An hp-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays

Abstract We present and analyze an h p -version of the discontinuous Galerkin (DG) time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bounds in the L 2 - and L ∞ -norm that are fully explicit with respect to the local time steps, the local approximation orders, and the local regularity of the exact solution. We further prove that the h p -DG scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start-up singularities. Moreover, we also propose an h p -DG scheme for the state dependent delay differential equations. We illustrate the theoretical results with a series of numerical experiments.

[1]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[2]  Yuan Zhang,et al.  Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations , 2009 .

[3]  Kassem Mustapha,et al.  An hp-Version Discontinuous Galerkin Method for Integro-Differential Equations of Parabolic Type , 2011, SIAM J. Numer. Anal..

[4]  Christopher T. H. Baker,et al.  Computing stability regions—Runge-Kutta methods for delay differential equations , 1994 .

[5]  A. Manitius,et al.  A fully-discrete spectral method for delay-differential equations , 1991 .

[6]  Marino Zennaro,et al.  Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method , 1985 .

[7]  Stefano Maset,et al.  Stability of Runge-Kutta methods for linear delay differential equations , 2000, Numerische Mathematik.

[8]  Thomas P. Wihler,et al.  An A Priori Error Analysis of the hp-Version of the Continuous Galerkin FEM for Nonlinear Initial Value Problems , 2005, J. Sci. Comput..

[9]  Ernst Hairer,et al.  Computing breaking points in implicit delay differential equations , 2008, Adv. Comput. Math..

[10]  Dominik Schötzau,et al.  hp-discontinuous Galerkin time stepping for parabolic problems , 2001 .

[11]  Ali Fuat Yeniçerioğlu.  The behavior of solutions of second order delay differential equations , 2007 .

[12]  Dirk Roose,et al.  Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations , 2000, SIAM J. Sci. Comput..

[13]  Lijun Yi,et al.  Legendre-Gauss Spectral Collocation Method for Second Order Nonlinear Delay Differential Equations , 2014 .

[14]  Dominik Schötzau,et al.  Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method , 2000, SIAM J. Numer. Anal..

[15]  Li,et al.  SUPERCONVERGENCE OF A DISCONTINUOUS GALERKIN METHOD FOR FIRST-ORDER LINEAR DELAY DIFFERENTIAL EQUATIONS , 2011 .

[16]  Lijun Yi,et al.  A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations , 2016 .

[17]  L. Berezansky,et al.  Some Oscillation Problems for a Second Order Linear Delay Differential Equation , 1998 .

[18]  Lijun Yi,et al.  An h-p Version of the Continuous Petrov-Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels , 2015, SIAM J. Numer. Anal..

[19]  D. Schötzau,et al.  Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations , 2014 .

[20]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[21]  Chengjun Guo,et al.  Existence of multiple periodic solutions for a class of second-order delay differential equations☆ , 2009 .

[22]  Alfredo Bellen,et al.  One-step collocation for delay differential equations , 1984 .

[23]  Dominik Schötzau,et al.  An hp a priori error analysis of¶the DG time-stepping method for initial value problems , 2000 .

[24]  Hehu Xie,et al.  Discontinuous Galerkin Methods for Delay Differential Equations of Pantograph Type , 2010, SIAM J. Numer. Anal..

[25]  Qiumei Huang,et al.  Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type , 2016 .

[26]  Chengjian Zhang,et al.  Superconvergence of a Dg Method for First-Order Linear Delay Differential Equations , 2011 .

[27]  Dominik Schötzau,et al.  hp-Discontinuous Galerkin Time-Stepping for Volterra Integrodifferential Equations , 2006, SIAM J. Numer. Anal..

[28]  H. Brunner,et al.  A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS * , 2009 .

[29]  Tingting Meng,et al.  An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations , 2017, Journal of Scientific Computing.

[30]  Hehu Xie,et al.  The hp Discontinuous Galerkin Method for Delay Differential Equations with Nonlinear Vanishing Delay , 2013, SIAM J. Sci. Comput..

[31]  Yunqing Huang,et al.  The Galerkin continuous finite element method for delay-differential equation with a variable term , 2007, Appl. Math. Comput..

[32]  Emiko Ishiwata,et al.  On the Attainable Order of Collocation Methods for Delay Differential Equations with Proportional Delay , 2000 .

[33]  Lijun Yi,et al.  An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems , 2015 .

[34]  Lina Wang,et al.  An h-p version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations , 2017, Advances in Computational Mathematics.

[35]  Zhongqing Wang,et al.  A LEGENDRE-GAUSS COLLOCATION METHOD FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS , 2010 .