Chapter 7: Analysis of finite volume methods for hyperbolic problems

In this chapter, we review theoretical foundations of finite volume schemes for hyperbolic conservation laws. So far, our presentation was focused on the design of numerical algorithms that possess certain properties. Some sufficient conditions for stability, positivity preservation, and validity of discrete maximum principles were formulated as theorems. The aspects of existence, uniqueness, and convergence received little attention in previous chapters. In Part II of this book, we present the theory that remained behind the scenes in Part I. This theory provides a deeper insight into the design criteria on which numerical methods for hyperbolic PDEs are based. Moreover, the results of detailed theoretical studies make it possible to develop more accurate and more reliable numerical approximations.