Geodesic
Resonance-based schemes for dispersive equations via decorated trees
Forum of Mathematics, Pi, Tome 10 (2022)

Voir la notice de l'article provenant de la source Cambridge University Press

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them. 
@article{10_1017_fmp_2021_13,
     author = {Yvain Bruned and Katharina Schratz},
     title = {Resonance-based schemes for dispersive equations via decorated trees},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {10},
     year = {2022},
     doi = {10.1017/fmp.2021.13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.13/}
}
TY  - JOUR
AU  - Yvain Bruned
AU  - Katharina Schratz
TI  - Resonance-based schemes for dispersive equations via decorated trees
JO  - Forum of Mathematics, Pi
PY  - 2022
VL  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.13/
DO  - 10.1017/fmp.2021.13
LA  - en
ID  - 10_1017_fmp_2021_13
ER  - 
%0 Journal Article
%A Yvain Bruned
%A Katharina Schratz
%T Resonance-based schemes for dispersive equations via decorated trees
%J Forum of Mathematics, Pi
%D 2022
%V 10
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.13/
%R 10.1017/fmp.2021.13
%G en
%F 10_1017_fmp_2021_13
Yvain Bruned; Katharina Schratz. Resonance-based schemes for dispersive equations via decorated trees. Forum of Mathematics, Pi, Tome 10 (2022). doi: 10.1017/fmp.2021.13

[1] Bao, W., Cai, Y. and Zhao, X., ‘A uniformly accurate multiscale time integrator pseudospectral method for the Klein–Gordon equation in the nonrelativistic limit regime’, SIAM J. Numer. Anal. 52(5) (2014), 2488–2511. doi:10.1137/130950665.Google Scholar | DOI

[2] Bao, W. and Dong, X., ‘Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime’, Numer. Math. 120 (2012), 189–229. doi:10.1007/s00211-011-0411-2.Google Scholar | DOI

[3] Bao, W. and Zhao, X., ‘A uniformly accurate multiscale time integrator spectral method for the Klein–Gordon–Zakharov system in the high-plasma-frequency limit regime’, J. Comput. Phys. 327 (2016), 270–293. doi:10.1016/j.jcp.2016.09.046.Google Scholar | DOI

[4] Baumstark, S., Faou, E. and Schratz, K., ‘Uniformly accurate oscillatory integrators for Klein–Gordon equations with asymptotic convergence to the classical NLS splitting’, Math. Comp. 87 (2018), 1227–1254. doi:10.1090/mcom/3263.Google Scholar | DOI

[5] Baumstark, S. and Schratz, K., ‘Uniformly accurate oscillatory integrators for the Klein–Gordon–Zakharov system from low- to high-plasma frequency regimes’, SIAM J. Numer. Anal. 57(1) (2019), 429–457. doi:10.1137/18M1177184.Google Scholar | DOI

[6] Berglund, N. and Bruned, Y., ‘BPHZ renormalisation and vanishing subcriticality limit of the fractional model’, (2019) Preprint .Google Scholar | arXiv

[7] Berland, H., Owren, B. and Skaflestad, B., ‘B-series and order conditions for exponential integrators’, SIAM J. Numer. Anal. 43(4) (2005), 1715–1727. doi:10.1137/040612683.Google Scholar | DOI

[8] Besse, C., Bidégaray, B. and Descombes, S., ‘Order estimates in time of splitting methods for the nonlinear Schrödinger equation’, SIAM J. Numer. Anal. 40(1) (2002), 26–40. doi:10.1137/S0036142900381497.Google Scholar | DOI

[9] Bourgain, J., ‘Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations’, Geom. Funct. Anal. 3(1) (1993), 209–262. doi:10.1007/BF01895688.Google Scholar | DOI

[10] Bruned, Y., ‘Recursive formulae in regularity structures’, Stoch. Partial Differ. Equ. Anal. Comput. 6(4) (2018), 525–564. doi:10.1007/s40072-018-0115-z.Google Scholar

