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D.C., Dougalis, V.A., Mitsotakis, D.: On error estimates for Galerkin finite element methods for the Camassa–Holm equation (2018). arXiv:1805.10744\nArtebrant, R., Schroll, H.J.: Numerical simulation of Camassa–Holm peakons by adaptive upwinding. Appl. Numer. Math. 56, 695–711 (2006)\nBenjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. A 272, 47–78 (1972)\nBona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. A 351, 107–164 (1995)\nBona, J.L., Pritchard, W.G., Scott, L.R.: Numerical schemes for a model for nonlinear dispersive waves. J. Comput. Phys. 60, 167–186 (1985)\nBressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)\nCamassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)\nCamassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)\nChertock, A., Liu, J.-G., Pendleton, T.: Convergence of a particle method and global weak solutions of a family of evolutionary PDEs. SIAM J. Numer. Anal. 50, 1–21 (2012)\nChertock, A., Liu, J.-G., Pendleton, T.: Elastic collisions among peakon solutions for the Camassa–Holm equation. Appl. Numer. Math. 93, 30–46 (2015)\nCoclite, G.M., Karlsen, K.H., Risebro, N.H.: A convergent finite difference scheme for the Camassa–Holm equation with general \\(H^{1}\\) initial data. SIAM J. Numer. Anal. 46, 1554–1579 (2008)\nConstanin, A., Lenells, J.: On the inverse scattering approach to the Camassa–Holm equation. J. Nonlinear Math. Phys. 10, 252–255 (2003)\nConstantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ. Equ. 141, 218–235 (1997)\nConstantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)\nConstantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa 26, 303–328 (1998)\nConstantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)\nConstantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)\nConstantin, A., Molinet, L.: Orbital stability of solitary waves for a shallow water equation. Physica D 57, 75–89 (2001)\nConstantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)\nConstantin, A., Strauss, W.: Stability of the Camassa–Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002)\nEscher, J., Yin, Z.: Initial boundary value problems of the Camassa–Holm equation. Commun. Partial Differ. Equ. 33, 377–395 (2008)\nFokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145–150 (1995)\nFuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation. Physica D 95, 229–243 (1996)\nFuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund tranformations and hereditary symmetries. Physica D 4, 47–66 (1981)\nHolden, H., Raynaud, X.: A convergent numerical scheme for the Camassa–Holm equation based on multipeakons. Discrete Contin. Dyn. Syst. 14, 505–523 (2006)\nHolden, H., Raynaud, X.: Convergence of a finite difference scheme for the Camassa–Holm equation. SIAM J. Numer. Anal. 44, 1655–1680 (2006)\nJohnson, R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)\nJohnson, R.S.: On solutions of the Camassa–Holm equation. Proc. R. Soc. Lond. A 459, 1687–1708 (2003)\nKalisch, H., Lenells, J.: Numerical study of traveling-wave solutions for the Camassa–Holm equation. Chaos Solitons Fractals 25, 287–298 (2005)\nKalisch, H., Raynaud, X.: Convergence of a spectral projection of the Camassa–Holm equation. Numer. Methods Partial Differ. Equ. 22, 1197–1215 (2006)\nKwek, K.H., Gao, H., Zhang, W., Qu, C.: An initial boundary value problem of Camassa–Holm equation. J. Math. Phys. 41, 8279–8285 (2000)\nLenells, J.: Traveling wave solutions of the Camassa–Holm equation. J. Differ. Equ. 217, 393–430 (2005)\nLi, Y.A., Olver, P.J.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27–63 (2000)\nLiu, H., Xing, Y.: An invariant preserving discontinuous Galerkin method for the Camassa–Holm Equation. SIAM J. Sci. Comput. 