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B-series, also known as Butcher series, are an algebraic tool for
analysing solutions to ordinary differential equations, including
approximate solutions. Through the formulation and manipulation of
these series, properties of numerical methods can be assessed.
Runge-Kutta methods, in particular, depend on B-series for a clean
and elegant approach to the derivation of high order and efficient
methods. However, the utility of B-series goes much further and
opens a path to the design and construction of highly accurate and
efficient multivalue methods. This book offers a self-contained
introduction to B-series by a pioneer of the subject. After a
preliminary chapter providing background on differential equations
and numerical methods, a broad exposition of graphs and trees is
presented. This is essential preparation for the third chapter, in
which the main ideas of B-series are introduced and developed. In
chapter four, algebraic aspects are further analysed in the context
of integration methods, a generalization of Runge-Kutta methods to
infinite index sets. Chapter five, on explicit and implicit
Runge-Kutta methods, contrasts the B-series and classical
approaches. Chapter six, on multivalue methods, gives a traditional
review of linear multistep methods and expands this to general
linear methods, for which the B-series approach is both natural and
essential. The final chapter introduces some aspects of geometric
integration, from a B-series point of view. Placing B-series at the
centre of its most important applications makes this book an
invaluable resource for scientists, engineers and mathematicians
who depend on computational modelling, not to mention computational
scientists who carry out research on numerical methods in
differential equations. In addition to exercises with solutions and
study notes, a number of open-ended projects are suggested. This
combination makes the book ideal as a textbook for specialised
courses on numerical methods for differential equations, as well as
suitable for self-study.
B-series, also known as Butcher series, are an algebraic tool for
analysing solutions to ordinary differential equations, including
approximate solutions. Through the formulation and manipulation of
these series, properties of numerical methods can be assessed.
Runge-Kutta methods, in particular, depend on B-series for a clean
and elegant approach to the derivation of high order and efficient
methods. However, the utility of B-series goes much further and
opens a path to the design and construction of highly accurate and
efficient multivalue methods. This book offers a self-contained
introduction to B-series by a pioneer of the subject. After a
preliminary chapter providing background on differential equations
and numerical methods, a broad exposition of graphs and trees is
presented. This is essential preparation for the third chapter, in
which the main ideas of B-series are introduced and developed. In
chapter four, algebraic aspects are further analysed in the context
of integration methods, a generalization of Runge-Kutta methods to
infinite index sets. Chapter five, on explicit and implicit
Runge-Kutta methods, contrasts the B-series and classical
approaches. Chapter six, on multivalue methods, gives a traditional
review of linear multistep methods and expands this to general
linear methods, for which the B-series approach is both natural and
essential. The final chapter introduces some aspects of geometric
integration, from a B-series point of view. Placing B-series at the
centre of its most important applications makes this book an
invaluable resource for scientists, engineers and mathematicians
who depend on computational modelling, not to mention computational
scientists who carry out research on numerical methods in
differential equations. In addition to exercises with solutions and
study notes, a number of open-ended projects are suggested. This
combination makes the book ideal as a textbook for specialised
courses on numerical methods for differential equations, as well as
suitable for self-study.
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