# Professor Peter Hydon

Professor of Mathematics
Director of Division of Computing, Engineering, and Mathematical Sciences

Peter had a broad mathematical education: BSc in Mathematics at Bath, followed by a master’s degree (Part III of the Mathematics Tripos) and PhD at the University of Cambridge. After three years’ postdoctoral work at the University of Leeds he spent two years in a research post at the University of Northumbria before joining the University of Surrey’s Department of Mathematics in 1996 as a Lecturer. He was promoted to Reader in 2000 and Professor in 2006. He held a wide variety of roles and responsibilities, and was Head of Department for the two years before he joined Kent in 2015 as Head of SMSAS.After five years in that role, in January 2020, Peter stepped into the role of Division of Computing, Engineering, and Mathematical Sciences.Member of Senate and various University and Faculty committees, Peter comments "I also have many opportunities to represent SMSAS nationally and more widely".

## Research interests


Most of Peter's research aims to develop methods for solving, simplifying or approximating a given system of equations, by exploiting the system’s algebraic and geometric structures.Multisymplectic systems, with applications to geometric mechanicsGeometric integration (developing numerical methods that preserve structural features of the system being approximated)Automorphisms of Lie algebras He has written two graduate textbooks, both published by Cambridge University Press, which introduce newcomers to these research areas. His website has details of these and other publications. 

## Supervision

Gianluca Frasca Caccia - Finite Difference Conservation Laws

## Publications

### Article

• Frasca-Caccia, G. and Hydon, P. (2019). Locally conservative finite difference schemes for the Modified KdV equation. Journal of Computational Dynamics [Online] 6:307-323. Available at: http://dx.doi.org/10.3934/jcd.2019015.
Finite diffrence schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the Modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.
• Mansfield, E., Rojo-Echeburua, A., Hydon, P. and Peng, L. (2019). Moving Frames and Noether’s Finite Difference Conservation Laws I. Transactions of Mathematics and its Applications [Online] 3. Available at: https://doi.org/10.1093/imatrm/tnz004.
We consider the calculation of Euler--Lagrange systems of ordinary difference equations, including the difference Noether's Theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions.

For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler--Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether's conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler--Lagrange equations, and
we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant.

Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded \textit{a priori}. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts.
We show the calculations for a discretization of the Lagrangian for Euler's elastica, and compare our discrete solution to that of its smooth continuum limit.
• Frasca-Caccia, G. and Hydon, P. (2018). Simple bespoke preservation of two conservation laws. IMA Journal of Numerical Analysis [Online]. Available at: https://doi.org/10.1093/imanum/dry087.
Conservation laws are among the most fundamental geometric properties of a partial differential equation
(PDE), but few known finite difference methods preserve more than one conservation law. All conservation
laws belong to the kernel of the Euler operator, an observation that was first used recently to
construct approximations symbolically that preserve two conservation laws of a given PDE. However,
the complexity of the symbolic computations has limited the effectiveness of this approach. The current
paper introduces some key simplifications that make the symbolic-numeric approach feasible. To
illustrate the simplified approach, we derive bespoke finite difference schemes that preserve two discrete
conservation laws for the Korteweg-de Vries (KdV) equation and for a nonlinear heat equation. Numerical
tests show that these schemes are robust and highly accurate compared to others in the literature.
• Grant, T. and Hydon, P. (2013). Characteristics of Conservation Laws for Difference Equations. Foundations of Computational Mathematics [Online] 13:667-692. Available at: http://dx.doi.org/10.1007/s10208-013-9151-2.
Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation.
• Fisher, D., Gray, R. and Hydon, P. (2013). Automorphisms of real Lie algebras of dimension five or less. Journal of Physics A: Mathematical and Theoretical [Online] 46:1-18. Available at: http://iopscience.iop.org/article/10.1088/1751-8113/46/22/225204.
The Lie algebra version of the Krull-Schmidt Theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form.
• Hydon, P. and Mansfield, E. (2011). Extensions of Noether’s Second Theorem: from continuous to discrete systems. Proceedings of the Royal Society A- Mathematical Physical and Engineering Sciences [Online] 467:3206-3221. Available at: http://dx.doi.org/10.1098/rspa.2011.0158.
A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler–Lagrange equations of any variational problem whose symmetries depend on a set of free or partly constrained functions. Our approach extends further to deal with finite-difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations.
• Mansfield, E. and Hydon, P. (2001). On a variational complex for difference equations. Contemporary Mathematics 285:195-205.

### Book section

• Mansfield, E. and Hydon, P. (2001). Towards approximations which preserve integrals. In: Mourrain, B. ed. Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation. New York, USA: ACM, pp. 217-222. Available at: http://dx.doi.org/10.1145/384101.384131.
We investigate the algorithmic approximation of ordinary differential equations having a known conservation law, with finite difference schemes which inherit a discrete version of the conservation law. We use the method of moving frames on a multispace due to Olver. We assume that the system of ODEs to be studied has a variational principle and that the conservation law arises from a variational symmetry via Noether's theorem.

### Book

• Hydon, P. (2014). Difference Equations by Differential Equation Methods. [Online]. Cambridge University Press: Cambridge University Press. Available at: http://www.cambridge.org/gb/academic/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/difference-equations-differential-equation-methods?format=HB&isbn=9780521878524.
Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.
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