asymptotical correctness of the proposed a posteriori error estimators.

[1]  L. Einkemmer.  Structure preserving numerical methods for the Vlasov equation , 2016, 1604.02616.

[2]  Mechthild Thalhammer,et al.  Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part II. Higher-order methods for linear problems , 2014, J. Comput. Appl. Math..

[3]  Mechthild Thalhammer,et al.  Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part I: The linear case , 2012, J. Comput. Appl. Math..

[4]  P. Gray,et al.  Sustained oscillations and other exotic patterns of behavior in isothermal reactions , 1985 .

[5]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[6]  Mechthild Thalhammer,et al.  Embedded exponential operator splitting methods for the time integration of nonlinear evolution equations , 2013 .

[7]  Weizhu Bao,et al.  ON THE GROSS-PITAEVSKII EQUATION WITH STRONGLY ANISOTROPIC CONFINEMENT: FORMAL ASYMPTOTICS AND NUMERICAL EXPERIMENTS , 2005 .

[8]  Hanquan Wang,et al.  An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates , 2006, J. Comput. Phys..

[9]  Jacob Kogan,et al.  The nonlinear case , 1986 .

[10]  Weizhu Bao.  Ground States and Dynamics of Multicomponent Bose-Einstein Condensates , 2004, Multiscale Model. Simul..

[11]  Moshe Sheintuch,et al.  Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach. , 2009, Physical chemistry chemical physics : PCCP.

[12]  W. Auzinger,et al.  Local error structures and order conditions in terms of Lie elements for exponential splitting schemes , 2014 .

[13]  W. Bao,et al.  Mathematical Models and Numerical Methods for Bose-Einstein Condensation , 2012, 1212.5341.

[14]  Yanzhi Zhang,et al.  Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation , 2007 .

[15]  O. Koch,et al.  Embedded Split-Step Formulae for the Time Integration of Nonlinear Evolution Equations , 2010 .

[16]  Stéphane Descombes,et al.  Splitting methods with complex times for parabolic equations , 2009 .

[17]  Yihong Du,et al.  On a nonlocal reaction–diffusion–advection equation modelling phytoplankton dynamics , 2010 .

[18]  David A. Fournier,et al.  An advection-diffusion-reaction model for the estimation of fish movement parameters from tagging data, with application to skipjack tuna (Katsuwonus pelamis) , 1999 .

[19]  Arjen Doelman,et al.  Pattern formation in the one-dimensional Gray - Scott model , 1997 .

[20]  Mechthild Thalhammer,et al.  Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case , 2015, J. Comput. Appl. Math..

[21]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[22]  Gheorghe Moroşanu,et al.  The Linear Case , 2007 .

[23]  W. Bao,et al.  MATHEMATICAL THEORY AND NUMERICAL METHODS FOR , 2012 .

[24]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .