Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are di?cult to mimic using conventional time integrators that are used to solve systems of ordinary di?erential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

Methods for parallel integration of stiff systems of odes

This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagnonal sub-system. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations.The main emphasis is on the formula of order 1, the decoupled implicit Euler formula. It is proved that this formula even for a wide range of multirate formulations has an asymptotic global error expansion permitting extrapolation. Besides, sufficient conditions for absolute stability are presented.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

Given a one-step numerical scheme,on which ordinary differential equations is it exact?

A necessary condition for a (non-autonomous) ordinary differential equation to be exactly solved by a one-step, finite difference method is that the principal term of its local truncation error be null. A procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed. Examples of differential equations exactly solved by the explicit Euler, implicit Euler, trapezoidal rule, second-order Taylor, third-order Taylor, van Niekerk’s second-order rational, and van Niekerk’s third-order rational methods are presented.     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

Asymptotically correct error estimation for collocation methods applied to singular boundary value problems

We discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind. As an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts. This requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class.     

Periodic orbits of delay differential equations under discretization

This paper deals with the long-time behaviour of numerical solutions of delay differential equations that have asymptotically stable periodic orbits. It is shown that Runge-Kutta discretizations of such equations have attractive invariant curves which approximate the periodic orbit with the order of the method. The research by this author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.     

Blow up oscillating solutions to some nonlinear fourth order differential equations

We give strong theoretical and numerical evidence that solutions to some nonlinear fourth order ordinary differential equations blow up in finite time with infinitely many wild oscillations. We exhibit an explicit example where this phenomenon occurs. We discuss possible applications to biharmonic partial differential equations and to the suspension bridges model. In particular, we give a possible new explanation of the collapse of bridges.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.