A First Course in the Numerical Analysis of Differential Equations
Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.
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Geometric numerical integration
Nonlinear algebraic systems
Finite diﬀerence schemes
The ﬁnite element method
Classical iterative methods for sparse linear equations
Fast Poisson solvers
The diﬀusion equation
Appendix Bluﬀers guide to useful mathematics
Gaussian elimination for sparse linear equations
A-stability Acta Numerica advection equation algebraic systems algorithm analytic applied approximation boundary conditions Chapter choose coeﬃcients compatible ordering vector computational convergence corresponding deduce deﬁned deﬁnition denote derivatives diﬃcult diﬀusion Dirichlet boundary conditions displays eﬀective eigenvalues eigenvectors error Euclidean norm Euler Euler’s method exact solution example Exercise explicit ﬁgure ﬁnite diﬀerences ﬁnite element ﬁrst ﬁve-point formula function Gauss–Seidel Gaussian elimination graph grid points Hamiltonian hence implicit initial condition integration Jacobi Lemma linear algebraic linear space linear system LU factorization mathematical matrix multigrid multistep method nonlinear nonsingular numerical analysis numerical method numerical solution ODE system operator orthogonal PDEs Poisson equation polynomial positive definite preconditioner problem proof prove requires Runge–Kutta methods SD scheme Section solve speciﬁc spectral methods stability step stiﬀ suﬃciently Suppose symmetric symplectic technique Theorem theory trapezoidal rule tridiagonal y(tn yn+1 zero