## Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent ProblemsThis book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples. Exercises and student projects are available on the book's webpage, along with Matlab mfiles for implementing methods. Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics. |

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absolute stability advection equation algorithm applied approach behavior boundary conditions boundary value problem bounded Chapter Chebyshev polynomial coefficient components compute consider constant convergence decay defined derivative diagonal diagonalizable difference equation differential equations diffusion discretization eigenvalues eigenvectors Euler’s method exact solution example explicit method finite difference methods Gaussian Gaussian elimination gives global error grid points heat equation hence higher order hyperbolic initial data initial guess interpolation interval iterative methods Jacobi kAnk Lax–Wendroff linear system Lipschitz Lipschitz continuous matrix exponential method is stable Newton’s method nonlinear nonzero norm Note numerical method obtain one-step method order method orthogonal Poisson problem requires Runge–Kutta method satisfied scalar second order accurate simply solve spectral stability region step stiff system of equations system of ODEs Taylor series theorem trapezoidal method tridiagonal true solution truncation error typically u.Nx UnC1 vector velocity wave numbers zero zero-stability