Mathematical Physics - Volume II - Numerical Methods

Chapter 3 Comparison of finite element method and finite difference method 49 3.1 Approximate analytical solution . . . . . . . . . . . . . . . . . . . 5.0 3.1.1 Finite difference method solution . . . . . . . . . . . . . . 5.1 3.1.2 Finite element method solution . . . . . . . . . . . . . . . . 5.2 3.2 Finite element method – two-dimensional problem . . . . . . . . . 5.5 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 3.2.2 Physical base of the problem . . . . . . . . . . . . . . . . . 5.5 3.2.3 Two-dimensional elliptical boundary problem . . . . . . . . 5.7 3.3 Variation formulation of the boundary problem . . . . . . . . . . . 6.0 3.4 Finite element interpolation . . . . . . . . . . . . . . . . . . . . . . 6.2 3.4.1 Interpolation within triangles . . . . . . . . . . . . . . . . . 6.4 3.4.2 Interpolation error . . . . . . . . . . . . . . . . . . . . . . 6.7 3.5 Finite element approximations . . . . . . . . . . . . . . . . . . . . 6.7 3.5.1 Determining of the finite element matrix . . . . . . . . . . . 7.4 3.5.2 Calculating of finite element matrices . . . . . . . . . . . . 7.9 3.6 Program for solving of elliptical problems . . . . . . . . . . . . . . 8.2 3.6.1 Uvodne napomene . . . . . . . . . . . . . . . . . . . . . . 8.2 3.6.2 Subprogram structure . . . . . . . . . . . . . . . . . . . . . 8.3 3.6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5

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Chapter 4 Finite volume method 109 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 0.9 4.2 Solution of Euler equations . . . . . . . . . . . . . . . . . . . . . 1. 1.0 4.2.1 Explicit numerical scheme . . . . . . . . . . . . . . . . . 1. 1.0 4.2.2 Implicit numerical scheme . . . . . . . . . . . . . . . . . 1. 1.5 4.2.3 Godunov scheme . . . . . . . . . . . . . . . . . . . . . . 1. 2.3 4.3 Solution of Navier–Stokes equations . . . . . . . . . . . . . . . . 1. 4.9 4.3.1 Implicit numerical scheme . . . . . . . . . . . . . . . . . 1. 4.9

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