[...]... Approximation MEthods (FLAME) 2.4 Some Classic Schemes for Initial Value Problems For completeness, this section presents a brief overview of a few popular timestepping schemes for Ordinary Differential Equations (ODE) 2.4 Some Classic Schemes for Initial Value Problems 19 Fig 2.2 Numerical errors for different one-step schemes Time step ∆t = 0.05 λ = −10 Fig 2.3 Numerical solution for the forward Euler... 5 There is a notable exception in variational methods: rigorous pointwise error bounds can, for some classes of problems, be established using dual formulations (see p 153 for more information) However, this requires numerical solution of a separate auxiliary problem for Green’s function at each point where the error bound is sought 1.3 How To Hone the Computational Tools 7 etc.) the mesh size h or... the computational side of nanoscale models Computational analysis is a mature discipline combining science, engineering and elements of art It includes general and powerful techniques such as finite difference, finite element, spectral or pseudospectral, integral equation and other methods; it has been applied to every physical problem and device imaginable Are these existing methods good enough for nanoscale. .. calculus of Flexible Local Approximation MEthods (FLAME) is a promising alternative (Chapter 4) This list could easily be extended to include other examples, but the main point is clear: a vast assortment of computational methods, both traditional and new, are very helpful for the efficient simulation of nanoscale systems 6 1 Introduction 1.3 How To Hone the Computational Tools A computer makes as many... equally fascinating nanoscale applications in numerous other areas could be given Like it or not, we live in interesting times 1.2 Why Special Models for the Nanoscale? A good model can advance fashion by ten years Yves Saint Laurent First, a general observation A simulation model consists of a physical and mathematical formulation of the problem at hand and a computational method The formulation tells... and memory size For several classes of problems, there exist divide-and-conquer or hierarchical strategies with either optimal O(n) or slightly suboptimal O(n log n) complexity The most notable examples are Fast Fourier Transforms (FFT), Fast Multipole Methods, multigrid methods, and FFT-based Ewald summation Clearly, the numerical factors c1,2 also affect the performance of the method For real-life problems,... underlying numerical methods work 1.4 So What? Avoid clich´s like the plague! e William Safire’s Rules for Writers Multisyllabic clich´s are probably the worst type, but I feel compelled to use e one: nanoscale science and technology are interdisciplinary The book is intended to be a bridge between two broad fields: computational methods, both traditional and new, on the one hand, and several nanoscale or molecularscale... the inner workings of computational methods, will find this book helpful for crossing the bridge between the disciplines Likewise, experts in computational methods may be interested in browsing the application-related chapters At the same time, readers who wish to stay on their side of the “bridge” may also find some topics in the book to be of interest An example of such a topic for numerical analysts... background on FD methods, this chapter is intended to set the stage for the generalized FD analysis with “Flexible Local Approximation” described in Chapter 4 The scope of the present chapter is limited, and for a more comprehensive treatment and analysis of 12 2 Finite-Difference Schemes FD methods – in particular, elaborate time-stepping schemes for ordinary differential equations, schemes for gas and fluid... (2.17) For the purposes of this introduction, we shall call a difference scheme stable if, for a given initial condition, the numerical solution remains bounded for all time steps; otherwise the scheme is unstable.2 It is clear that in the second and third case above the numerical solution is qualitatively incorrect The forward Euler scheme is stable only for sufficiently small time steps – namely, for ∆t . Europe (Sankt Augustin, Germany) provided not only financial sup- port but also an excellent opportunity to work with Achim Basermann, an expert in high performance computing, on parallel implementation. Toronto), for his help, cooperation and continuing support over many years. • Fritz Keilmann (the Max-Planck-Institut f¨ur Biochemie in Martinsried, Germany), for providing an excellent opportunity for. David Smith (UCSD and Duke) for famil- iarizing me with plasmonic effects a decade ago. I appreciate the help, support and opportunities provided by the Interna- tional Compumag Society through a