38, A1919–A1934 (2016)\nMolinet, L.: On well-posedness results for Camassa–Holm equation on the line: a survey. J. Nonlinear Math. Phys. 11, 521–533 (2004)\nParker, A.: On the Camassa–Holm equation and a direct method of solution I. Bilinear form and solitary waves. Proc. R. Soc. Lond. A 460, 2929–2957 (2004)\nParker, A.: On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions. Proc. R. Soc. Lond. A 461, 3611–3632 (2005)\nParker, A.: On the Camassa–Holm equation and a direct method of solution. III. N-soliton solutions. Proc. R. Soc. Lond. A 461, 3893–3911 (2005)\nParker, A.: Wave dynamics for peaked solitons of the Camassa–Holm equation. Chaos Solitons Fractals 35, 220–237 (2008)\nThomée, V., Wendroff, B.: Convergence estimates for Galerkin methods for variable coefficient initial value problems. SIAM J. Numer. Anal. 11, 1059–1068 (1974)\nXu, Y., Shu, C.-W.: A local discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Numer. Anal. 46, 1998–2021 (2008)",{"EN":301},"We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order \n                  \n                    \n                  \n                  $$L^{2}$$\n                  \n                    \n                  \n                -error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.",{"EN":303},"Error estimates for Galerkin finite element methods for the Camassa–Holm equation",{"VOID":305},"10.1007\u002Fs00211-019-01045-7","PUBLICATION","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00211-019-01045-7",[309,338,357],{"id":310,"sortIndex":19,"researcher":18,"roles":311,"affiliations":313,"properties":335},"7c86d445-c62f-4289-83de-8c80e7d111fc",[312],"AUTHOR",[314,327],{"id":315,"sortIndex":316,"affiliation":317,"properties":326},"985fb1f6-a8b9-4d7e-92b4-4de8a0a4fffa",1,{"id":318,"createTime":319,"updateTime":320,"relativeEntities":321,"slug":322,"properties":323,"entityType":47,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"29ef7fa5-ca2b-4b8e-985a-b1c1cd67b71e","2024-04-06T18:41:29.225+00:00","2024-09-24T05:06:03.661+00:00",[],"Institute-of-Applied-and-Computational-Mathematics-FORTH-Heraklion-Greece",{"title":324},{"VI":325},"Institute of Applied and Computational Mathematics, FORTH, Heraklion, Greece",{},{"id":18,"sortIndex":19,"affiliation":328,"properties":18},{"id":329,"createTime":330,"updateTime":330,"relativeEntities":331,"slug":18,"properties":332,"entityType":47,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"f2b6a879-bf67-4c72-928a-0953f23435a6","2023-12-19T23:59:54.330+00:00",[],{"title":333},{"VI":334},"Mathematics Department, National and Kapodistrian University of Athens, Zographou, Greece",{"title":336},{"VI":337},"D. 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A.: Handbook of mathematical functions. New York: Dover Publications Inc. 1965\nBulirsch, R.: Bemerkungen zur Romberg-Integration. Numer. Math.6, 6–16 (1964)\nDavis, P. J., Rabinowitz, P.: Numerical integration. Blaisdell 1967\nFox, L.: Romberg integration for a class of singular integrands. Computer J.10, 87–93 (1967)\nGragg, W. B.: On extrapolation algorithms for ordinary initial value problems. J. SIAM (Num. Anal.)2, 384–403 (1965)\nHunter, D. B.: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals. Numer. Math.19, 419–424 (1972)\nKress, R.: Ein ableitungsfreies Restglied für die trigonometrische Interpolation periodischer analytischer Funktionen. Numer. Math.16, 389–396 (1971)\nLongman, I. M.: On the numerical evaluation of Cauchy principal values of integrals. MTAC12, 205–207 (1958)\nLuke, Y. L.: Simple formulas for the evaluation of some higher transcendental functions. J. Math. and Phys.34, 298–307 (1956)\nLyness, J. N., Ninham, B. W.: Numerical quadrature and asymptotic expansions. Math. Comp.21, 162–178 (1967)\nNavot, I.: An extension of the Euler-Maclaurin summation formula to functions with a branch singularity. J. Math. and Phys.40, 271–276 (1961)\nNavot, I.: A further extension of the Euler-Maclaurin summation formula. J. Math. and Phys.41, 155–163 (1962)\nNinham, B. W., Lyness, J. N.: Further asymptotic expansions for the error functional. Math. Comp.23, 71–83 (1969)\nPaget, D. F., Elliott, D.: An algorithm for the numerical evaluation of certain Cauchy principal value integrals. Numer. Math.19, 373–385 (1972)\nPiessens, R.: Numerical evaluation of Cauchy principal values of integrals. Nordisk Tidskr. Informationsbehandling (BIT)10, 476–480 (1970)\nRomberg, W.: Vereinfachte numerische Integration, Det. Kong. Norske