in Chapter 4 for problems with Dirichlet, Neumann and Robin boundary conditions, persist for problems with absorbing boundary conditions. We therefore expect the error of the Strang splitting (5.9) to satisfy an estimate similar to Theorem 4.4.3. Hence, we conjecture that for f =f(x) independent of u, the Strang splitting (5.9) satisfies
ku(tn) unkL2(⌦) C⌧2 tn
, (5.17)
with C independent of ⌧, n and tn. Under the simplified assumption that f = f(x) has compact support in ⌦, it is shown in [60] that the midpoint rule (5.8), which is equivalent to the Strang splitting (5.9) for this specific case (see remark (5.1.1)), satisfies the estimate ku(tn) unkL2(⌦)C⌧2, (5.18) with C independent of ⌧, n and tn. The proof of (5.18) uses a standard Taylor series approach which is only possible because f = f(x) has compact support in ⌦. The proof cannot therefore be easily generalize to the more complex case where f = f(x) is not compactly supported in ⌦ or if f = f(x, u) is solution dependent (see Chapter 2 for a discussion on Taylor series in the parabolic PDE context). For a solution dependent source term f = f(x, u), the Strang splitting appears numerically, in Figure 5.2, second order convergent without order reduction at final timeT and we expect the estimate (5.17) to hold in this case. It is not straightforward to construct an analytic semigroup as in the preceding chapters. This is one of the major difficulties we encounter in our attempt to prove the conjecture (5.17). Following the methodology of Chapter 3 and Chapter 4, it is natural to define the operatorAas the restriction of the Laplace operator to some linear spaceD(A) which includes the boundary conditions. However, the absorbing boundary conditions (5.2) are time dependent, which suggests thatD(A) should be also time dependent. This makes it difficult to generalize the methodology explained in Chapter 2 for the heat equation with absorbing boundary conditions (5.2). We recall that the difficulties we encounter do not arise from the discretization of @t12 and would also occur if for instance a Pad´e approximation of @t12 is chosen.
5.4 Conclusion
In this thesis, we studied the Strang splitting in the context of semilinear parabolic equa-tions with various noncanonical boundary condiequa-tions. We observed that, in this setting, the standard Strang splitting su↵ers in general from a reduction of order. We presented first a modification of the Strang splitting and we proved that it allows us to recover the order two of accuracy. The advantage of this modification is that it allows us the modi-fied Strang splitting to be factorized, similarly to the classical Strang splitting, using the semigroup property of the exact flows. It reveals to be easy to implement and it requires no additional evaluation of the source term f or the di↵usion term. It seems furthermore that this modification remains useful for a sti↵ nonlinear source term.
In the second part of the thesis, we investigated a remarkable interaction between the Strang splittingand the Crank-Nicolson scheme. We proved that, if the di↵usion flow is
86 CHAPTER 5. CONCLUSION AND OUTLOOK approximated with the Crank-Nicolson scheme, then the resulting Strang splitting method is second order convergent outside a neighbourhood of timet= 0 and thus su↵ers from no order reduction. We proved this property in the specific case where the source f = f(x) does not depend on the solutionu. We performed numerical experiments which show that this property also holds whenf is solution dependent.
Finally, we presented some ongoing work on the Strang splitting method applied to the semilinear heat equation with absorbing boundary conditions. We explained how to im-plement discrete absorbing boundary conditions together with the Strang splitting method when the di↵usion is approximated with the Crank-Nicolson scheme. We proved the sta-bility of this Strang splitting method. We presented numerical experiments which suggest that, even for absorbing boundary conditions, the Strang splitting is second order conver-gent outside a neighbourhood of t = 0, similarly to the convergence analysis presented in Chapter 4.
We focused our research on the solution of semilinear parabolic equations. This allowed us to take benefit of the smoothing property of the di↵usion exact flow in our proofs thanks to the classical theory of analytic semigroups. We hope that the material we presented in this thesis can also be extended to more general PDE problems. For example, in [9], it is shown with numerical experiments that the modification techniques developed in [24, 25]
for Dirichlet, Neumann and Robin boundary conditions in the parabolic setting can be applied to avoid the order reduction of the Schr¨odinger equation with absorbing boundary conditions. Note, with this respect, that the modification presented in Chapter 3 was designed to be simple and hence easily adaptable to more complex problems. As for the results in Chapter 4, they show that working directly with numerical flows instead of exact flows in time may allow us to design more accurate splitting methods. We hope that these new tools and analysis will inspire new ideas and help develop the theory of splitting methods in more general contexts.
