In this paper, we present a positivity-preserving high order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations based on the framework for constructing uniformly high order accurate positivity-preserving discontinuous Galerkin and finite volume schemes for Euler equations proposed in [20]. The major advantages of the HWENO schemes is their...

This is an extension of our earlier work [9] in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. An algorithm of building a suitable polygonal mesh based on the random points was given and the traditional discontinuous Galerkin (DG) method was adopted on the constructed polygonal mesh. Numerical results in [9] sho...

In this paper, we survey our recent work on designing high order positivitypreserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solut...

In this paper, we consider several high-order schemes in one space dimension. In particular, we compare the second-order relaxation ( << 1) or “relaxed” ( = 0) schemes of Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 235–277] with the second-order Lax–Friedrichs scheme of Nessyahu and Tadmor [J. Comp. Phys., 87 (1990), pp. 408–463] and with higher-order essentially nonoscillatory (ENO) an...