In this paper we introduce a definition of the local conservation property for numerical methods solving time dependent conservation laws, which generalizes the classical local conservation definition. The motivation of our definition is the Lax-Wendroff theorem, and thus we prove it for locally conservative numerical schemes per our definition in one and two space dimensions. Several numerical methods, including continuous Galerkin methods and compact schemes, which do not fit the classical local conservation definition, are given as examples of locally conservative methods under our generalized definition.

This is an extension of our earlier work \cite{DS} in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. %The initial condition is given on such a point cloud, %and the algorithm solves for point values of the solution at %later time also on this point cloud. 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 \cite{DS} show that the current scheme will generate spurious numerical oscillations when dealing with solutions containing strong shocks. In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for DG schemes on two-dimensional unstructured triangular meshes \cite{ZZSQ}, to our high order method on polygonal meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the original method in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt is particularly simple to implement and will not harm the conservativeness and compactness of the original method. Moreover, we also extend the maximum-principle-satisfying limiter for the scalar case and the positivity-preserving limiter for the Euler system to our method. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of these limiters.

For a Lagrangian scheme solving the compressible Euler equations in cylindrical coordinates, two important issues are whether the scheme can maintain spherical symmetry (symmetry-preserving) and whether the scheme can maintain positivity of density and internal energy (positivity-preserving). While there were previous results in the literature either for symmetry-preserving in the cylindrical coordinates or for positivity-preserving in cartesian coordinates, the design of a Lagrangian scheme in cylindrical coordinates, which is high order in one-dimension and second order in two-dimensions, and can maintain both spherical symmetry-preservation and positivity-preservation simultaneously, is challenging. In this paper we design such a Lagrangian scheme and provide numerical results to demonstrate its good behavior.

A new class of high order weighted essentially non-oscillatory (WENO) schemes [J. Comput. Phys., 318 (2016), 110-121] is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes [J. Comput. Phys., 126 (1996), 202-228] might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in [J. Comput. Phys., 126 (1996), 202-228] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.

In this article, we present a second-order in time implicit-explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn-Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn-Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn-Hilliard system modeled by the IMEX LDG method can inherit the nonlinear stability of the continuous model. We apply a stabilization technique and prove unconditional energy stability of our scheme. Numerical experiments are performed to validate the analysis. Computational efficiency can be significantly enhanced by using this IMEX LDG method with a large time step.