Confederated Modular Differential Equation APIs for Accelerated Algorithm Development and Benchmarking

Performant numerical solving of differential equations is required for large-scale scientific modeling. In this manuscript we focus on two questions: (1) how can researchers empirically verify theoretical advances and consistently compare methods in production software settings and (2) how can users (scientific domain experts) keep up with the state-of-the-art methods to select those which are most appropriate? Here we describe how the confederated modular API of DifferentialEquations.jl addresses these concerns. We detail the package-free API which allows numerical methods researchers to readily utilize and benchmark any compatible method directly in full-scale scientific applications. In addition, we describe how the complexity of the method choices is abstracted via a polyalgorithm. We show how scientific tooling built on top of DifferentialEquations.jl, such as packages for dynamical systems quantification and quantum optics simulation, both benefit from this structure and provide themselves as convenient benchmarking tools.Comment: 4 figures, 3 algorithm

Some aspects of algorithm performance and modeling in transient analysis of structures

The status of an effort to increase the efficiency of calculating transient temperature fields in complex aerospace vehicle structures is described. The advantages and disadvantages of explicit algorithms with variable time steps, known as the GEAR package, is described. Four test problems, used for evaluating and comparing various algorithms, were selected and finite-element models of the configurations are described. These problems include a space shuttle frame component, an insulated cylinder, a metallic panel for a thermal protection system, and a model of the wing of the space shuttle orbiter. Results generally indicate a preference for implicit over explicit algorithms for solution of transient structural heat transfer problems when the governing equations are stiff (typical of many practical problems such as insulated metal structures)

A composite integration scheme for the numerical solution of systems of ordinary differential equations

AbstractA generalization of a composite linear multistep method [2] is developed and applied to the approximate integration of systems of ordinary differential equations. The proposed scheme is second-order accurate and L-stable. An algorithm, based on the integration formula derived in this paper, is applied to approximate the solutions of a number of standard test problems. The numerical results indicate that the method is competitive with other fixed-order methods particularly in terms of computational overhead and could provide the basis for efficient temporal integration in the semidiscretization of time dependent partial differential equations

Numerical investigations on global error estimation for ordinary differential equations

AbstractFour techniques of global error estimation, which are Richardson extrapolation (RS), Zadunaisky's technique (ZD), Solving for the Correction (SC) and Integration of Principal Error Equation (IPEE) have been compared in different integration codes (DOPRI5, DVODE, DSTEP). Theoretical aspects concerning their implementations and their orders are first given. Second, a comparison of them based on a large number of tests is presented. In terms of cost and precision, SC is a method of choice for one-step methods. It is much more precise and less costly than RS, and leads to the same precision as ZD for half its cost. IPEE can provide the order of the error for a cheap cost in codes based on one-step methods. In multistep codes, only RS and IPEE have been implemented since they are the only ones whose theoretical justification has been extended to this case. There, RS still provides a more reliable estimation than IPEE. However, as these techniques are based on variations of the global error, irrespective of the numerical method used, they fail to provide any more usefull information once the numerical method has reached its limit of accuracy due to the finite arithmetic

Solitons from sine waves: Analytical and numerical methods for non-integrable solitary and cnoidal waves

The "FKDV" equation, ut+uux-uxxxxx=0, is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of "non-integrable" differential equations, that is to say, to equations which cannot be solved by the inverse scattering transform. The basic tools are (i) Pade approximants formed from power series in the amplitude; (ii) a Newton-Kantorovich/pseudospectral Fourier/continuation numerical method; (iii) singular perturbation theory for two interacting solitons of almost identical phase speed; (iv) bifurcation and branch-switching methods; (v) the imbricate-soliton series. A number of new results for the FKDV equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak ("cnoidal wave") and two peaks ("bicnoidal wave") per period, an analytical expression for the double-peaked soliton ("bion"), calculation of both the limit and bifurcation points for the bicnoidal wave, and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes. More important, all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform. Work now in progress will apply these methods to non-integrable equations in two space dimensions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26057/1/0000131.pd