Bibliography
[1] A. Abdulle. Fourth order Chebyshev methods with recurrence relation. SIAM J. Sci.
Comput., 23(6):2041–2054, 2002.
[2] A. Abdulle and A. A. Medovikov. Second order Chebyshev methods based on orthog-onal polynomials. Numer. Math., 90(1):1–18, 2001.
[3] I. Alonso-Mallo, B. Cano, and N. Reguera. Avoiding order reduction when integrating reaction-di↵usion boundary value problems with exponential splitting methods. J.
Comput. Appl. Math., 357:228–250, 2019.
[4] I. Alonso-Mallo, B. n. Cano, and N. Reguera. Comparison of efficiency among di↵erent techniques to avoid order reduction with Strang splitting. Numer. Methods Partial Di↵erential Equations, 37(1):854–873, 2021.
[5] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Sch¨adle. A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schr¨odinger equations. Commun. Comput. Phys., 4(4):729–796, 2008.
[6] X. Antoine, C. Besse, and P. Klein. Absorbing boundary conditions for the one-dimensional Schr¨odinger equation with an exterior repulsive potential. J. Comput.
Phys., 228(2):312–335, 2009.
[7] X. Antoine, C. Besse, and P. Klein. Absorbing boundary conditions for the two-dimensional Schr¨odinger equation with an exterior potential. Part I: Construction and a priori estimates. Math. Models Methods Appl. Sci., 22(10):1250026, 38, 2012.
[8] X. Antoine, C. Besse, and P. Klein. Absorbing boundary conditions for the two-dimensional Schr¨odinger equation with an exterior potential. Part II: Discretization and numerical results. Numer. Math., 125(2):191–223, 2013.
[9] G. Bertoli. Splitting methods for the Schr¨odinger equation with absorbing boundary conditions. Master thesis, University of Geneva, 2017.
[10] G. Bertoli, C. Besse, and G. Vilmart. The strang splitting method for semilinear parabolic equation with absorbing boundary conditions. In preparation, 2021.
87
88 BIBLIOGRAPHY [11] G. Bertoli, C. Besse, and G. Vilmart. Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic pdes with Dirichlet and oblique bound-ary conditions. to appear in Math. Comp, 2021.
[12] G. Bertoli and G. Vilmart. Strang splitting method for semilinear parabolic prob-lems with inhomogeneous boundary conditions: a correction based on the flow of the nonlinearity. SIAM J. Sci. Comput., 42(3):A1913–A1934, 2020.
[13] S. Blanes and F. Casas. On the necessity of negative coefficients for operator splitting schemes of order higher than two. Appl. Numer. Math., 54(1):23–37, 2005.
[14] S. Blanes and F. Casas. A concise introduction to geometric numerical integration.
Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.
[15] S. Blanes, F. Casas, P. Chartier, and A. Murua. Optimized high-order splitting meth-ods for some classes of parabolic equations. Math. Comp., 82(283):1559–1576, 2013.
[16] C. J. Budd and M. D. Piggott. Geometric integration and its applications. In Hand-book of numerical analysis, Vol. XI, Handb. Numer. Anal., XI, pages 35–139. North-Holland, Amsterdam, 2003.
[17] J. C. Butcher. The numerical analysis of ordinary di↵erential equations. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1987. Runge-Kutta and general linear methods.
[18] J. C. Butcher. B-Series, volume 55 ofSpringer Series in Computational Mathematics.
Springer Nature, 2021. Algebraic Analysis of Numerical Methods.
[19] F. Castella, P. Chartier, S. Descombes, and G. Vilmart. Splitting methods with complex times for parabolic equations. BIT, 49(3):487–508, 2009.
[20] M. Crouzeix. Approximation of parabolic equations. Lecture notes available at http://perso.univ-rennes1.fr/michel.crouzeix/, 2005.
[21] S. Descombes. Convergence of a splitting method of high order for reaction-di↵usion systems. Math. Comp., 70(236):1481–1501, 2001.
