Lectures for VII Mexican School on Gravitation and Mathematical Physics Relativistic Astrophysics and Numerical Relativity Numerical Analysis for Numerical Relativists
نویسنده
چکیده
1 Basic Finite Difference Techniques for Time Dependent PDEs 2 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Types of IVPs (by example) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Wave and “Wave-Like” (“Hyperbolic”): The 1-d Wave Equation . . . . . . . . . . . . 3 1.2.2 Diffusion (“Parabolic”): The 1-d Diffusion Equation . . . . . . . . . . . . . . . . . . . 3 1.2.3 Schrödinger: The 1-d Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Some Basic Concepts, Definitions and Techniques . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Truncation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.5 Order of an FDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.6 Solution Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.7 Relation Between Truncation Error and Solution Error . . . . . . . . . . . . . . . . . 5 1.3.8 Deriving Finite Difference Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Sample Discretizations / FDAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.1 1-d Wave equation with fixed (Dirichlet) boundary conditions . . . . . . . . . . . . . . 6 1.4.2 1-d Diffusion equation with Dirichlet boundary conditions . . . . . . . . . . . . . . . . 8 1.5 The 1-D Wave Equation in More Detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.1 Heuristic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6.2 Von-Neumann (Fourier) Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Dispersion and Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 The Leap-Frog Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.9 Error Analysis and Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.9.1 Sample Analysis: The Advection Equation . . . . . . . . . . . . . . . . . . . . . . . . 18 1.10 Dispersion and Dissipation in FDAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
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تاریخ انتشار 2006