This monograph presents, in an attractive and self-contained form,
techniques based on the L1 stability theory derived at the end of the
1990s by A. Bressan, T.-P. Liu and T. Yang that yield original error
estimates for so-called well-balanced numerical schemes solving 1D
hyperbolic systems of balance laws. Rigorous error estimates are
presented for both scalar balance laws and a position-dependent
relaxation system, in inertial approximation. Such estimates shed light
on why those algorithms based on source terms handled like local
scatterers can outperform other, more standard, numerical schemes.
Two-dimensional Riemann problems for the linear wave equation are also
solved, with discussion of the issues raised relating to the treatment
of 2D balance laws. All of the material provided in this book is highly
relevant for the understanding of well-balanced schemes and will
contribute to future improvements.