Numerical Solutions of Partial Differential Equations: Advanced Courses in Mathematics - CRM Barcelona

Wavelets and Partial Differential Equations.- What is a Wavelet?.- The Fundamental Property of Wavelets.- Wavelets for Partial Differential Equations.- High-Order Shock-Capturing Schemes for Balance Laws.- Upwind Scheme for Systems.- The Numerical Flux Function.- Nonlinear Reconstruction and High-Order Schemes.- Central Schemes.- Systems with Stiff Source.- Discontinuous Galerkin Methods: General Approach and Stability.- Time Discretization.- Discontinuous Galerkin Method for Conservation Laws.- Discontinuous Galerkin Method for Convection-Diffusion Equations.- Discontinuous Galerkin Method for PDEs Containing Higher-Order Spatial Derivatives.

From the reviews:
“It includes an extended version of the lectures given by the four authors at the Advanced School on Numerical Solutions of Partial Differential Equations: New Trends and Applications, held at the CRM – Barcelona between November 15 – 22, 2007. … The book is addressed to masterands, for PhD candidates and to research workers in the fields of applied mathematics and engineering.” (Constantin I. Popovici, IASI Polytechnic Magazine, Vol. 22 (1/4), March-December, 2010)
“The book is well written and organized at a difficulty level that precisely meets the target audiences’ needs. … PhD students as well as engineers and researchers in the field of applied mathematics or scientific computing and interested graduate students will find this book an excellent resource to rapid introduction into the field of modern numerical methods … . serve as a textbook for graduate – level courses in numerical methods … . be useful for experienced researchers and practitioners both from academia or industry.” (Andrzej Myśliński, Control and Cybernetics, Vol. 39 (3), 2010)

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.

The review on central schemes, on error estimates for discontinuous Galerkin methods and on the use of wavelets in scientific computing form an excellent teaching material for graduate students Includes supplementary material: sn.pub/extras