# Article

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Keywords:
Rothe method; stability; comparison
Summary:
We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in $L^\infty$ and $L^1$ norms. Proofs of these results are based on comparison inequalities. Our theory is illustrated by numerical experiments. Our research is motivated by certain models of mathematical biology.
References:
[1] Abia, L. M., López-Marcos, J. C.: On the numerical integration of non-local terms for age-structured population models. Math. Biosci. 157 (1999), 147-167. DOI 10.1016/S0025-5564(98)10080-9 | MR 1686472
[2] Jr., J. Douglas, Milner, F. A.: Numerical methods for a model of population dynamics. Calcolo 24 (1987), 247-254. DOI 10.1007/BF02679109 | MR 1004520 | Zbl 0658.65145
[3] Feichtinger, G., Prskawetz, A., Veliov, V. M.: Age-structured optimal control in population economics. Theor. Popul. Biol. 65 (2004), 373-387. DOI 10.1016/j.tpb.2003.07.006 | Zbl 1110.92035
[4] Gurtin, M. E., MacCamy, R. C.: Non-linear age-dependend population dynamics. Arch. Ration. Mech. Anal. 54 (1974), 281-300. DOI 10.1007/BF00250793 | MR 0354068
[5] Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics, and Epidemics. CBMS-NSF Regional Conference Series in Applied Mathematics 20. SIAM Philadelphia (1975). MR 0526771
[6] Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Applied Mathematics Monographs 7. Giardini Editori e Stampatori Pisa (1994).
[7] Jossey, J., Hirani, A. N.: Equivalence theorems in numerical analysis: integration, differentiation and interpolation. http://arxiv.org/abs/0709.4046 (2007).
[8] Kačur, J.: Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities. Czech. Math. J. 34 (1984), 92-106. MR 0731982 | Zbl 0554.35086
[9] Kikuchi, N., Kačur, J.: Convergence of Rothe's method in Hölder spaces. Appl. Math., Praha 48 (2003), 353-365. DOI 10.1023/B:APOM.0000024481.01947.da | MR 2008889 | Zbl 1099.65079
[10] Kostova, T. V.: Numerical solutions of a hyperbolic differential-integral equation. Comput. Math. Appl. 15 (1988), 427-436. DOI 10.1016/0898-1221(88)90270-2 | MR 0953556 | Zbl 0651.65099
[11] Lax, P. D., Richtmyer, R. D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9 (1956), 267-293. DOI 10.1002/cpa.3160090206 | MR 0079204 | Zbl 0072.08903
[12] Manfredi, P., Williams, J. R.: Realistic population dynamics in epidemiological models: the impact of population decline on the dynamics of childhood infectious diseases. Measles in Italy as an example. Math. Biosci. 192 (2004), 153-175. DOI 10.1016/j.mbs.2004.11.006 | MR 2120645 | Zbl 1073.92046
[13] Milner, F. A.: A finite element method for a two-sex model of population dynamics. Numer. Methods Partial Differ. Equations 4 (1988), 329-345. DOI 10.1002/num.1690040406 | MR 1012488 | Zbl 0661.65149
[14] Murray, J. D.: Mathematical Biology. Vol. 1: An introduction. 3rd ed. Interdisciplinary Applied Mathematics 17. Springer New York (2002). MR 1908418
[15] Ostermann, A., Thalhammer, M.: Convergence of Runge-Kutta methods for nonlinear parabolic equations. Appl. Numer. Math. 42 (2002), 367-380. DOI 10.1016/S0168-9274(01)00161-1 | MR 1921348 | Zbl 1004.65093
[16] Rektorys, K.: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications (East European Series) vol. 4 Reidel, Dordrecht (1982). MR 0689712 | Zbl 0522.65059
[17] Sanz-Serna, J. M., Palencia, C.: A general equivalence theorem in the theory of discretization methods. Math. Comput. 45 (1985), 143-152. DOI 10.1090/S0025-5718-1985-0790648-7 | MR 0790648 | Zbl 0599.65034
[18] Slodička, M.: Semigroup formulation of Rothe's method: application to parabolic problems. Commentat. Math. Univ. Carol. 33 (1992), 245-260. MR 1189655 | Zbl 0756.65121
[19] Spigler, R., Vianello, M.: Convergence analysis of the semi-implicit Euler method for abstract evolution equations. Numer. Funct. Anal. Optimization 16 (1995), 785-803. DOI 10.1080/01630569508816645 | MR 1341112 | Zbl 0827.65061
[20] Tchuenche, J. M.: Convergence of an age-physiology dependent population model. Sci., Ser. A, Math. Sci. (N.S.) 15 (2007), 23-30. MR 2367910 | Zbl 1138.92366
[21] Walter, W.: Ordinary Differential Equations Transl. from the German by Russell Thompson. Graduate Texts in Mathematics. Readings in Mathematics 182. Springer New York (1998). MR 1629775
[22] al., J. A. J. Metz et: The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics 68. Springer Berlin (1986). MR 0860959

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