We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials sc...

In this paper we formulate a high order algorithm for hyperbolic conservation laws using a combination of Fourier expansions and unstructured spectral/hp elements. The unstructured part of the algorithm is formulated using a discontinuous Galerkin approximation. The conservation properties of the unstructured algorithm using a variable polynomial order are examined and an orthogonal projection ...

Many applications involve hyperbolic systems of conservation laws with source terms. The numerical solution of such systems may be challenging, especially when the source terms are stiff. Uniform accuracy with respect to the stiffness parameter is a highly desirable property but it is, in general, very difficult to achieve using underresolved discretizations. For such problems we develop differ...

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L∞ as t→ 0+ and satisfy the entropy inequality in the sense of distributions for t > 0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solut...

We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semi-linear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. ...

In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in L2, provided that the sums of the initial...