Extrapolation-based implicit-explicit general linear methods

For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order. We look for methods with large absolute stability region, assuming that the implicit part of the method is A- or L-stable. We provide examples of IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings

Extrapolated Implicit–Explicit Runge–Kutta Methods

We investigate a new class of implicit–explicit singly diagonally implicit Runge–Kutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicit–explicit general linear methods

Optimization-based search for nordsieck methods of high order with quadratic stability polynomials

We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7

Extrapolated Implicit–Explicit Runge–Kutta Methods

We investigate a new class of implicit–explicit singly diagonally implicit Runge–Kutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicit–explicit general linear methods

A Finite Difference Spectral-Collocation Method for Fractional Reaction-Diffusion Systems

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

Numerical schemes specially tuned for some evolutionary problems

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)

Problem-based numerical methods for some local and non-local models

The numerical solution of real-life models cannot disregard the behavior of the analytical solution and/or the preservation of its special properties, such as for example the periodicity, the stiffness, the (lack of) smoothness in some intervals. In this talk it will be illustrated how this approach led to effective numerical methods, both for differential and integral models. Many practical problems in science and engineering are modeled by large systems of ordinary differential equations which arise from discretization in space of partial differential equations. For such systems there are often natural splittings of the right hand sides of the differential systems into two parts, one of which is non-stiff or mildly stiff, and suitable for explicit time integration, and the other part is stiff, and suitable for implicit time integration. Thus, here it is proposed an implicit-explicit (IMEX) scheme based on GLM methods, which has the advantage of preserving the order of the composing methods and to have enough free parameters to optimize the stability properties. On the side of non local models, it will be illustrated the numerical discretization of Volterra integral equations (VIEs) with periodic solution and of fractional differential equations (FDEs), both suitable to model problems with memory. VIEs with periodic solution can represent periodic phenomena with memory, like the spread of seasonal diseases. Classical methods are able to follow the oscillations of the solution at a high computational cost, while the exponentiallyfitted methods that we propose can considerably reduce this cost by exploiting the knowledge of an estimation of the frequency. FDEs can model the anomalous kinetics of some processes in physics, chemistry, pharmacokinetics. It will be illustrated a spectral collocation method, which takes into account the non-local nature of the equation, with a function basis suitably chosen to reproduce the behavior of the analytical solution. This presentation is based on the research work carried out with M. Bras (AGH Univ., Poland), K. Burrage (Oxford Univ.), D. Conte (Univ. of Salerno), R. D’Ambrosio (Univ. of L’Aquila), L.Gr. Ixaru (“Horia Hulubei” Nat.Inst. Physics & Nuclear Eng., Romania), Z. Jackiewicz (Arizona State Univ.), B. Paternoster (Univ. of Salerno), A. Sandu (Virginia Polytechnic Inst. & State Univ.), G.Santomauro (ENEA) and H. Zhang (Argonne Nat. Lab.)

Explicit Nordsieck methods with quadratic stability

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