The first part of this paper introduces sufficient conditions to determine conservation laws of d... more The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have conservation laws that approximate the continuous ones. In the second part of the paper, we propose a method that combines a finite difference method in space with a spectral integrator in time. The time integrator has already been applied in literature to solve time fractional equations with Caputo fractional derivative of order α ∈ (0, 1). It is here generalised to approximate Caputo and Riemann-Liouville fractional derivatives of arbitrary order. We apply the method to subdiffusion and superdiffusion equations with Riemann-Liouville fractional derivative and derive its conservation laws. Finally, we present a range of numerical experiments to show the convergence of the method and its conservation properties.
International Conference on Applied Mathematics, 2017
We describe the derivation of highly stable general linear methods for the numerical solution of ... more We describe the derivation of highly stable general linear methods for the numerical solution of initial value problems for systems of ordinary differential equations. In particular we describe the construction of explicit Nordsiek methods and implicit two step Runge Kutta methods with stability properties determined by quadratic stability functions. We aim for methods which have wide stability regions in the explicit case and which are Aand L-stable in the implicit one case. We moreover describe the construction of algebraically stable and G-stable two step Runge Kutta methods. Examples of methods are then provided.
Journal of Difference Equations and Applications, Sep 1, 2012
ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete ... more ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.
We present a collection of recent results on the numerical approximation of Volterra integral equ... more We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.
We describe the construction of explicit Nordsieck methods with s stages of order p = s − 1 and s... more We describe the construction of explicit Nordsieck methods with s stages of order p = s − 1 and stage order q = p with inherent quadratic stability and quadratic stability with large regions of absolute stability. Stability regions of these methods compare favorably with stability regions of corresponding general linear methods of the same order with inherent Runge-Kutta stability.
Innsbruck Contents Contents Plenary Talks The art of computing global manifolds 27 Hinke M Osinga... more Innsbruck Contents Contents Plenary Talks The art of computing global manifolds 27 Hinke M Osinga Computational methods for blood flow simulation and personalized medicine in cardiovascular disease 28 Alison Marsden Multiscale modelling of particles in membranes 29 Carsten Gräser Recent advance on numerical methods for oscillatory dispersive PDEs 30 Yongyong Cai Nonlinear Fourier integrators for dispersive equations 31 Katharina Schratz Numerics of charged particle dynamics in a magnetic field 32 Ernst Hairer Long time accuracy of some MCMC and Bayesian sampling schemes 33 Jonathan Mattingly Special Session 45 years of B-series 35 John Charles Butcher How numerical analysis emerged in Innsbruck 36 Gerhard Wanner MS 01: Part 1 Multiscale methods and analysis for oscillatory PDEs Uniformly accurate methods for highly-oscillatory problems 39
This presentation deals with the numerical solution of a reaction-diffusion problems, where the t... more This presentation deals with the numerical solution of a reaction-diffusion problems, where the time derivative is of fractional order. Since the fractional derivative of a function depends on its past history, these systems can successfully model evolutionary problems with memory, as for example electrochemical processes, porous or fractured media, viscoelastic materials, bioengineering applications. On the side of numerical simulation, the research mainly focused on suitable extensions of methods for PDE. This approach often produced low accuracy and/or high computational methods, due to the lack of smoothness of the analytical solution and to the longrange history dependence of the fractional derivative. Here we consider a finite difference scheme along space, to discretize the integer-order spatial derivatives, while we adopt a spectral collocation method through time. A suitable choice of the function basis produces an exponential convergence though time at a low computational cost, since the spectral method avoids the step-by-step method
The time fractional derivative of a function y(t) depends on the past history of the function y(t... more The time fractional derivative of a function y(t) depends on the past history of the function y(t), and so time fractional differential systems are naturally suitable to describe evolutionary processes with memory. Fractional models are increasingly used in many modelling situations including, for example, viscoelastic materials in mechanics, anomalous diffusion in transport dynamics of complex systems and some biological processes in rheology. Here we consider a time-fractional reaction diusion problem [2]. This is a non-local model and as the solution depends on all its past history, numerical step-by-step methods are computationally expensive. We propose a mixed method, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods and is exponentially convergent [3]. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given [1]. References [1] Burrage, K., Cardone, A., D'Ambrosio, R. and Paternoster, B. 2017 Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116 8294. [2] Gafiychuk, V., Datsko, B. and Meleshko, V. 2008 Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1-2) 215225. [3] Zayernouri, M. and Karniadakis, G. Em 2014 Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1) A40A62
The talk presents a model of β-cell transitions for diabetes. The normal β cells may die through ... more The talk presents a model of β-cell transitions for diabetes. The normal β cells may die through apoptosis (a naturally occurring programmed death) or necrosis (where the cells die in an uncontrolled way resulting in a build-up of dead tissue and cell debris) which can lead to premature death in patients. The model is based on a system of delay-differential equations and the aim of the project is to study the interactions and to understand how one may control cell death resulting from necrosis through the values of parameters of the system.