[22] N. Dunford and J. T. Schwartz. Linear operators. Part I. Wiley Classics Library.
John Wiley & Sons, Inc., New York, 1988.
[23] L. Einkemmer, M. Moccaldi, and A. Ostermann. Efficient boundary corrected Strang splitting. Appl. Math. Comput., 332:76–89, 2018.
[24] L. Einkemmer and A. Ostermann. Overcoming order reduction in di↵usion-reaction splitting. Part 1: Dirichlet boundary conditions. SIAM J. Sci. Comput., 37(3):A1577–
A1592, 2015.
[25] L. Einkemmer and A. Ostermann. Overcoming order reduction in di↵usion-reaction splitting. Part 2: Oblique boundary conditions. SIAM J. Sci. Comput., 38(6):A3741–
A3757, 2016.
BIBLIOGRAPHY 89 [26] L. Einkemmer and A. Ostermann. A comparison of boundary correction methods for
Strang splitting. Discrete Contin. Dyn. Syst. Ser. B, 23(7):2641–2660, 2018.
[27] L. Einkemmer, A. Ostermann, and M. Residori. A pseudo-spectral splitting method for linear dispersive problems with transparent boundary conditions.J. Comput. Appl.
Math., 385:113240, 13, 2021.
[28] K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000.
[29] D. Fujiwara. Concrete characterization of the domains of fractional powers of some elliptic di↵erential operators of the second order. Proc. Japan Acad., 43:82–86, 1967.
[30] D. Goldman and T. J. Kaper. Nth-order operator splitting schemes and nonreversible systems. SIAM J. Numer. Anal., 33(1):349–367, 1996.
[31] P. Grisvard. Caract´erisation de quelques espaces d’interpolation.Arch. Rational Mech.
Anal., 25:40–63, 1967.
[32] E. Hairer. Polycopi´es du cours ”analyse num´erique”. Lecture notes available at http://www.unige.ch/ hairer/polycop.html, 2005.
[33] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010. Structure-preserving algorithms for ordinary di↵erential equations, Reprint of the second (2006) edition.
[34] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary di↵erential equations. I, volume 8 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 1993. Nonsti↵ problems.
[35] E. Hairer and G. Wanner. Solving ordinary di↵erential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2010. Sti↵
and di↵erential-algebraic problems.
[36] L. Halpern and J. Rauch. Absorbing boundary conditions for di↵usion equations.
Numer. Math., 71(2):185–224, 1995.
[37] A. Hansbo. Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space. Calcolo, 36(2):75–101, 1999.
[38] E. Hansen and A. Ostermann. High order splitting methods for analytic semigroups exist. BIT, 49(3):527–542, 2009.
[39] W. Hundsdorfer and J. Verwer. Numerical solution of time-dependent advection-di↵usion-reaction equations, volume 33 of Springer Series in Computational Math-ematics. Springer-Verlag, Berlin, 2003.
[40] W. Hundsdorfer and J. G. Verwer. A note on splitting errors for advection-reaction equations. Appl. Numer. Math., 18(1-3):191–199, 1995. Seventh Conference on the Numerical Treatment of Di↵erential Equations (Halle, 1994).
90 BIBLIOGRAPHY [41] T. Jahnke and C. Lubich. Error bounds for exponential operator splittings. BIT,
40(4):735–744, 2000.
[42] P. Klein. Construction et analyse de conditions aux limites artificielles pour des
´equations de Schr¨odinger avec potentiels et non lin´earit´es. PhD thesis, Universit´e Henri Poincar´e, Nancy I, 4 2010.
[43] B. Kov´acs, B. Li, and C. Lubich. A-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal., 54(6):3600–3624, 2016.
[44] S. Larsson, V. Thom´ee, and L. B. Wahlbin. Finite-element methods for a strongly damped wave equation. IMA J. Numer. Anal., 11(1):115–142, 1991.
[45] B. Leimkuhler and S. Reich. Simulating Hamiltonian dynamics, volume 14 of Cam-bridge Monographs on Applied and Computational Mathematics. CamCam-bridge Univer-sity Press, Cambridge, 2004.
[46] J. L¨ofstr¨om. Interpolation of boundary value problems of Neumann type on smooth domains. J. London Math. Soc. (2), 46(3):499–516, 1992.
