InthispaperwedevelopaconservativelocaldiscontinuousGalerkin(LDG) method for the Schro ̈ dinger-Korteweg-de Vries (Sch-KdV) system, which arises in var- ious physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical ex- periments illustrate the accuracy and capability of the method.

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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.

We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules \cite{Tchen2017entropy} for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov \cite{godunov1972symmetric}, the entropy stable DG framework with suitable quadrature rules \cite{Tchen2017entropy}, the entropy conservative flux in \cite{chandrashekar2016entropy} inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. {\color{blue}{ The main difficulty in the generalization of the results in \cite{Tchen2017entropy} is the appearance of the non-conservative ``source terms'' added in the modified MHD model introduced by Godunov \cite{godunov1972symmetric}, which do not exist in the general hyperbolic system studied in \cite{Tchen2017entropy}. Special care must be taken to discretize these ``source terms'' adequately so that the resulting DG scheme satisfies entropy stability. }} Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples.

Intrinsic curvature flows can be used to design Riemannian metrics by prescribed curvatures. This chapter presents three discrete curvature flow methods that are recently introduced into the engineering fields: the discrete Ricci flow and discrete Yamabe flow for surfaces with various topology, and the discrete curvature flow for hyperbolic 3-manifolds with boundaries. For each flow, we introduce its theories in both the smooth setting and the discrete setting, plus the numerical algorithms to compute it. We also provide a brief survey on their history and their link to some of the engineering applications in computer graphics, computer vision, medical imaging, computer aided design and others.