[11] Bruned, Y., Chandra, A., Chevyrev, I. and Hairer, M., ‘Renormalising SPDEs in regularity structures’, J. Eur. Math. Soc. (JEMS) 23(3) (2021), 869–947. doi:10.4171/JEMS/1025.Google Scholar | DOI

[12] Bruned, Y., Gabriel, F., Hairer, M. and Zambotti, L., ‘Geometric stochastic heat equations’, J. Amer. Math. Soc. 35(1) (2022), 1–80. doi:10.1090/jams/977.Google Scholar

[13] Bruned, Y., Hairer, M. and Zambotti, L., ‘Algebraic renormalisation of regularity structures’, Invent. Math. 215(3) (2019), 1039–1156. doi:10.1007/s00222-018-0841-x.Google Scholar | DOI

[14] Bruned, Y., Hairer, M. and Zambotti, L., ‘Renormalisation of stochastic partial differential equations’, EMS Newsl. 115(3) (2020), 7–11. doi: 10.4171/NEWS/115/3.Google Scholar | DOI

[15] Burq, N., Gérard, P. and Tzvetkov, N., ‘Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds’, Amer. J. Math. 126(3) (2004), 569–605. doi:10.1353/ajm.2004.0016.Google Scholar | DOI

[16] Butcher, J. C., ‘An algebraic theory of integration methods’, Math. Comp. 26 (1972), 79–106. doi:10.2307/2004720.Google Scholar | DOI

[17] Butcher, J. C., Numerical Methods for Ordinary Differential Equations, 3rd ed. (Wiley, Hoboken, 2016).Google Scholar | DOI

[18] Butcher Trees, J. C., ‘B-series and exponential integrators’, IMAJNA 30(1) (2010), 131–140. doi:10.1093/imanum/drn086.Google Scholar

[19] Calaque, D., Ebrahimi-Fard, K. and Manchon, D., ‘Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series’, Adv. Appl. Math. 47(2) (2011), 282–308. doi:10.1016/j.aam.2009.08.003.Google Scholar | DOI

[20] Cano, B. and González-Pachón, A., ‘Exponential time integration of solitary waves of cubic Schrödinger equation’, Appl. Numer. Math. 91 (2015), 26–45. doi:10.1016/j.apnum.2015.01.001.Google Scholar | DOI

[21] Celledoni, E., Cohen, D. and Owren, B., ‘Symmetric exponential integrators with an application to the cubic Schrödinger equation’, Found. Comput. Math. 8 (2008), 303–317. doi:10.1007/s10208-007-9016-7.Google Scholar | DOI

[22] Chandra, A. and Hairer, M., ‘An analytic BPHZ theorem for regularity structures’, (2016) Preprint, .Google Scholar | arXiv

[23] Chandra, A., Hairer, M. and Shen, H., ‘The dynamical sine-Gordon model in the full subcritical regime’, (2018) Preprint, .Google Scholar | arXiv

[24] Chandra, A., Moinat, A. and Weber, H., ‘A priori bounds for the equation in the full sub-critical regime’, (2019) Preprint, .Google Scholar | arXiv

[25] Chartier, P., Crouseilles, N., Lemou, M. and Méhats, F., ‘Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödingier equations’, Numer. Math. 129 (2015), 211–250. doi:10.1007/s00211-014-0638-9.Google Scholar | DOI

[26] Chartier, P., Hairer, E. and Vilmart, G., ‘Algebraic structures of B-series’, Found. Comput. Math. 10(4) (2010), 407–427. doi:10.1007/s10208-010-9065-1.Google Scholar | DOI

[27] Christ, M., ‘Power series solution of a nonlinear Schrödinger equation’, in Mathematical Aspects of Nonlinear Dispersive Equations, Vol. 163 of Ann. of Math. Stud. (Princeton Univ. Press, Princeton, NJ, 2007), 131–155.Google Scholar

