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Functional analysis --- General relativity (Physics) --- Generalized spaces. --- Mathematics. --- 519.63 --- Generalized spaces --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- Geometry of paths --- Minkowski space --- Spaces, Generalized --- Weyl space --- Calculus of tensors --- Geometry, Differential --- Geometry, Non-Euclidean --- Hyperspace --- Relativity (Physics) --- Mathematics --- General relativity (Physics) - Mathematics.

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Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

519.63 --- 519.6 --- 517.95 --- Numerical methods for solution of partial differential equations --- Computational mathematics. Numerical analysis. Computer programming --- Partial differential equations --- Differential equations, Hyperbolic. --- Differential equations, Partial. --- Spectral theory (Mathematics) --- Spectral theory (Mathematics). --- 517.95 Partial differential equations --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 519.63 Numerical methods for solution of partial differential equations --- Differential equations, Hyperbolic --- Differential equations, Partial --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Hyperbolic differential equations

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Geometric integrators are time-stepping 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 N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, 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 self-instruction or as a text for a graduate course on the subject.

Hamiltonian systems --- 519.62 --- 519.63 --- 681.3*I6 --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- 681.3*I6 Simulation and modeling (Computing methodologies)--See also {681.3*G3} --- Simulation and modeling (Computing methodologies)--See also {681.3*G3} --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Hamiltonian systems.

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Numerical solutions of differential equations --- Differential equations, Partial --- Numerical solutions --- 519.63 --- -681.3 *G18 --- Partial differential equations --- Numerical methods for solution of partial differential equations --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Numerical solutions. --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 519.63 Numerical methods for solution of partial differential equations --- 681.3 *G18 --- Numerical analysis --- Differential equations, Partial - Numerical solutions --- Analyse numerique --- Equations aux derivees partielles

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This book discusses a family of computational methods, known as discontinuous Galerkin methods, for solving partial differential equations. While these methods have been known since the early 1970s, they have experienced an almost explosive growth interest during the last ten to fifteen years, leading both to substantial theoretical developments and the application of these methods to a broad range of problems. These methods are different in nature from standard methods such as finite element or finite difference methods, often presenting a challenge in the transition from theoretical developments to actual implementations and applications. This book is aimed at graduate level classes in applied and computational mathematics. The combination of an in depth discussion of the fundamental properties of the discontinuous Galerkin computational methods with the availability of extensive software allows students to gain first hand experience from the beginning without eliminating theoretical insight. Jan S. Hesthaven is a professor of Applied Mathematics at Brown University. Tim Warburton is an assistant professor of Applied and Computational Mathematics at Rice University.

Numerical analysis --- Computer. Automation --- algoritmen --- Mathematical physics --- numerieke analyse --- differentiaalvergelijkingen --- wiskunde --- Partial differential equations --- fysica --- Differential equations, Partial --- Finite element method --- Galerkin methods --- 519.63 --- 681.3 *G18 --- Sinc-Galerkin methods --- Sinc methods --- FEA (Numerical analysis) --- FEM (Numerical analysis) --- Finite element analysis --- Isogeometric analysis --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis)