Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations

Summary.We prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions, analytic nonlinearity, and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize. Our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic. The method of proof is that of backward error analysis, whereby we construct a modified equation which is itself Lagrangian and translation invariant, and therefore also conserves momentum. This modified equation interpolates the semidiscrete system for all time, and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system. These properties directly imply approximate momentum conservation for the semidiscrete system. We also consider discretizations that are not variational as well as discretizations on non-uniform grids. Through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum.

[1]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[2]  S. Reich.  Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations , 2000 .

[3]  Robert D. Skeel,et al.  Does variable step size ruin a symplectic integrator , 1992 .

[4]  E. Hairer.  Backward analysis of numerical integrators and symplectic methods , 1994 .

[5]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[6]  Karsten Matthies.  Backward error analysis of a full discretization scheme for a class of semilinear parabolic partial differential equations , 2003 .

[7]  J. M. Sanz-Serna,et al.  Symplectic integrators for Hamiltonian problems: an overview , 1992, Acta Numerica.

[8]  Georges-Henri Cottet,et al.  A new approach for the analysis of Vortex Methods in two and three dimensions , 1988 .

[9]  Ernst Hairer,et al.  The life-span of backward error analysis for numerical integrators , 1997 .

[10]  S. Reich.  Backward Error Analysis for Numerical Integrators , 1999 .

[11]  J. Sanz-Serna,et al.  Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation , 1997 .

[12]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[13]  Andrew M. Stuart,et al.  Discrete Gevrey regularity attractors and uppers–semicontinuity for a finite difference approximation to the Ginzburg–Landau equation , 1995 .

[14]  Exponential averaging for Hamiltonian evolution equations , 2002 .

[15]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

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

[17]  Aldo H. Romero,et al.  Regular Article: Generation of Short and Long Range Temporal Correlated Noises , 1999 .

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

[19]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[20]  A. Hohmann,et al.  Numerical Analysis in Modern Scientific Computing: An Introduction , 2003 .

[21]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[22]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[23]  A. Neishtadt.  The separation of motions in systems with rapidly rotating phase , 1984 .

[24]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[25]  M. Reed.  Higher order estimates and smoothness of nonlinear wave equations , 1975 .

[26]  Klaus Kirchgässner,et al.  Travelling Waves in a Chain¶of Coupled Nonlinear Oscillators , 2000 .

[27]  G. Benettin,et al.  On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms , 1994 .

[28]  J. Marsden,et al.  Asynchronous Variational Integrators , 2003 .

[29]  M. Golubitsky,et al.  Singularities and Groups in Bifurcation Theory: Volume I , 1984 .

[30]  Edriss S. Titi,et al.  GEVREY REGULARITY FOR NONLINEAR ANALYTIC PARABOLIC EQUATIONS , 1998 .

[31]  S. Shkoller,et al.  THE VORTEX BLOB METHOD AS A SECOND-GRADE NON-NEWTONIAN FLUID , 1999, math/9910088.