Approximation of Non-Linear Hyperbolic Problems and Conservation Properties
Explore a comprehensive lecture on the approximation of non-linear hyperbolic problems and conservation properties. Delve into the mathematical formulation of problems involving partial differential equations, focusing on hyperbolic systems with flux functions. Examine the Lax-Wendroff theorem and its implications for numerical schemes in conservation form. Investigate various numerical methods, including finite volume schemes, discontinuous Galerkin methods, and continuous stabilized finite element approaches. Learn about a novel reformulation of the conservation property that allows for the reinterpretation of seemingly non-flux-form schemes. Discover how this reformulation can be applied to construct schemes compatible with additional conservation constraints, such as entropy inequalities and thermodynamic compatibility. Gain insights into advanced topics like schemes on staggered grids and the unique characteristics of the Active Flux scheme.