Numerical Solutions of Partial Differential Equations

✍ Scribed by Silvia Bertoluzza, Giovanni Russo, Silvia Falletta, Chi-Wang Shu (auth.)

• Publisher: Birkhäuser Basel
• Year: 2009
• Tongue: English
• Leaves: 195
• Series: Advanced Courses in Mathematics - CRM Barcelona
• Edition: 1
• Category: Library

✦ Synopsis

This volume offers researchers the opportunity to catch up with important developments in the field of numerical analysis and scientific computing and to get in touch with state-of-the-art numerical techniques.

The book has three parts. The first one is devoted to the use of wavelets to derive some new approaches in the numerical solution of PDEs, showing in particular how the possibility of writing equivalent norms for the scale of Besov spaces allows to develop some new methods. The second part provides an overview of the modern finite-volume and finite-difference shock-capturing schemes for systems of conservation and balance laws, with emphasis on providing a unified view of such schemes by identifying the essential aspects of their construction. In the last part a general introduction is given to the discontinuous Galerkin methods for solving some classes of PDEs, discussing cell entropy inequalities, nonlinear stability and error estimates.

✦ Table of Contents

Front Matter....Pages i-ix
Front Matter....Pages 1-1
Introduction....Pages 3-3
What is a Wavelet?....Pages 5-21
The Fundamental Property of Wavelets....Pages 23-36
Wavelets for Partial Differential Equations....Pages 37-53
Back Matter....Pages 55-57
Front Matter....Pages 59-59
Introduction....Pages 61-81
Upwind Scheme for Systems....Pages 83-88
The Numerical Flux Function....Pages 89-95
Nonlinear Reconstruction and High-Order Schemes....Pages 97-108
Central Schemes....Pages 109-123
Systems with Stiff Source....Pages 125-142
Back Matter....Pages 143-147
Front Matter....Pages 149-151
Introduction....Pages 153-154
Time Discretization....Pages 155-156
Discontinuous Galerkin Method for Conservation Laws....Pages 157-174
Discontinuous Galerkin Method for Convection-Diffusion Equations....Pages 175-181
Discontinuous Galerkin Method for PDEs Containing Higher-Order Spatial Derivatives....Pages 183-195
Back Matter....Pages 197-201

✦ Subjects

Numerical Analysis; Partial Differential Equations