The course focuses on the design of numerical methods for molecular dynamics simulation, in both its deterministic and stochastic forms. The primary source of material is my book with Charlie Matthews Molecular Dynamics, supplemented by the articles mentioned below. Copies of the book will be available onsite. The course is divided into four lectures. The lectures will be given using data projector. Slides will be posted here as soon as available. It is recommended to obtain the free software MD.M (works with MATLAB, Octave) and to perform some of the demo simulations on small systems, especially if this is your first exposure to molecular dynamics.
Lecture 1. Molecular dynamics models goals and purposes, numerical methods by splitting, analysis using BakerCampbellHausdorff expansion, and examples.
Materials for Lecture 1. Chapters 13 of Molecular Dynamics.
Slides for Lecture 1 [uploaded 3.4.2017].
Lecture 2. Ensembles for molecular simulation (microcanonical, canonical), stochastic differential equations (Brownian/Langevin dynamics), numerical methods for Langevin dynamics (especially splitting methods), error analysis, examples in biomolecular dynamics.
Materials for Lecture 2. Chapters 67 of Molecular Dynamics, ref. [1].
Slides for Lecture 2 [uploaded 3.4.2017].
Lecture 3. The timestep problem in MD, holonomic constraints, SHAKE and RATTLE discretization, stochastic schemes, geodesic integration, isokinetic MD, examples.
Materials for Lecture 3. Chapters 4, 6 of Molecular Dynamics, ref. [2,3].
Slides for Lecture 3 [uploaded 5.4.2017].
Lecture 4. Extended systems, thermostats, Nose dynamics, pairwise adaptive schemes and shear flows, an application in Bayesian inference.
Materials for Lecture 4. Chapter 8 of Molecular Dynamics, ref. [4,5].
Benedict Leimkuhler
Studies
PhD 1988, MS 1986, University of Illinois
BS 1983, Purdue University
Career History
2006   Chair of Applied Mathematics, University of Edinburgh 
2000  2006  Professor of Applied Mathematics, University of Leicester 
1996  1999  Associate Professor of Mathematics, University of Kansas 
1990  1996  Assistant Professor of Mathematics, University of Kansas 
1988  1990  Researcher, Helsinki University of Technology 
1986  Researcher, Lawrence Livermore National Laboratory 
1983  1988  Research Assistant, University of Illinois 
Research Interests
My research is in the broad area of the computational modelling of dynamical systems, such systems ranging in scale from computer simulation of the motion of atoms and molecules to the modelling of celestial mechanics. This involves the development of appropriate numerical methods to solve the system of equations driving the dynamics. It is desirable to develop approximation schemes that preserve important qualitative features, socalled geometric integrators, and this has been the focus of much of my work. Of particular importance in this regard are the symplectic integrators for Hamiltonian systems which preserve the symplectic structure of phase space and have superior stability properties, particularly for long time computations.
While many of my articles are related to Hamiltonian systems and the development of geometrypreserving integration methods, I have lately been more focussed on stochastic differential equations. Results in this direction have been obtained on both formulation and numerical solution of SDE models for thermodynamic modelling, including proving the ergodicity of degenerate diffusion techniques and studying the perturbation of dynamics by stochastic methods. Most recently, I have focused on the design of Langevin dynamics integration strategies, including the construction of superconvergent Langevin dynamics methods for invariant measures relevant for molecular dynamics.
From 2009 to 2014 I was heavily involved with the EPSRC (Science and Innovation) Centre for Numerical Algorithms and Intelligent Software (NAIS) which links Edinburgh, HeriotWatt and Strathclyde Universities. I was also part of the ExTASY Project (Extensible Toolkit for Advanced Sampling and analYsis) which is cofunded by the UK’s EPSRC and the US NSF to develop advanced methods for the study of biological molecular energy surfaces.
In recent years I have been working mor eand more in the areas of data science. I was involved with setting up the Alan Turing Institute and I am currently a Faculty Fellow of the ATI. This means that I spend part of my time at the new Alan Turing Institute headquarters in the British Library. The directions in data science that I am exploring include the use of stochastic differential equations to explore Bayesian parameterization of complicated distributions for data analysis. This work has potential high impact in neural network training and applications in "big data". Much of my work on molecular simulation can be applied in data science through this approach.
I currently hold a new EPSRC grant in DataDriven CoarseGraining using SpaceTime Diffusion Maps. This is an exciting project that bridges the data science and molecular modelling worlds. I also have an ERCfunded collaboration in Biological Modelling with V. Danos in Informatics.
Associations, Positions, Roles
2016  Research Director, School of Mathematics 
2016  CoDirector, Maxwell Institute for Mathematical Sciences 
