Open AccessBook
Difference Equations by Differential Equation Methods
Peter E. Hydon
- 29 Sep 2014
93
TL;DR: In this paper, the authors give an introduction to elementary solution methods for differential equations and give readers a clear explanation of exact techniques for ordinary and partial difference equations, which is suitable for anyone who is familiar with standard differential equation methods.
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Abstract: 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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Citations
Structure-Preserving Geometric Particle-in-Cell Methods for Vlasov-Maxwell Systems
TL;DR: In this article, a new generation of structure-preserving geometric particle-in-cell (PIC) algorithms for Vlasov-Maxwell systems is presented, which utilizes modern mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure.
59
Structure-preserving geometric particle-in- cell methods for Vlasov-Maxwell systems
TL;DR: A structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discretecharge conservation law are presented.
47
The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations
TL;DR: The multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations was proposed in this article, which is based on discretizing conservation law multipliers and their associated density and flux functions.
31
Symmetry-Preserving Numerical Schemes
TL;DR: In this article, two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations are reviewed. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames.
26
Discrete shallow water equations preserving symmetries and conservation laws
TL;DR: In this article, an invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed and the developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.
21
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Peter J. Olver
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Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
Ernst Hairer,Christian Lubich,Gerhard Wanner +2 more
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TL;DR: In this article, the authors present a model for symmetric integration of non-Canonical Hamiltonian systems and a model of symmetric Hamiltonian integration with symmetric integrators.
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Group analysis of differential equations
Willard Miller,L. V. Ovsiannikov +1 more
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3.1K
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An introduction to difference equations
Saber Elaydi
- 01 Dec 1995
TL;DR: In this article, the authors combine both analytic and geometric (topological) approaches to studying difference equations and integrate both classical and modern treatments of the subject, offering material stability, z-transform, discrete control theory and symptotic theory.
2.3K