Talks

Uniqueness for weak solutions of conservation laws is based on the sign of the entropy production across discontinuos solutions. Although the entropy plays a fundamental role in the theory of hyperbolic systems, it is generally not used as a computational tool.
In this talk I describe how the numerical production of entropy induced by the discretization of the equations is a reliable indicator of the quality of the numerical solution. Thus the entropy production can be used as a regularity indicator, identifying the cells in which non linear limiters must be used to prevent the onset of spurious oscillations.
More quantitatively, when the solution is smooth, the entropy production has the same size of the local truncation error and can therefore be used as an a-posteriori error indicator to drive the construction of adaptive grids.

We start with a brief introduction to Conservation Laws and to their numerical solutions. Then we discuss the construction and manipulation of non-uniform meshes, using geometric properties of the numerical solution under consideration. Next, we examine properties (such as consistency, stability and order of accuracy) of numerical schemes over both uniform and non-uniform meshes. Finally, we combine a proper mesh selection mechanism with Entropy Conservative or oscillatory numerical schemes for the evolution step.