Chi-Wang Shu

This book captures the state-of-the-art in the field of Strong Stability Preserving (SSP) time stepping methods, which have significant advantages for the time evolution of partial differential equations describing a wide range of physical phenomena. This comprehensive book describes the development of SSP methods, explains the types of problems which require the use of these methods and demonstrates the efficiency of these methods using a variety of numerical examples. Another valuable feature of this book is that it collects the most useful SSP methods, both explicit and implicit, and presents the other properties of these methods which make them desirable (such as low storage, small error coefficients, large linear stability domains). This book is valuable for both researchers studying the field of time-discretizations for PDEs, and the users of such methods.

This book contains an expanded and smoothed version of lecture notes delivered by the authors at the Advanced School on Numerical Solutions of Partial Di?- ential Equations: New Trends and Applications, which took place at the Centre de Recerca Matem' atica (CRM) in Bellaterra (Barcelona) from November 15th to 22nd, 2007. The book has three parts. The ?rst part, by Silvia Bertoluzza and Silvia Falletta, 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 wr- ing equivalent norms for the scale of Besov spaces allows to write down some new methods. Thesecondpart,byGiovanniRusso,providesanoverviewofthemodern finite-volume and finite-difference shock-capturing schemes for systems of cons- vationandbalancelaws,with emphasisingiving auni?ed viewofsuchschemesby identifying the essential aspects of their construction. In the last part Chi-Wang Shugivesageneralintroductionto thediscontinuousGalerkinmethods forsolving some classes of PDEs, discussing cell entropy inequalities, nonlinear stability and error estimates. The school that originated these notes was born with the objective of p- viding an opportunity for PhD students, recent PhD doctorates and researchers in general in ?elds of applied mathematics and engineering to catch up with - portant developments in the ?elds and/or to get in touch with state-of-the-art numerical techniques that are not covered in usual courses at graduate level.