[28] Cohen, D. and Gauckler, L., ‘One-stage exponential integrators for nonlinear Schrödinger equations over long times’, BIT 52 (2012), 877–903. doi:10.1007/s10543-012-0385-1.Google Scholar | DOI

[29] Cohen, D., Hairer, E. and Lubich, C., ‘Modulated Fourier expansions of highly oscillatory differential equations’, Found. Comput. Math. 3 (2003), 327–345. doi:10.1007/s10208-002-0062-x.Google Scholar | DOI

[30] Coifman, R. and Meyer, Y., ‘On commutators of singular integrals and bilinear singular integrals’, Trans. Amer. Math. Soc. 212 (1975), 315–331. doi:10.1090/S0002-9947-1975-0380244-8.Google Scholar | DOI

[31] Connes, A. and Kreimer, D., ‘Hopf algebras, renormalization and noncommutative geometry’, Comm. Math. Phys. 199(1) (1998), 203–242. doi:10.1007/s002200050499.Google Scholar | DOI

[32] Connes, A. and Kreimer, D., ‘Renormalization in quantum field theory and the Riemann–Hilbert problem I: the Hopf algebra structure of graphs and the main theorem’, Commun. Math. Phys. 210 (2000), 249–73. doi:10.1007/s002200050779.Google Scholar | DOI

[33] Dujardin, G., ‘Exponential Runge–Kutta methods for the Schrödinger equation’, Appl. Numer. Math. 59(8) (2009), 1839–1857. doi:10.1016/j.apnum.2009.02.002.Google Scholar | DOI

[34] Ecalle, J., Les fonctions résurgentes. Tome I, II and III [Mathematical Publications of Orsay 81 and 85] (Université de Paris-Sud, Département de Mathématique, Orsay, 1981 and 1985).Google Scholar

[35] Ecalle, J. ‘Singularités non abordables par la géométrie’, [Singularities that are inaccessible by geometry] Ann. Inst. Fourier (Grenoble) 42 (1992)(1–2, 73–164.Google Scholar | DOI

[36] Engquist, B., Fokas, A., Hairer, E. and Iserles, A., Highly Oscillatory Problems (Cambridge University Press, Berlin, 2009).Google Scholar | DOI

[37] Faou, E., Geometric Numerical Integration and Schrödinger Equations (European Math. Soc., Zürich, 2012).Google Scholar | DOI

[38] Fauvet, F. and Menous, F., ‘Ecalle’s arborification coarborification transforms and Connes Kreimer Hopf algebra’, Ann. Sc. de l’Ecole Normale Sup. 50(1) (2017), 39–83. doi:10.24033/asens.2315.Google Scholar | DOI

[39] Fornberg, B., ‘Generation of finite difference formulas on arbitrarily spaced grids’, Math. Comp. 51 (1988), 699–706. doi:10.1090/S0025-5718-1988-0935077-0.Google Scholar | DOI

[40] Gauckler, L. and Lubich, C., ‘Nonlinear Schrödinger equations and their spectral semi-discretisations over long times’, Found. Comput. Math. 20 (2010), 141–169. doi:10.1007/s10208-010-9059-z.Google Scholar | DOI

[41] Gubinelli, M., ‘Controlling rough paths’, J. Funct. Anal. 216(1) (2004), 86–140. doi:10.1016/j.jfa.2004.01.002.Google Scholar | DOI

[42] Gubinelli, M., ‘Ramification of rough paths’, J. Differ. Equ. 248(4) (2010), 693 – 721. doi:10.1016/j.jde.2009.11.015.Google Scholar | DOI

[43] Gubinelli, M., ‘Rough solutions for the periodic Korteweg–de Vries equation’, Commun. Pure Appl. Anal. 11(4) (2012), 709–733. doi:10.3934/cpaa.2012.11.709.Google Scholar | DOI

