6 research outputs found
Explicit Runge-Kutta algorithm to solve non-local equations with memory effects: case of the Maxey-Riley-Gatignol equation
A standard approach to solve ordinary differential equations, when they
describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such
schemes, however, are not applicable to the large class of equations which do
not constitute dynamical systems. In several physical systems, we encounter
integro-differential equations with memory terms where the time derivative of a
state variable at a given time depends on all past states of the system.
Secondly, there are equations whose solutions do not have well-defined Taylor
series expansion. The Maxey-Riley-Gatignol equation, which describes the
dynamics of an inertial particle in nonuniform and unsteady flow, displays both
challenges. We use it as a test bed to address the questions we raise, but our
method may be applied to all equations of this class. We show that the
Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system
which is constructed by introducing a new dynamical co-evolving state variable
that encodes memory of past states. We develop a Runge-Kutta algorithm for the
resultant Markovian system. The form of the kernels involved in deriving the
Runge-Kutta scheme necessitates the use of an expansion in powers of .
Our approach naturally inherits the benefits of standard time-integrators,
namely a constant memory storage cost, a linear growth of operational effort
with simulation time, and the ability to restart a simulation with the final
state as the new initial condition.Comment: 26 pages, 5 figures, 1 table (v2) Typos correcte
Generalized exponential time differencing methods for fractional order problems
The main aim of this paper is to discuss the generalization of exponential integrators to differential equations of non-integer orders. Two methods of this kind are devised and the accuracy and stability are investigated. Some numerical experiments are presented to validate the theoretical findings. (C) 2011 Elsevier Ltd. All rights reserved
Exponential Integrator Methods for Nonlinear Fractional Reaction-diffusion Models
Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this work, we propose an exponential integrator method for nonlinear fractional reaction-diffusion equations. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order FETD schemes. This numerical scheme combined with fractional centered differencing is used for simulating many important nonlinear fractional models in applications. We demonstrate the superiority of our method over competing second order FETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional centered differencing and matrix transfer technique in discretization of Riesz fractional derivatives.
The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative. Furthermore, we present a preliminary result of the application of the M-L RDP approximation to develop a generalized exponetial integrator scheme for time-fractional nonlinear reaction-diffusion equation