Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...

Background

The theory of integrable systems has advanced with the connection between

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

. For example, the discovery of solitons came from the numerical experiments to the KdV equation by

Norman Zabusky Norman J. Zabusky was an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work ...

and

Martin David Kruskal Martin David Kruskal (; September 28, 1925 – December 26, 2006) was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and ...

. Today, various relations between numerical analysis and integrable systems have been found (

Toda lattice The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and ...

and

numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics ...

, discrete soliton equations and

series acceleration In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve ...

), and studies to apply integrable systems to numerical computation are rapidly advancing.

Integrable difference schemes

Generally, it is hard to accurately compute the solutions of nonlinear differential equations due to its non-linearity. In order to overcome this difficulty, R. Hirota has made discrete versions of integrable systems with the viewpoint of "Preserve mathematical structures of integrable systems in the discrete versions". At the same time, Mark J. Ablowitz and others have not only made discrete soliton equations with discrete

Lax pair In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss s ...

but also compared numerical results between integrable difference schemes and ordinary methods. As a result of their experiments, they have found that the accuracy can be improved with integrable difference schemes at some cases.

Background

Integrable difference schemes

References

See also