2016  Faculty Fellow, Alan Turing Institute 
20152016  Science Committee, Alan Turing Institute 
201415  IMA Leslie Fox Prize Adjudication Committee 
2014  Fellowships subpanel, Royal Society of Edinburgh 
2014  Fellow of the Royal Society of Edinburgh (FRSE) 
2012  Fellow of the Institute of Mathematics and its Applications (FIMA) 
2012  2014 CoDirector, Maxwell Institute for Mathematical Sciences 
2012  Senior Research Fellow, Dutch Science Foundation 
2011  JT Oden Fellowship, University of Texas 
2009  Member, Steering Committee of the Centre for Numerical Algorithms (NAIS) 
2009  11 Director, NAIS 
2009  SIAM Dahlquist Prize Selection Committee 
200811  Deputy Director, Maxwell Institute for Mathematical Sciences 
20072015  Member, Board of the International Centre for Mathematical Sciences (ICMS) 
2007  Member, Programme Committee, ICMS 
20045  Leverhulme Trust Research Fellow 
20045  Visiting Researcher, Institute for Mathematics and Its Applications, Minneapolis 
1998  Member, Mathematical Sciences Research Insitute, Berkeley 
19967  Visiting Researcher, Cambridge University 
Editorial Boards
2014  Proceedings of the Royal Society A 
2013  European Journal of Applied Mathematics 
2012  Journal of Computational Dynamics (AIMS) 
2008  IMA Journal on Numerical Analysis 
200913  Nonlinearity 
200913  SIAM Journal on Numerical Analysis 
20017  SIAM Journal on Scientific Computing 
Grants and Projects
2016  EPSRC International Centre for Mathematical Sciences (coI) 
2016   EPSRC DataDriven CoarseGraining using SpaceTime Diffusion Maps (PI) 
2014  Adaptive Collective Variables: Automatic Identification and Application of Multiresolution Modelling (coI) 
2014  Adaptive QM/MM simulations (coI) 
2014  Highly efficient timedomain quantum chemistry algorithms (coI) 
2013  2016  NSFEPSRC SI2CHE:ExTASY Extensible Tools for Advanced Sampling and analYsis (Edinburgh PI) 
2013  2018  ERC Advanced Grant in RuledBased Modelling for Biology (CoI) 
2009  2014  EPSRC (S&I)/SFC Numerical Algorithms and Intelligent Software for the Evolving HPC Platform (coI) 
2008  2010  EPSRC Network (Bath, Bristol, Edinburgh, Warwick) “Mathematical Challenges of Molecular Dynamics” 
2004  2007  Australian RC Geometric Integration (coI) 
2004  US NIH Algorithms for Macromolecular Modelling (coI) 
2004  2005  EPSRC Algorithms for Macromolecular Modelling (PI) 
2004  2007  EPSRC Developing an Efficient Method for Locating Periodic Orbits (coI) 
2003  2005  SRIF Advanced Computing Facility (HPC) (coI) 
2002  2004  Australian RC Geometric Numerical Integration (coI) 
2001  2004  EPSRC A Mixed Atomistic and Continuum Model for Crossing Multiple Length and Time Scales (coI) 
2001  2004  EPSRC Geometric Integrators for Switched and Multiple Timescale Dynamics (PI) 
2000  2004  EU Research Training Network MASIE (Mechanics and Symmetry in Europe) 
1994  99  Multiple grants awarded whilst at the University of Kansas, mostly by US National Science Foundation 
Conferences and Workshops [examples]
Principal Organiser of the inaugural conference in 1994 of the series on Algorithms for Macromolecular Modelling (AM3) in Lawrence, Kansas, followed by service on the Organising Committee for subsequent meetings in the series (Berlin, 1997; New York City, 2000; Leicester 2004 as Principal Organiser; Austin, 2009)
Organising Committee, LMS Durham Symposium, 2000
CoOrganiser, Advanced Integration Methods for Molecular Dynamics, CECAM, Lyon, 2000
Scientific Committee, Prestissimo/DFG Conference on Molecular Simulation, Inst. Henri Poincaré, Paris, 2004
CoOrganiser, Workshop on Astrophysical Nbody Problems, Inst. for Pure and Applied Mathematics, UCLA, 2005
CoOrganiser, The Interplay between Mathematical Theory and Applications, Newton Institute, 2007
CoOrganiser, NSFNAIS Workshop on Intelligent Software, Edinburgh 2009
CoOrganiser, Capstone Conference, EPSRC Conference on Challenges in Scientific Computing, Warwick 2009
Principal Organiser, Multiscale Molecular Modelling, Edinburgh, 2010
Principal Organiser, Stateoftheart Algorithms for Molecular Dynamics, Edinburgh 2012
Coorganiser, Complex Molecular Systems, Lorentz Center, Leiden, 2012
Organiser, Multiscale Computational Methods in Materials Modelling, Edinburgh 2014
Coorganiser, three Alan Turing Institute ``scoping workshops'', 201516
Coorganiser, Stochastic numerical algorithms, multiscale modeling and highdimensional data analytics, ICERM/Brown University, 2016
Coorganiser, Trends and advances in Monte Carlo sampling algorithms, SAMSI/Duke University, 2017
Books
Molecular Dynamics (Springer, 2015) one of the first mathematical books on the subject.
Simulating Hamiltonian Dynamics (Cambridge University Press, 2005), coauthored with S. Reich (Potsdam) is an introduction to the subject of Geometric Integration for undergraduate and graduate students in mathematics and cognate disciplines.
Please see the separate "books" page for more details on these.
Research Articles
Most of my articles are uploaded to the arXiv. For complete, uptodate lists of publications, please refer to my papers page or see my page in Google Scholar.
Molecular Dynamics
B. Leimkuhler and C. Matthews Springer, Berlin, 2015. Link to Publisher's Website: SpringerVerlag 

