Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks. Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tracking. The theory is illustrated by numerous examples. There is a dedicated Web page that provides MATLAB® codes for many of the examples. The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. ® MATLAB, The MathWorks, Inc., Natick, MA.
Author(s): Helge Holden, Kenneth H. Karlsen, Knut-andreas Lie, Nils Henrik Risebro
Series: EMS Series of Lectures in Mathematics
Publisher: European Mathematical Society
Year: 2010
Language: English
Pages: 235
Contents ......Page 8
1 Introduction ......Page 10
1.1 Purpose of the book ......Page 13
1.2 The class of PDEs discussed in the book ......Page 14
1.3 Operator splitting for initial-value problems ......Page 15
1.5 Rigorous analysis of operator-splitting methods ......Page 17
1.6 Topics not treated in the book ......Page 19
1.7 Organization of the book ......Page 20
1.8 Matlab programs ......Page 21
1.9 A guide to the reader ......Page 22
1.10 Notation ......Page 23
2 Simple Examples of Semi-Discrete Operator Splitting ......Page 26
3 General Convergence Theory ......Page 35
3.1 Mathematical preliminaries ......Page 36
3.2 Degenerate parabolic equations ......Page 40
3.3 Weakly coupled systems of degenerate parabolic equations ......Page 45
3.4 A general convergence theory ......Page 47
4 Convergence Results for Convection-Diffusion Problems ......Page 60
4.1 A semi-discrete splitting method ......Page 61
4.2 A fully discrete splitting method ......Page 65
4.2.1 Convergence in the discrete L^1 norm ......Page 67
4.2.2 Convergence analysis ......Page 70
4.3 Nonlinear error mechanisms ......Page 83
4.4 Viscous splitting with a posteriori flux splitting ......Page 87
5.1 Multi-dimensional scalar conservation laws ......Page 97
5.2 Weakly coupled systems of conservation laws ......Page 125
6.1 Operator splitting in porous media flow ......Page 145
6.1.1 Dimensional splitting ......Page 147
6.1.2 Dimensional splitting combined with viscous splitting ......Page 150
6.1.3 Streamline methods ......Page 155
6.2 Dimensional splitting for systems of conservation laws ......Page 159
6.3.1 Geometric source terms ......Page 168
6.3.2 Equilibrium states ......Page 171
6.3.3 Reactive flows ......Page 172
6.3.4 External forces ......Page 176
6.4 Final remarks ......Page 179
A.1 Hyperbolic conservation laws ......Page 182
A.2 Finite-volume methods ......Page 184
A.3 Conservative methods ......Page 185
A.4 A few classical schemes ......Page 186
A.5 Convergence of conservative methods ......Page 190
A.6 High-resolution Godunov methods ......Page 193
A.6.1 High-resolution central schemes ......Page 195
A.6.2 High-resolution upwind schemes ......Page 201
A.7 Front tracking ......Page 202
References ......Page 210
Index ......Page 234