[44] Guo, Z., Kwon, S. and Oh, T., ‘Poincaré–Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS’, Commun. Math. Phys. 322(1) (2013), 19–48. doi:10.1007/s00220-013-1755-5.Google Scholar | DOI

[45] Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Vol. 31 of Springer Series in Computational Mathematics (Springer, Berlin, 2006).Google Scholar

[46] Hairer, E., Nørsett, S. and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems (Springer, Berlin, 1987).Google Scholar | DOI

[47] Hairer, M., ‘A theory of regularity structures’, Invent. Math. 198(2) (2014), 269–504. doi:10.1007/s00222-014-0505-4.Google Scholar | DOI

[48] Hairer, M. and Schönbauer, P., ‘The support of singular stochastic PDEs’, (2019) Preprint, .Google Scholar | arXiv

[49] Hochbruck, M. and Ostermann, A., ‘Exponential integrators’, Acta Numer. 19 (2010), 209–286. doi:10.1017/S0962492910000048.Google Scholar | DOI

[50] Hofmanová, M. and Schratz, K., ‘An oscillatory integrator for the KdV equation’, Numer. Math. 136 (2017), 1117–1137. doi:10.1007/s00211-016-0859-1.Google Scholar | DOI

[51] Holden, H., Karlsen, K. H., Lie, K.-A. and Risebro, N. H., Splitting for Partial Differential Equations with Rough Solutions (European Math. Soc., Zürich, 2010).Google Scholar | DOI

[52] Holden, H., Karlsen, K. H. and Risebro, N. H., ‘Operator splitting methods for generalized Korteweg–de Vries equations’, J. Comput. Phys. 153(1) (1999), 203–222. doi:10.1006/jcph.1999.6273.Google Scholar | DOI

[53] Holden, H., Karlsen, K. H., Risebro, N. H. and Tao, T., ‘Operator splitting methods for the Korteweg–de Vries equation’, Math. Comp. 80 (2011), 821–846. doi:10.1090/S0025-5718-2010-02402-0.Google Scholar | DOI

[54] Holden, H., Lubich, C. and Risebro, N. H., ‘Operator splitting for partial differential equations with Burgers nonlinearity’, Math. Comp. 82 (2012), 173–185. doi:10.1090/S0025-5718-2012-02624-X.Google Scholar | DOI

[55] Ignat, L. and Zuazua, E., ‘Numerical dispersive schemes for the nonlinear Schrödinger equation’, SIAM J. Numer. Anal. 47(2) (2009), 1366–1390. doi:10.1137/070683787.Google Scholar | DOI

[56] Iserles, A., Quispel, G. R. W. and Tse, P. S. P., ‘B-series methods cannot be volume-preserving’, BIT 47 (2007), 351–378. doi:10.1007/s10543-006-0114-8.Google Scholar | DOI

[57] Jahnke, T. and Lubich, C., ‘Error bounds for exponential operator splittings’, BIT, 40 (2000), 735–744. doi:10.1023/A:1022396519656.Google Scholar | DOI

[58] Keel, M. and Tao, T., ‘Endpoint Strichartz estimates’, Amer. J. Math. 120(5) (1998), 955–980. doi:10.1353/ajm.1998.0039.Google Scholar | DOI

[59] Klein, C., ‘Fourth order time-stepping for low dispersion Korteweg–de Vries and nonlinear Schrödinger equation’, ETNA 29 (2008), 116–135. http://eudml.org/doc/117659.Google Scholar

[60] Knöller, M., Ostermann, A. and Schratz, K., ‘A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data’, SIAM J. Numer. Anal. 57(4) (2019), 1967–1986. doi:10.1137/18M1198375.Google Scholar | DOI

[61] Leimkuhler, B. and Reich, S., Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, Vol. 14 (Cambridge University Press, Cambridge, 2004).Google Scholar

[62] Lubich, C., ‘On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations’, Math. Comp. 77(4) (2008), 2141–2153. doi:10.1090/S0025-5718-08-02101-7.Google Scholar | DOI

