The effective numerical integration of evolutionary problems arising from real-life applications requires the analysis of the characteristics of the phenomenon and of the corresponding mathematical model. The resulting numerical methods will therefore be able to reproduce the behavior of the analytical solution and to exploit the knowledge on the problem to reduce the computational effort. This approach has been developed for some classes of differential systems and for some classes of problems with memory modeled by integral or fractional equations. Problems like advection-diffusion or reaction-diffusion problems are usually solved by a semidiscretization along space, which gives raise to (large) systems of ordinary differential systems characterized by a stiff part and a non-stiff one. IMEX methods treat implicitly the stiff part and explicitly the non-stiff one, in order to have strong stability properties and to reduce the computational cost. We introduce a class of IMEX general linear methods which have no coupling order conditions, do not suffer of the order reduction phenomenon thanks to the high stage order, and have optimal stability properties. Periodic phenomena with memory, like the spread of seasonal diseases, are modeled by Volterra integral equations with periodic solution. Classical methods require a small stepsize to follow the oscillations.We apply the exponential fitting technique [8] to derive direct quadrature methods with parameters depending on an estimate of the frequency. The error is smaller than the error of classical methods, when periodic problems are treated; the numerical stability is not affected by the accuracy of the estimate of the frequency. Fractional models can represent memory effects of natural processes and also the anomalous kinetics of some processes in physics, chemistry, pharmacokinetis. Here we focus on the numerical solution of time-fractional reaction-diffusion systems, by a spectral technique along time and a finite difference scheme along space, which are specially designed to reproduce the behavior of the analytical solution and to simplify the overall computation. The results presented here have been obtained by various collaborations, with K. Burrage, R. D’Ambrosio, L.Gr. Ixaru, Z. Jackiewicz, B. Paternoster, A. Sandu, G. Santomauro, H. Zhang. References [1] Ascher, U.M., Ruuth, S.J., Spiteri, R.J. Implicit-explicit Runge-Kutta methods for timedependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997). [2] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Extrapolated implicit-explicit Runge-Kutta methods. Math. Model. Anal. 19, 18–43 (2014). [3] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Extrapolation-based implicit-explicit general linear methods. Numer. Algorithms 65, 377–399 (2014). [4] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Construction of highly stable implicit-explicit general linear methods, accepted for publication in Discrete Contin. Dyn. Systs. [5] A. Cardone, L. Gr. Ixaru, and B. Paternoster, Exponential fitting direct quadrature methods for Volterra integral equations, Numer. Algorithms 55, no. 4, 467-480 (2010). [6] A. Cardone, L.Gr. Ixaru, B. Paternoster, and G. Santomauro, Ef-gaussian direct quadrature methods for Volterra integral equations with periodic solution, Math. Comput. Simul., in press. [7] V. Gafiychuk, B. Datsko, and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math. 220(1-2), 215-225 (2008). [8] L.Gr. Ixaru, G. Vanden Berghe, (2004) Exponential Fitting. Kluwer Academic Publishers, Dordrecht. [9] L. Pareschi, G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25(1-2), 129–155 (2005).

Numerical schemes specially tuned for some evolutionary problems

CARDONE, Angelamaria
2015

Abstract

The effective numerical integration of evolutionary problems arising from real-life applications requires the analysis of the characteristics of the phenomenon and of the corresponding mathematical model. The resulting numerical methods will therefore be able to reproduce the behavior of the analytical solution and to exploit the knowledge on the problem to reduce the computational effort. This approach has been developed for some classes of differential systems and for some classes of problems with memory modeled by integral or fractional equations. Problems like advection-diffusion or reaction-diffusion problems are usually solved by a semidiscretization along space, which gives raise to (large) systems of ordinary differential systems characterized by a stiff part and a non-stiff one. IMEX methods treat implicitly the stiff part and explicitly the non-stiff one, in order to have strong stability properties and to reduce the computational cost. We introduce a class of IMEX general linear methods which have no coupling order conditions, do not suffer of the order reduction phenomenon thanks to the high stage order, and have optimal stability properties. Periodic phenomena with memory, like the spread of seasonal diseases, are modeled by Volterra integral equations with periodic solution. Classical methods require a small stepsize to follow the oscillations.We apply the exponential fitting technique [8] to derive direct quadrature methods with parameters depending on an estimate of the frequency. The error is smaller than the error of classical methods, when periodic problems are treated; the numerical stability is not affected by the accuracy of the estimate of the frequency. Fractional models can represent memory effects of natural processes and also the anomalous kinetics of some processes in physics, chemistry, pharmacokinetis. Here we focus on the numerical solution of time-fractional reaction-diffusion systems, by a spectral technique along time and a finite difference scheme along space, which are specially designed to reproduce the behavior of the analytical solution and to simplify the overall computation. The results presented here have been obtained by various collaborations, with K. Burrage, R. D’Ambrosio, L.Gr. Ixaru, Z. Jackiewicz, B. Paternoster, A. Sandu, G. Santomauro, H. Zhang. References [1] Ascher, U.M., Ruuth, S.J., Spiteri, R.J. Implicit-explicit Runge-Kutta methods for timedependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997). [2] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Extrapolated implicit-explicit Runge-Kutta methods. Math. Model. Anal. 19, 18–43 (2014). [3] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Extrapolation-based implicit-explicit general linear methods. Numer. Algorithms 65, 377–399 (2014). [4] Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H., Construction of highly stable implicit-explicit general linear methods, accepted for publication in Discrete Contin. Dyn. Systs. [5] A. Cardone, L. Gr. Ixaru, and B. Paternoster, Exponential fitting direct quadrature methods for Volterra integral equations, Numer. Algorithms 55, no. 4, 467-480 (2010). [6] A. Cardone, L.Gr. Ixaru, B. Paternoster, and G. Santomauro, Ef-gaussian direct quadrature methods for Volterra integral equations with periodic solution, Math. Comput. Simul., in press. [7] V. Gafiychuk, B. Datsko, and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math. 220(1-2), 215-225 (2008). [8] L.Gr. Ixaru, G. Vanden Berghe, (2004) Exponential Fitting. Kluwer Academic Publishers, Dordrecht. [9] L. Pareschi, G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25(1-2), 129–155 (2005).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4646144
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