Videnskabers Forhandlinger28, 30–36 (1955)\nRutishauser, H.: Ausdehnung des Rombergschen Prinzips. Numer. Math.5, 48–54 (1963)",{"EN":422},"The problem considered is that of evaluating numerically an integral of the form\n                  \n                    \n                  \n                 where the integrand has one or more simple poles in the interval (O,p). Modified forms of the trapezoidal and mid-ordinate rules, taking account of the singularities, are obtained; it is then shown that the resulting approximations can be extrapolated by Romberg's method. Further modifications to deal with the case when the integrand has an integrable branch singularity at one or both ends of the interval of integration are also briefly discussed.",{"EN":424},"The numerical evaluation of cauchy principal values of integrals by Romberg integration",{"VOID":426},"10.1007\u002FBF01436622","VERIFIED","Auto Verify","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF01436622",[431],{"id":432,"sortIndex":19,"researcher":18,"roles":433,"affiliations":434,"properties":443},"2714b618-72d0-458a-ae2d-ed90713d6b09",[312],[435],{"id":18,"sortIndex":19,"affiliation":436,"properties":18},{"id":437,"createTime":438,"updateTime":438,"relativeEntities":439,"slug":18,"properties":440,"entityType":47,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"a16afbe9-eb77-457f-9c73-1e8b348f2846","2023-12-13T19:01:35.436+00:00",[],{"title":441},{"VI":442},"School of Mathematics, University of Bradford, Yorks, England",{"title":444},{"VI":445},"D. 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However, our numerical experiment suggests\nthat they are divergent when \n                  \n$|c|$\n                 is large. In\norder to obtain convergent numerical solutions when\n                  \n$|c|\\geq 1$\n                \n,\nwe use\n                  \n$\\theta$\n                \n-methods to obtain approximants\nto some high order derivative\nof the exact solution, then we use the Taylor expansion with integral\nremainder to obtain approximants to the exact solution. Since\nthe equation under consideration has unbounded time lags, it\nis in general difficult to investigate numerically the long time\ndynamical behaviour of the exact solution due to limited computer\n(random access) memory. To avoid this problem we\ntransform the equation under consideration into a neutral\nequation with constant time lags. Using the\nlater equation as a test model, we prove that the linear\n                  \n$\\theta$\n                \n-method\nis \n                  \n$\\Lambda$\n                -stable, i.e., the numerical\nsolution tends to zero for\nany constant stepsize as long as\n                  \n${\\rm Re} a\u003C0$\n                \n and \n                  \n$|a|>|b|$\n                , if and only\nif \n                  \n$\\theta\\geq 1\u002F2$\n                , and that the\none-leg \n                  \n$\\theta$\n                -method  is \n                  \n$\\Lambda$\n                -stable if\n                  \n$\\theta=1$\n                \n. We also\nfind out that inappropriate stepsize causes spurious solution in the\nmarginal case where \n                  \n${\\rm Re} a\u003C0$\n                 and \n                  \n$|a|=|b|$\n                .\n ",{"EN":489},"Stability analysis of $\\theta$ -methods for neutral functional-differential equations",{"VOID":491},"10.1007\u002Fs002110050129","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs002110050129",[494],{"id":495,"sortIndex":19,"researcher":18,"roles":496,"affiliations":497,"properties":506},"e8fb1089-7cc7-4da6-9b37-04aa770c58eb",[312],[498],{"id":18,"sortIndex":19,"affiliation":499,"properties":18},{"id":500,"createTime":501,"updateTime":501,"relativeEntities":502,"slug":18,"properties":503,"entityType":47,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"3b1fdeca-068c-4709-a78d-b6d3440654cc","2024-01-12T23:59:14.035+00:00",[],{"title":504},{"VI":505},"DAMTP, University 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W.