An almost symmetric Strang splitting scheme for nonlinear evolution equations

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described the classic Strang splitting scheme, while still a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation

Lo - stable methods for parabolic partial differential equations

In recent years much attention has been devoted in the literature to the development, analysis and implementation of extrapolation methods for the numerical solution of partial differential equations with mixed initial and boundary values specified, see, for example, Lawson and Morris [5], Lawson and Swayne [6] and Gourlay and Morris [3]. The essential theme of these papers was to develop Lo-stable methods for the solution of parabolic partial differential equations in which splitting methods, such as the Crank-Nicolson method, are less than satisfactory when a time discretization is used with time steps which are too large relative to the spatial discretization. In the present paper a family of new Lo-stable methods based on Padé approximants to the exponential function is developed, and. higher accuracy is achieved. The methods are tested on heat equations in one and two space dimensions in which discontinuities exist between the initial and boundary conditions

Higher order parallel splitting methods for parabolic partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The thesis develops two families of numerical methods, based upon new rational approximations to the matrix exponential function, for solving second-order parabolic partial differential equations. These methods are L-stable, third- and fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by new difference approximations. Then parallel algorithms are developed and tested on one-, two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions with discontinuities between initial and boundary conditions. The schemes are seen to have high accuracy. A family of cubic polynomials, with a natural number dependent coefficients, is also introduced. Each member of this family has real zeros. Third- and fourth-order methods are also developed for one-dimensional heat equation subject to time-dependent boundary conditions, approximating the integral term in a new way, and tested on a variety of problems from the literature.Government of Pakistan (Central Overseas Training Scholarship

Numerical methods for ordinary differential equations with applications to partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The thesis develops a number of algorithms for the numerical solution of ordinary differential equations with applications to partial differential equations. A general introduction is given; the existence of a unique solution for first order initial value problems and well known methods for analysing stability are described. A family of one-step methods is developed for first order ordinary differential equations. The methods are extrapolated and analysed for use in PECE mode and their theoretical properties, computer implementation and numerical behaviour, are discussed. Lo-stable methods are developed for second order parabolic partial differential equations 1n one space dimension; second and third order accuracy is achieved by a splitting technique in two space dimensions. A number of two-time level difference schemes are developed for first order hyperbolic partial differential equations and the schemes are analysed for Ao-stability and Lo-stability. The schemes are seen to have the advantage that the oscillations which are present with Crank-Nicolson type schemes, do not arise. A family of two-step methods 1S developed for second order periodic initial value problems. The methods are analysed, their error constants and periodicity intervals are calculated. A family of numerical methods is developed for the solution of fourth order parabolic partial differential equations with constant coefficients and variable coefficients and their stability analyses are discussed. The algorithms developed are tested on a variety of problems from the literature.British Governmen

