Mathematical Physics - Volume II - Numerical Methods

Contents

Numerical method - FDM-FEM

I

9

Chapter 1 Finite difference method and Finite element method 11 1.1 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.1.1 Finite difference method for parabolic partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.1.2 Consistency and convergence . . . . . . . . . . . . . . . . 1.2 1.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 1.1.4 Parabolic equations – application to diffusion equation . . . 1.4 1.1.5 Explicit finite difference method . . . . . . . . . . . . . . . 1.5 1.1.6 The program . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 1.1.7 Application of the finite difference method to hyperbolic partial differential equations – one-dimensional wavelength 2.2 1.1.8 Application of the finite difference method to elliptical differential equations . . . . . . . . . . . . . . . . . . . . . 2.3 1.1.9 Neumann problem . . . . . . . . . . . . . . . . . . . . . . 2.4 1.1.10 Curvilinear boundaries . . . . . . . . . . . . . . . . . . . . 2.6 29 2.1 Finite element application to solving of one-dimensional problems . 2.9 2.1.1 Variation formulation . . . . . . . . . . . . . . . . . . . . . 3.5 2.1.2 Galerkin’s method . . . . . . . . . . . . . . . . . . . . . . 3.6 2.1.3 Finite element base functions . . . . . . . . . . . . . . . . . 3.7 2.2.1 Interpolation error . . . . . . . . . . . . . . . . . . . . . . 4.0 2.3 Finite element approximation . . . . . . . . . . . . . . . . . . . . . 4.2 2.4 Determining of finite element matrices and finite element system matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 2.4.1 Application of finite element method to parabolic and hy perbolic partial differential equations . . . . . . . . . . . . 4.5 Chapter 2 Finite element method