The aim of our research is the construction of efficient and accurate numerical methods for the s... more The aim of our research is the construction of efficient and accurate numerical methods for the solution of Volterra Integro-Differential Equations (VIDEs). In order to increase the order of convergence of classical one-step collocation methods, we propose multistep collocation methods, which have been successfully introduced for Volterra integral equations in [1; 2]. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. In this talk we describe the derivation of multistep collocation methods for VIDEs and the analysis of convergence and stability properties. We show some examples of methods which compare favorably with respect to existing one-step methods. This is a joint work with B. Paternoster and D. Conte from University of Salerno. REFERENCES [1] D. Conte, Z. Jackiewicz, B. Paternoster. Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput., 204 :839{853, 2008. [2] D. Conte, B. Paternoster. Multistep collocation methods for Volterra Integral Equations. Appl. Math. Comput., 59 :1721-1736, 2009
The talk is focused on the numerical solution of differential problems, whose qualitative behavio... more The talk is focused on the numerical solution of differential problems, whose qualitative behaviour is known a-priori. In particular, the investigation is oriented to problems with oscillatory solutions, with special reference to reaction-diffusion problems and stochastic differential equations. As it regards, reaction-diffusion systems, the investigation is devoted to nonlinear problems generating periodic wavefronts, with special interest to λ − ω problems. In this case, an adapted numerical approach results to be more efficient, accurate and stable in comparison to a general purpose numerical scheme ignoring the qualitative behaviour of the problem. An adapted method of lines will be presented, relying on trigonometrically fitted finite differences. The numerical scheme will also be analyzed in terms of accuracy and stability properties, also in comparison with its classical counterpart. Concerning stochastic differential equations, the presentation will mainly be devoted to the numerical treatment of damped stochastic oscillators. For such problems, mainly given by second order problems in time, the long terms dynamics is known to be distributed according to a normal distribution, for which is it possible to easily compute long term statistics. The talk analyzes the properties of stochastic linear multistep methods in retaining this invariance law for long time.
The first part of this paper introduces sufficient conditions to determine conservation laws of d... more The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have conservation laws that approximate the continuous ones. In the second part of the paper, we propose a method that combines a finite difference method in space with a spectral integrator in time. The time integrator has already been applied in literature to solve time fractional equations with Caputo fractional derivative of order α ∈ (0, 1). It is here generalised to approximate Caputo and Riemann-Liouville fractional derivatives of arbitrary order. We apply the method to subdiffusion and superdiffusion equations with Riemann-Liouville fractional derivative and derive its conservation laws. Finally, we present a range of numerical experiments to show the convergence of the method and its conservation properties.
International Conference on Applied Mathematics, 2017
We describe the derivation of highly stable general linear methods for the numerical solution of ... more We describe the derivation of highly stable general linear methods for the numerical solution of initial value problems for systems of ordinary differential equations. In particular we describe the construction of explicit Nordsiek methods and implicit two step Runge Kutta methods with stability properties determined by quadratic stability functions. We aim for methods which have wide stability regions in the explicit case and which are Aand L-stable in the implicit one case. We moreover describe the construction of algebraically stable and G-stable two step Runge Kutta methods. Examples of methods are then provided.
Journal of Difference Equations and Applications, Sep 1, 2012
ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete ... more ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.
We present a collection of recent results on the numerical approximation of Volterra integral equ... more We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.
We describe the construction of explicit Nordsieck methods with s stages of order p = s − 1 and s... more We describe the construction of explicit Nordsieck methods with s stages of order p = s − 1 and stage order q = p with inherent quadratic stability and quadratic stability with large regions of absolute stability. Stability regions of these methods compare favorably with stability regions of corresponding general linear methods of the same order with inherent Runge-Kutta stability.