This book describes the mathematical underpinnings of algorithms used for molecular dynamics simulation, including both deterministic and stochastic numerical methods. Molecular dynamics is one of the most versatile and powerful methods of modern computational science and engineering and is used widely in chemistry, physics, materials science and biology. Understanding the foundations of numerical methods means knowing how to select the best one for a given problem (from the wide range of techniques on offer) and how to create new, efficient methods to address particular challenges as they arise in complex applications. Aimed at a broad audience, this book presents the basic theory of Hamiltonian mechanics and stochastic differential equations, as well as topics including symplectic numerical methods, the handling of constraints and rigid bodies, the efficient treatment of Langevin dynamics, thermostats to control the molecular ensemble, multiple timestepping, and the dissipative particle dynamics method.  
Contents 1. Introduction; 2. Numerical integrators; 3. Analyzing geometric integrators; 4. The stability threshold; 5. Phase space distributions and microcanonical averages; 6. The canonical distribution and stochastic differential equations; 7. Numerical methods for stochastic differential equations; 8. Extended variable methods. 

Reviews

Simulating Hamiltonian Dynamics
B. Leimkuhler and S. Reich Cambridge University Press, 2005. Link to Publisher's Website: Cambridge University Press 

Geometric integrators are timestepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the Nbody problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include highorder and variable stepsize methods, schemes for treating problems involving multiple timescales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for selfinstruction or as a text for a graduate course on the subject. • Thorough treatment of a relatively new subject, covers theory, applications and also gives practical advice on implementing the techniques • Emphasis on 'efficient' numerical methods • Large number of examples and exercises 

Contents 1. Introduction; 2. Numerical methods; 3. Hamiltonian mechanics; 4. Geometric integrators; 5. The modified equations; 6. Higher order methods; 7. Contained mechanical systems; 8. Rigid Body dynamics; 9. Adaptive geometric integrators; 10. Highly oscillatory problems; 11. Molecular dynamics; 12. Hamiltonian PDEs. 

Reviews