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

[EN] This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection-diffusion-reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers' and Maxwell-Stefan diffusion equations.This research was funded by German Academic Exchange Service grant number 91588469. We acknowledge support by the DFG Open Access Publication Funds of the Ruhr-Universität of Bochum, Germany and by Ministerio de Economía y Competitividad, Spain, under grant PGC2018-095896-B-C21-C22.Geiser, J.; Hueso, JL.; Martínez Molada, E. (2020). Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations. Mathematics. 8(3):1-22. https://doi.org/10.3390/math8030302S12283Auzinger, W., & Herfort, W. (2014). Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opuscula Mathematica, 34(2), 243. doi:10.7494/opmath.2014.34.2.243Auzinger, W., Koch, O., & Quell, M. (2016). Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions. Numerical Algorithms, 75(1), 261-283. doi:10.1007/s11075-016-0206-8Descombes, S., & Massot, M. (2004). Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction. Numerische Mathematik, 97(4), 667-698. doi:10.1007/s00211-003-0496-3Descombes, S., Dumont, T., Louvet, V., & Massot, M. (2007). On the local and global errors of splitting approximations of reaction–diffusion equations with high spatial gradients. International Journal of Computer Mathematics, 84(6), 749-765. doi:10.1080/00207160701458716McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053Trotter, H. F. (1959). On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4), 545-545. doi:10.1090/s0002-9939-1959-0108732-6Strang, G. (1968). On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517. doi:10.1137/0705041Jahnke, T., & Lubich, C. (2000). Bit Numerical Mathematics, 40(4), 735-744. doi:10.1023/a:1022396519656Nevanlinna, O. (1989). Remarks on Picard-Lindelöf iteration. BIT, 29(2), 328-346. doi:10.1007/bf01952687Farago, I., & Geiser, J. (2007). Iterative operator-splitting methods for linear problems. International Journal of Computational Science and Engineering, 3(4), 255. doi:10.1504/ijcse.2007.018264DESCOMBES, S., DUARTE, M., DUMONT, T., LOUVET, V., & MASSOT, M. (2011). ADAPTIVE TIME SPLITTING METHOD FOR MULTI-SCALE EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATIONS. Confluentes Mathematici, 03(03), 413-443. doi:10.1142/s1793744211000412Geiser, J. (2008). Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations. Journal of Computational and Applied Mathematics, 217(1), 227-242. doi:10.1016/j.cam.2007.06.028Dimov, I., Farago, I., Havasi, A., & Zlatev, Z. (2008). Different splitting techniques with application to air pollution models. International Journal of Environment and Pollution, 32(2), 174. doi:10.1504/ijep.2008.017102Karlsen, K. H., Lie, K.-A., Natvig, J. ., Nordhaug, H. ., & Dahle, H. . (2001). Operator Splitting Methods for Systems of Convection–Diffusion Equations: Nonlinear Error Mechanisms and Correction Strategies. Journal of Computational Physics, 173(2), 636-663. doi:10.1006/jcph.2001.6901Geiser, J. (2010). Iterative operator-splitting methods for nonlinear differential equations and applications. Numerical Methods for Partial Differential Equations, 27(5), 1026-1054. doi:10.1002/num.20568Geiser, J., & Wu, Y. H. (2015). Iterative solvers for the Maxwell–Stefan diffusion equations: Methods and applications in plasma and particle transport. Cogent Mathematics, 2(1), 1092913. doi:10.1080/23311835.2015.1092913Geiser, J., Hueso, J. L., & Martínez, E. (2017). New versions of iterative splitting methods for the momentum equation. Journal of Computational and Applied Mathematics, 309, 359-370. doi:10.1016/j.cam.2016.06.002Boudin, L., Grec, B., & Salvarani, F. (2012). A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete & Continuous Dynamical Systems - B, 17(5), 1427-1440. doi:10.3934/dcdsb.2012.17.1427Duncan, J. B., & Toor, H. L. (1962). An experimental study of three component gas diffusion. AIChE Journal, 8(1), 38-41. doi:10.1002/aic.69008011

New versions of iterative splitting methods for the momentum equation

[EN] In this paper we propose some modifications in the schemes for the iterative splitting techniques defined in Geiser (2009) for partial differential equations and introduce the parallel version of these modified algorithms. Theoretical results related to the order of the iterative splitting for these schemes are obtained. In the numerical experiments we compare the obtained results by applying iterative methods to approximate the solutions of the nonlinear systems obtained from the discretization of the splitting techniques to the mixed convection-diffusion Burgers' equation and a momentum equation that models a viscous flow. The differential equations in each splitting interval are solved by the back-Euler-Newton algorithm using sparse matrices. (C) 2016 Elsevier B.V. All rights reserved.This work has been supported by Ministerio de Economía y Competitividad de España MTM2014-52016-02-2-PGeiser, J.; Hueso Pagoaga, JL.; Martínez Molada, E. (2017). New versions of iterative splitting methods for the momentum equation. Journal of Computational and Applied Mathematics. 309:359-370. https://doi.org/10.1016/j.cam.2016.06.002S35937030