Relative Entropy for the Finite Volume Approximation of Strong Solutions to Systems of Conservation Laws

Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate meth...

Monte Carlo and multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws

A mathematical formulation of conservation and of balance laws with random input data, specifically with random initial conditions, random source terms and random flux functions, is reviewed. The concept of random entropy solution is specified. For scalar conservation laws in multi-dimensions, recent results on the existence and on the uniqueness of random entropy solutions with finite variance...

Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws

A mathematical formulation of conservation and of balance laws with random input data, specifically with random initial conditions, random source terms and random flux functions, is reviewed. The concept of random entropy solution is specified. For scalar conservation laws in multi-dimensions, recent results on the existence and on the uniqueness of random entropy solutions with finite variance...

Self-similar solutions‎ ‎of the Riemann problem for two-dimensional systems of conservation‎ ‎laws

In this paper, a new approach is applied to study the self-similar solutions of 2×2 systems of nonlinear hyperbolic conservation laws. A notion of characteristic directions is introduced and then used to construct local and smooth solutions of the associated Riemann problem

Error Estimate for Time-Explicit Finite Volume Approximation of Strong Solutions to Systems of Conservation Laws