Innsbruck Contents Contents Plenary Talks The art of computing global manifolds 27 Hinke M Osinga... more Innsbruck Contents Contents Plenary Talks The art of computing global manifolds 27 Hinke M Osinga Computational methods for blood flow simulation and personalized medicine in cardiovascular disease 28 Alison Marsden Multiscale modelling of particles in membranes 29 Carsten Gräser Recent advance on numerical methods for oscillatory dispersive PDEs 30 Yongyong Cai Nonlinear Fourier integrators for dispersive equations 31 Katharina Schratz Numerics of charged particle dynamics in a magnetic field 32 Ernst Hairer Long time accuracy of some MCMC and Bayesian sampling schemes 33 Jonathan Mattingly Special Session 45 years of B-series 35 John Charles Butcher How numerical analysis emerged in Innsbruck 36 Gerhard Wanner MS 01: Part 1 Multiscale methods and analysis for oscillatory PDEs Uniformly accurate methods for highly-oscillatory problems 39
This presentation deals with the numerical solution of a reaction-diffusion problems, where the t... more This presentation deals with the numerical solution of a reaction-diffusion problems, where the time derivative is of fractional order. Since the fractional derivative of a function depends on its past history, these systems can successfully model evolutionary problems with memory, as for example electrochemical processes, porous or fractured media, viscoelastic materials, bioengineering applications. On the side of numerical simulation, the research mainly focused on suitable extensions of methods for PDE. This approach often produced low accuracy and/or high computational methods, due to the lack of smoothness of the analytical solution and to the longrange history dependence of the fractional derivative. Here we consider a finite difference scheme along space, to discretize the integer-order spatial derivatives, while we adopt a spectral collocation method through time. A suitable choice of the function basis produces an exponential convergence though time at a low computational cost, since the spectral method avoids the step-by-step method
The time fractional derivative of a function y(t) depends on the past history of the function y(t... more The time fractional derivative of a function y(t) depends on the past history of the function y(t), and so time fractional differential systems are naturally suitable to describe evolutionary processes with memory. Fractional models are increasingly used in many modelling situations including, for example, viscoelastic materials in mechanics, anomalous diffusion in transport dynamics of complex systems and some biological processes in rheology. Here we consider a time-fractional reaction diusion problem [2]. This is a non-local model and as the solution depends on all its past history, numerical step-by-step methods are computationally expensive. We propose a mixed method, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods and is exponentially convergent [3]. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given [1]. References [1] Burrage, K., Cardone, A., D'Ambrosio, R. and Paternoster, B. 2017 Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116 8294. [2] Gafiychuk, V., Datsko, B. and Meleshko, V. 2008 Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1-2) 215225. [3] Zayernouri, M. and Karniadakis, G. Em 2014 Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1) A40A62
The talk presents a model of β-cell transitions for diabetes. The normal β cells may die through ... more The talk presents a model of β-cell transitions for diabetes. The normal β cells may die through apoptosis (a naturally occurring programmed death) or necrosis (where the cells die in an uncontrolled way resulting in a build-up of dead tissue and cell debris) which can lead to premature death in patients. The model is based on a system of delay-differential equations and the aim of the project is to study the interactions and to understand how one may control cell death resulting from necrosis through the values of parameters of the system.
The aim of our research is the construction of efficient and accurate numerical methods for the s... more The aim of our research is the construction of efficient and accurate numerical methods for the solution of Volterra Integro-Differential Equations (VIDEs). In order to increase the order of convergence of classical one-step collocation methods, we propose multistep collocation methods, which have been successfully introduced for Volterra integral equations in [1; 2]. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. In this talk we describe the derivation of multistep collocation methods for VIDEs and the analysis of convergence and stability properties. We show some examples of methods which compare favorably with respect to existing one-step methods. This is a joint work with B. Paternoster and D. Conte from University of Salerno. REFERENCES [1] D. Conte, Z. Jackiewicz, B. Paternoster. Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput., 204 :839{853, 2008. [2] D. Conte, B. Paternoster. Multistep collocation methods for Volterra Integral Equations. Appl. Math. Comput., 59 :1721-1736, 2009
The talk is focused on the numerical solution of differential problems, whose qualitative behavio... more The talk is focused on the numerical solution of differential problems, whose qualitative behaviour is known a-priori. In particular, the investigation is oriented to problems with oscillatory solutions, with special reference to reaction-diffusion problems and stochastic differential equations. As it regards, reaction-diffusion systems, the investigation is devoted to nonlinear problems generating periodic wavefronts, with special interest to λ − ω problems. In this case, an adapted numerical approach results to be more efficient, accurate and stable in comparison to a general purpose numerical scheme ignoring the qualitative behaviour of the problem. An adapted method of lines will be presented, relying on trigonometrically fitted finite differences. The numerical scheme will also be analyzed in terms of accuracy and stability properties, also in comparison with its classical counterpart. Concerning stochastic differential equations, the presentation will mainly be devoted to the numerical treatment of damped stochastic oscillators. For such problems, mainly given by second order problems in time, the long terms dynamics is known to be distributed according to a normal distribution, for which is it possible to easily compute long term statistics. The talk analyzes the properties of stochastic linear multistep methods in retaining this invariance law for long time.
Uploads
Papers by Angelamaria Cardone