Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

by: David Cohen, Ernst Hairer, Christian Lubich

Numerische Mathematik, Vol. 110, No. 2. (2008), pp. 113-143.

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Abstract

Abstract For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.

@article{citeulike:3044980, abstract = {Abstract\ \ For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the St\"{o}rmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.}, author = {Cohen, David and Hairer, Ernst and Lubich, Christian }, citeulike-article-id = {3044980}, doi = {http://dx.doi.org/10.1007/s00211-008-0163-9}, journal = {Numerische Mathematik}, keywords = {energy-conservation, geometric-integration, modulated-fourier-expansion, numerics}, number = {2}, pages = {113--143}, posted-at = {2008-07-26 21:52:27}, priority = {2}, title = {Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations}, url = {http://dx.doi.org/10.1007/s00211-008-0163-9}, volume = {110}, year = {2008} }

RIS record

TY - JOUR ID - citeulike:3044980 L3 - citeulike-article-id:3044980 TI - Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations JF - Numerische Mathematik VL - 110 IS - 2 SP - 113 EP - 143 N2 - Abstract  For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time. KW - energy-conservation KW - geometric-integration KW - modulated-fourier-expansion KW - numerics AU - Cohen, David AU - Hairer, Ernst AU - Lubich, Christian PY - 2008//01/ UR - http://dx.doi.org/10.1007/s00211-008-0163-9 ER -

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