Mathematical Physics - Volume II - Numerical Methods

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Cover	1
Part I — Numerical method - FDM-FEM	11
1 Finite difference method and Finite element method	13
1.1 Finite difference method	13
2 Finite element method	31
2.1 Finite element application to solving of one-dimensional problems	31
2.3 Finite element approximation	44
2.4 Determining of finite element matrices and finite element system matrices	44
3 Comparison of finite element method and finite difference method	51
3.1 Approximate analytical solution	52
3.2 Finite element method – two-dimensional problem	57
3.3 Variation formulation of the boundary problem	62
3.4 Finite element interpolation	64
3.5 Finite element approximations	69
3.6 Program for solving of elliptical problems	84
Bibliography	107
Part II — FVM	109
4 Finite volume method	111
4.1 Introduction	111
4.2 Solution of Euler equations	112
4.3 Solution of Navier–Stokes equations	151
Bibliography	157
Part III — Numerical method - SPH	161
5 Review of Development of the Smooth Particle Hydrodynamics (SPH) Method	163
5.1 Introduction	163
5.2 Basic Formulation	167
5.3 Conservation Equations	169
5.4 Kernel Function	171
5.5 Variable Smoothing Length	172
5.6 Neighbour Search	173
5.7 SPH Shortcomings	174
5.8 Derivation of Normalised Corrected Gradient SPH formula	174
5.9 Tensile Instability	178
5.10 Zero-Energy Modes	183
5.11 Non-Local properties of SPH	184
5.12 Summary	203
Bibliography	205
Part IV — Numerical method - CMD	215
6 Introduction to Computational Mechanics of Discontinua	217
6.1 Introduction	217
6.2 Molecular Dynamics	223
6.3 Particle Dynamics	241
6.4 Lattices	246
6.5 Discrete Element Methods	259
Bibliography	275
Index	285

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