[63] Lyons, T., ‘On the nonexistence of path integrals’, Proc. Roy. Soc. London Ser. A 432(1885) (1991), 281–290. doi:10.1098/rspa.1991.0017.Google Scholar

[64] Lyons, T. J., ‘Differential equations driven by rough signals’, Rev. Mat. Iberoamericana 14(2) (1998), 215–310. doi:10.4171/RMI/240.Google Scholar | DOI

[65] Maday, Y. and Quarteroni, A., ‘Error analysis for spectral approximation of the Korteweg–de Vries equation’, RAIRO - Modélisation mathématique et analyse numérique 22 (1988), 821–846. http://www.numdam.org/item/M2AN_1988__22_3_499_0/.Google Scholar

[66] Manchon, D., ‘Hopf algebras, from basics to applications to renormalization’, in Proceedings of the 5th Mathematical Meeting of Glanon: Algebra, Geometry and Applications to Physics (Glanon, Burgundy, France, 2001), 2–6.Google Scholar

[67] Mclachlan, R. I. and Quispel, G. R. W., ‘Splitting methods’, Acta Numer. 11 (2002), 341–434. doi:10.1017/S0962492902000053.Google Scholar | DOI

[68] Munthe-Kaas, H. and Føllesdal, K., Lie–Butcher Series, Geometry, Algebra and Computation. Springer Lectures in Mathematics and Statistics Cambridge, (2017).Google Scholar

[69] Murua, A. and Sanz-Serna, J. M., ‘Word series for dynamical systems and their numerical integrators’, Found. Comp. Math. 17 (2017), 675–712. doi:10.1007/s10208-015-9295-3.Google Scholar | DOI

[70] Ostermann, A., Rousset, F. and Schratz, K., ‘Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity’, Found. Comput. Math. 21 (2021), 725–765. doi:10.1007/s10208-020-09468-7.Google Scholar | DOI

[71] Ostermann, A., Rousset, F. and Schratz, K., ‘Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces’, to appear in J. Eur. Math. Soc. (JEMS), .Google Scholar | arXiv

[72] Ostermann, A. and Schratz, K., ‘Low regularity exponential-type integrators for semilinear Schrödinger equations’, Found. Comput. Math. 18 (2018), 731–755. doi:10.1007/s10208-017-9352-1.Google Scholar | DOI

[73] Ostermann, A. and Su, C., ‘Two exponential-type integrators for the”good”Boussinesq equation’, Numer. Math. 143 (2019), 683–712. doi:10.1007/s00211-019-01064-4.Google Scholar | DOI

[74] Rousset, F. and Schratz, K., ‘A general framework of low regularity integrators’, to appear in SIAM J. Numer. Anal. .Google Scholar | arXiv

[75] Sanz-Serna, J. M. and Calvo, M. P., Numerical Hamiltonian Problems (Chapman and Hall, London, 1994).Google Scholar | DOI

[76] Schratz, K., Wang, Y. and Zhao, X., ‘Low-regularity integrators for nonlinear Dirac equations’, Math. Comp. 90 (2021), 189–214. doi:10.1090/mcom/3557.Google Scholar | DOI

[77] Strichartz, R. S., ‘Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations’, Duke Math. J. 44(3) (1977), 705–714. doi:10.1215/S0012-7094-77-04430-1.Google Scholar | DOI

[78] Tao, T., Nonlinear Dispersive Equations. Local and Global Analysis (Amer. Math. Soc., Providence RI, 2006).Google Scholar

[79] Tappert, F., ‘Numerical solutions of the Korteweg–de Vries equation and its generalizations by the split-step Fourier method’, in (Newell, A. C., editor) Nonlinear Wave Motion (Amer. Math. Soc., 1974), 215–216.Google Scholar

[80] Thalhammer, M., ‘Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations’, SIAM J. Numer. Anal. 50(6) (2012), 3231–3258. doi:10.1137/120866373.Google Scholar | DOI

Cité par Sources :