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.     

Energy-preserving methods for Poisson systems

We present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type that are able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see Brugnano et al. [10] and references therein). It is seen that the proposed approach allows us to obtain the methods recently derived in Cohen and Hairer (2011) [17], giving an alternative derivation of such methods and a new proof of their order. Sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given. A study of the implementation of the methods is provided. In particular, order and preservation properties when the involved integrals are approximated by means of a quadrature formula, are derived.     

On Multistep Methods for Differential Equations of Fractional Order

This paper concerns with numerical methods for the treatment of differential equations of fractional order. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which order conditions and stability properties are investigated. Dedicated to the memory of Professor Aldo Cossu     

Numerical simulation of fluid-structure interaction problems on hybrid meshes with algebraic multigrid methods

Fluid-structure interaction problems arise in many fields of application such as flows around elastic structures and blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincaré operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization, implicit first-order methods are used for both sub-problems. The discretized equations are solved by algebraic multigrid methods.     

An algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy

In this paper, we introduce an algorithm and a computer code for numerical differentiation of discrete functions. The algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points. The algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas, in particular when the derivatives are required with high approximation accuracy. Moreover, the given Matlab computer code can be implemented to solve boundary-value ordinary and partial differential equations with high numerical accuracy. The numerical technique is based on the undetermined coefficient method in conjunction with Taylor’s expansion. To avoid the difficulty of solving a system of linear equations, an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions. This is done by using an explicit closed formula for the Vandermonde matrix inverse. Moreover, the code is designed to give a unified approximation order throughout the given domain. A numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code. It is found that the method and the code work properly for any degree of derivative and any order of accuracy.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

Implementation of chebyshevian linear multistep formulas

A Chebyshevian linear multistep formula is a formula fitted to a Chebyshev set of basis functions. This paper presents a unified approach for the implementation of Chebyshevian backward differentiation and Adams formulas for solving ordinary differential equations. The approach is based on generalized scaled differences, derived from generalized divided differences, and it includes the generalized Newton interpolation formula as predictor for the Chebyshevian implicit backward differentiation formula and Chebyshevian Adams-Bashforth-Moulton formulas. The local truncation errors are estimated by means of the scaled differences providing information for the control of order and steplength.     

On the solvability of the systems of equations arising in implicit Runge-Kutta methods

In this paper we present a new condition under which the systems of equations arising in the application of an implicit Runge-Kutta method to a stiff initial value problem, has unique solutions. We show that our condition is weaker than related conditions presented previously. It is proved that the Lobatto IIIC methods fulfil the new condition.     

Equilibrium attractivity of Krylov-W-methods for nonlinear stiff ODEs

Since implicit integration schemes for differential equations which use Krylov methods for the approximate solution of linear systems depend nonlinearly on the actual solution a classical stability analysis is difficult to perform. A different, weaker property of autonomous dissipative systemsy′=f(y) is that the norm ‖f(y(t))‖ decreases for any solutiony(t). This property can also be analysed for W-methods using a Krylov-Arnoldi approximation. We discuss different additional assumptions onf and conditions on the Arnoldi process that imply this kind of attractivity to equilibrium points for the numerical solution. One assumption is general enough to cover quasilinear parabolic problems. This work was supported by Deutsche Forschungsgemeinschaft.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

Local error control inSDIRK-methods

This paper describes some problems that are encountered in the implementation of a class of Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. The contribution to the local error from the local truncation error and the residual error from the algebraic systems involved are analysed. A section describes a special interpolation formula. This is used as a prediction stage in the iterative solution of the algebraic equations. A strategy for computing a starting stepsize is presented. The techniques are applied to numerical examples.     

A variable step implicit block multistep method for solving first-order ODEs

A new four-point implicit block multistep method is developed for solving systems of first-order ordinary differential equations with variable step size. The method computes the numerical solution at four equally spaced points simultaneously. The stability of the proposed method is investigated. The Gauss-Seidel approach is used for the implementation of the proposed method in the PE(CE)^m mode. The method is presented in a simple form of Adams type and all coefficients are stored in the code in order to avoid the calculation of divided difference and integration coefficients. Numerical examples are given to illustrate the efficiency of the proposed method.     

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.     

Least squares completions for nonlinear differential algebraic equations

Summary A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays using nonlinear singular least squares methods. The theoretical foundations, generality, and limitations of this approach remain to be determined. This paper carefully examines several key aspects of this approach. The emphasis is on general results rather than specific results based on the structure of various applications.Research supported in part by the U.S. Army Research Office under DAALO3-89-D-0003 and the National Science Foundation under ECS-9012909 and DMS-9122745     

The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations

Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Received February 10, 1997 / Revised version received January 5, 1998     

Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.      

A Test Set for stiff Initial Value Problem Solvers in the open source software R: Package deTestSet

In this paper we present the R package deTestSet that includes challenging test problems written as ordinary differential equations (ODEs), differential algebraic equations (DAEs) of index up to 3 and implicit differential equations (IDEs). In addition it includes 6 new codes to solve initial value problems (IVPs). The R package is derived from the Test Set for Initial Value Problem Solvers available at http://www.dm.uniba.it/~testset which includes documentation of the test problems, experimental results from a number of proven solvers, and Fortran subroutines providing a common interface to the defining problem functions. Many of these facilities are now available in the R package deTestSet, which comprises an R interface to the test problems and to most of the Fortran solvers. The package deTestSet is free software which is distributed under the GNU General Public License, as part of the R open source software project.     

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.