-J.: On the numerical approximation of phase portraits near stationary points. SIAM J. Numer. Anal. (to appear)\nBrezzi, F., Ushiki, S., Fujii, H.: Real and ghost bifurcation dynamics in difference schemes for ODEs, in Numerical Methods for Bifurcation Problems (T. Küpper, H.D. Mittelmann, H. Weber, eds.). pp. 79–104. Boston: Birkhäuser 1984\nBraun, M., Hershenov, J.: Periodic solutions of finite difference equations. Quart. Appl. Math.35, 139–147 (1977)\nDoan, H.T.: Invariant curves for numerical methods. Quart. Appl. Math.43, 385–393 (1985)\nEirola, T.: Two concepts for numerical periodic solutions of ODE's. Appl. Math. Comput. (to appear)\nEirola, T.: Invariant circles of one-step methods. BIT (Submitted)\nGrigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen 1. Stuttgart: Teubner 1972\nGuckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math. Sci.42. Berlin, Heidelberg, New York: Springer 1983\nHale, J.: Ordinary Differential Equations. New York: John Wiley 1969\nIooss, G.: Bifurcation of Maps and Applications. Math. Stud.36. Amsterdam: North Holland 1979\nIrwin, M.C.: Smooth Dynamical Systems. New York: Academic Press 1980\nKloeden, P.E., Lorenz, J.: Stable attracting sets in dynamical systems and their one step discretizations. SIAM J. Numer. Anal.23, 986–995 (1986)\nSell, G.R.: What is a dynamical system? In: Studies in Ordinary Differential Equations (J. Hale, ed.), pp. 32–51. Math. Assoc. Am. 1977\nStetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer 1973",{"EN":554},"We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. 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M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, 1970\nBezhaev, A.Y., Vasilenko, V.A.: Variational theory of splines. Kluwer Academic\u002FPlenum Publishers, New York, 2001\nBrownlee, R., Light, W.: Approximation orders for interpolation by surface splines to rough functions. IMA J. Numer. Anal. To appear\nDuchon, J.: Splines minimizing rotation-invariant seminorms in Sobolev spaces. Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571 W. Schempp, K. Zeller, (eds.), Springer-Verlag, Berlin, 1977, pp. 85–100\nDuchon, J.: Sur l’erreur d’interpolation des fonctions de plusieur variables par les Dm-splines. RAIRO Analyse Numerique 12, 325–334 (1978)\nvon Golitschek, M., Light, W.: Interpolation by polynomials and radial basis functions on spheres. Constr. Approx. 17, 1–18 (2001)\nJohnson, M.J.: Overcoming the boundary effects in surface spline interpolation. IMA J. Numer. Anal. 20, 405–422 (2000)\nLight, W., Wayne, H.: On power functions and error estimates for radial basis function interpolation. J. Approx. Th. 92, 245–266 (1998)\nLight, W., Wayne, H.: Spaces of distributions, interpolation by translates of a basis function and error estimates. Numer. Math. 81, 415–450 (1999)\nMadych, W., Nelson, S.: Multivariate interpolation and conditionally positive definite functions II. Math. Comp. 54, 211–230 (1990)\nMicchelli, C.A.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22 (1986)\nNarcowich, F.J., Ward, J.D.: Scattered-data interpolation on Error estimates for radial basis and band-limited functions. Manuscript\nWendland, H.: Sobolev-type error estimates for interpolation by radial basis functions. Surface Fitting and Multiresolution Methods, A. LeMéhauté, C. Rabut, L.L. Schumaker, (eds.), Vanderbilt University Press, Nashville, Tenessee USA, 1997, pp. 337–344\nWu, Z., Schaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13–27 (1993)\nYoon, J.: L p -error estimates for ‘shifted’ surface spline interpolation on Sobolev space. Math. Comp. 72, 1349–1367 (2003)",{"EN":621},"Radial basis function interpolation refers to a method of interpolation which writes the interpolant to some given data as a linear combination of the translates of a single function ϕ and a low degree polynomial. We develop an error analysis which works well when the Fourier transform of ϕ has a pole of order 2m at the origin and a zero at ∞ of order 2κ. In case 0≤m≤κ, we derive error estimates which fill in some gaps in the known theory; while in case m>κ we obtain previously unknown error estimates. 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