| Preface |
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ix | |
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Chapter 1 Ordinary Differential Equations |
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1 | (140) |
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1.1 Introduction to the theory of ordinary differential equations |
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1 | (26) |
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1.1.1 Existence--uniqueness of first-order ordinary differential equations |
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1 | (10) |
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1.1.2 The concept of maximal solution |
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11 | (5) |
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1.1.3 Linear systems with constant coefficients |
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16 | (4) |
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1.1.4 Higher-order differential equations |
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20 | (1) |
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1.1.5 Inverse function theorem and implicit function theorem |
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21 | (6) |
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1.2 Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability |
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27 | (75) |
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27 | (2) |
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1.2.2 Fundamental notions for the analysis of numerical ODE methods |
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29 | (4) |
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1.2.3 Analysis of explicit and implicit Euler schemes |
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33 | (17) |
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1.2.4 Higher-order schemes |
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50 | (1) |
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1.2.5 Leslie's equation (Perron--Frobenius theorem, power method) |
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51 | (27) |
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1.2.6 Modeling red blood cell agglomeration |
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78 | (9) |
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87 | (6) |
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1.2.8 A chemotaxis problem |
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93 | (9) |
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102 | (39) |
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1.3.1 The pendulum problem |
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106 | (6) |
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1.3.2 Symplectic matrices; symplectic schemes |
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112 | (13) |
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125 | (4) |
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129 | (12) |
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Chapter 2 Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems |
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141 | (126) |
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141 | (25) |
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2.1.1 The ID model problem; elements of modeling and analysis |
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144 | (11) |
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2.1.2 A radiative transfer problem |
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155 | (8) |
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2.1.3 Analysis elements for multidimensional problems |
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163 | (3) |
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2.2 Finite difference approximations to elliptic equations |
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166 | (14) |
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2.2.1 Finite difference discretization principles |
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166 | (7) |
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2.2.2 Analysis of the discrete problem |
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173 | (7) |
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2.3 Finite volume approximation of elliptic equations |
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180 | (11) |
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2.3.1 Discretization principles for finite volumes |
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180 | (7) |
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2.3.2 Discontinuous coefficients |
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187 | (2) |
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2.3.3 Multidimensional problems |
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189 | (2) |
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2.4 Finite element approximations of elliptic equations |
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191 | (13) |
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2.4.1 P1 approximation in one dimension |
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191 | (6) |
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2.4.2 P2 approximations in one dimension |
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197 | (3) |
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2.4.3 Finite element methods, extension to higher dimensions |
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200 | (4) |
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2.5 Numerical comparison of FD, FV and FE methods |
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204 | (1) |
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205 | (12) |
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2.7 Poisson--Boltzmann equation; minimization of a convex function, gradient descent algorithm |
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217 | (7) |
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2.8 Neumann conditions: the optimization perspective |
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224 | (4) |
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2.9 Charge distribution on a cord |
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228 | (7) |
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235 | (32) |
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Chapter 3 Numerical Simulations of Partial Differential Equations: Time-dependent Problems |
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267 | (140) |
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267 | (24) |
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3.1.1 L2 stability (von Neumann analysis) and L∞ stability: convergence |
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269 | (7) |
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276 | (5) |
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3.1.3 Finite element discretization |
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281 | (2) |
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3.1.4 Numerical illustrations |
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283 | (8) |
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3.2 From transport equations towards conservation laws |
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291 | (54) |
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291 | (4) |
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3.2.2 Transport equation: method of characteristics |
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295 | (4) |
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3.2.3 Upwinding principles: upwind scheme |
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299 | (2) |
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3.2.4 Linear transport at constant speed; analysis of FD and FV schemes |
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301 | (25) |
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3.2.5 Two-dimensional simulations |
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326 | (3) |
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3.2.6 The dynamics of prion proliferation |
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329 | (16) |
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345 | (9) |
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3.4 Nonlinear problems: conservation laws |
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354 | (53) |
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3.4.1 Scalar conservation laws |
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354 | (33) |
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3.4.2 Systems of conservation laws |
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387 | (6) |
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393 | (14) |
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407 | (40) |
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409 | (8) |
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417 | (10) |
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427 | (6) |
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433 | (10) |
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443 | (4) |
| Bibliography |
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447 | (8) |
| Index |
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455 | |
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