[47] C. Lubich. Discretized fractional calculus. SIAM J. Math. Anal., 17(3):704–719, 1986.
[48] C. Lubich. Convolution quadrature and discretized operational calculus. I. Numer.
Math., 52(2):129–145, 1988.
[49] C. Lubich. Convolution quadrature and discretized operational calculus. II. Numer.
Math., 52(4):413–425, 1988.
[50] A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Modern Birkh¨auser Classics. Birkh¨auser/Springer Basel AG, Basel, 2013.
[51] R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier. B-series methods are exactly the affine equivariant methods. Numer. Math., 133(3):599–622, 2016.
[52] R. I. McLachlan and G. R. W. Quispel. Splitting methods. Acta Numer., 11:341–434, 2002.
[53] J. Niesen and W. M. Wright. Algorithm 919: a Krylov subspace algorithm for evaluat-ing the -functions appearevaluat-ing in exponential integrators.ACM Trans. Math. Software, 38(3):Art. 22, 19, 2012.
[54] A. Pazy. Semigroups of linear operators and applications to partial di↵erential equa-tions, volume 44 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1983.
[55] I. Podlubny. Fractional di↵erential equations, volume 198 of Mathematics in Science and Engineering. Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional di↵erential equations, to methods of their solution and some of their applications.
[56] J. Szeftel. Absorbing boundary conditions for reaction-di↵usion equations. IMA J.
Appl. Math., 68(2):167–184, 2003.
BIBLIOGRAPHY 91 [57] J. Szeftel. Calcul pseudodi↵´erentiel et paradi↵´erentiel pour l’´etude de conditions aux limites absorbantes et de propri´et´es qualitatives d’´equations aux d´eriv´ees partielles non lin´eaires. PhD thesis, Universit´e Paris 13, 4 2004.
[58] V. Thom´ee. Galerkin finite element methods for parabolic problems, volume 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edi-tion, 2006.
[59] A. S. Vasudeva Murthy and J. G. Verwer. Solving parabolic integro-di↵erential equa-tions by an explicit integration method. J. Comput. Appl. Math., 39(1):121–132, 1992.
[60] C. Zheng. Approximation, stability and fast evaluation of exact artificial boundary condition for the one-dimensional heat equation. J. Comput. Math., 25(6):730–745, 2007.
List of Figures
1.1 One dimensional problem with absorbing boundary conditions . . . 5 1.2 Second order convergence of the Strang splitting for absorbing boundary
con-ditions . . . 6 1.3 Reduction of order of the Strang splitting with exact flows. No order Reduction
when the Crank-Nicolson scheme is used to approximate the di↵usion . . . 7 1.4 The new modification is more stable for sti↵ reactions . . . 11 1.5 The Crank-Nicolson removes the order reduction but other Runge-Kutta
meth-ods do not. No Runge-Kutta method, not even Crank-Nicolson, allows to re-move the order reduction if the di↵usion and source term flows are inverted in the splitting. . . 12 1.6 The splitting with Crank-Nicolson is also of order two for solution dependent
source terms. . . 13 3.1 Comparison between the classical Strang splitting and the modified Strang
split-tings for a quadratic source term. . . 44 3.2 Comparison between the classical Strang splitting and the modified Strang
split-ting for an integral source term. . . 45 3.3 Comparison between the classical Strang splitting and the modified Strang
split-ting for a sti↵ reaction in two dimensions. . . 47 4.1 Comparison between the Strang splitting with exact flows and the splitting
with Crank-Nicolson applied to a simple one dimensional problem. Order of convergence for the infinity norm. . . 63 4.2 The di↵usion flow which appears in the Strang splitting with Crank-Nicolson is
solved with various Runge-Kutta methods. . . 63 4.3 Comparison between the Strang splitting with exact flows and the splitting with
Crank-Nicolson applied to a 2d problem with f =f(u). . . 64 4.4 The splitting with Crank-Nicolson applied to a stationary problem withf =f(x). 66 4.5 The splitting with Crank-Nicolson applied to a stationary problem withf =f(u). 66 4.6 The Strang splitting method in the case of the Schr¨odinger equation. . . 67 5.1 Convergence of the Strang splitting applied to a one dimensional problem with
various norms. . . 83 5.2 Strang splitting with absorbing boundary conditions and f =f(u) . . . 84
92