Woonjae Hwang

Numerical simulations of vortex sheet evolution in stratified shear flow

The nonlinear evolution of an interface subject to a parallel shear flow is studied on the basis of the vortex sheet model. We perform numerical simulations of the vortex sheet in density stratified fluids by using the point vortex method. The numerical algorithm based on two types of regularization method improves the stability of the vortex method and provides solutions with much higher resolutions than previous results. The numerical results show that, for a smaller density ratio, the interface has a larger growth rate and a more singular vortex sheet strength and curvature at late times. It is also found that, for an infinite density ratio, the interface has a solution of a travelling wave with no deformations.

Adaptive mesh refinement for weighted essentially non-oscillatory schemes

Abstract. In this paper, we describe the application procedure of theadaptive mesh refinement (AMR) for the weighted essentially non-oscill-atory schemes (WENO), and observe the effects of the derived algorithmwhen problems have piecewise smooth solutions containing discontinu-ities. We find numerically that the dissipation of the WENO scheme canbe lessened by the implementation of AMR while the accuracy is main-tained. We deduce from the experiments that the AMR-implementedWENO scheme captures shocks more efficiently than the WENO methodusing uniform grids. 1. IntroductionIn this study, we describe numerical schemes for one-dimensional conserva-tion laws(1) ∂u∂t + ∂f ( u ) ∂x = 0 , ( x,t ) ∈ I × [0 ,∞ ) , where I isanintervalinspace. Theweightedessentiallynon-oscillatory(WENO)scheme is a high order accurate finite difference scheme for problems with piece-wise smooth solutions containing discontinuities [1, 2, 3]. There are two ap-proaches of the WENO scheme to solve (1). One is the finite difference WENO,which solves discrete version of (1), while the finite volume approach does notsolve (1) directly but its integrated version. That is, the finite difference ap-proach evolves point values whereas the finite volume approach evolves cellaverages. Balsara and Shu [4] derived finite difference WENO schemes with or-ders from 7 to 13 for one dimensional problems. For high dimensional problems,the computational cost for the finite volume WENO approach is more expen-sive than that of the finite difference WENO approach [3, 5]. In addition, the

Distance three labellings for Kn×K2

In a wireless communication network, the channel assignment problem addresses the correct assignment of a frequency to each transmitter in the network. To avoid interference between two nearby transmitters, the assigned frequencies must satisfy certain conditions related to the distance between transmitters. We can use only a limited number of channels; hence, we have to minimize the range of frequencies used. The distance labelling of a graph is a mathematical model of the channel assignment problem derived from the work of Hale [Frequency assignment: Theory and application, Proc. IEEE 68 (1980), pp. 1497–1514]. Lam et al. [L(j, k)-labelings for the products of complete graphs, J. Comb. Optim. 14 (2007), pp. 219–227] addressed distance two labelling for the direct product K n ×K m of complete graphs K n and K m . In this paper, we solve the distance three labelling problem for K n ×K 2.

CLASSIFICATION OF TWO-REGULAR DIGRAPHS WITH MAXIMUM DIAMETER

The Klee-Quaife problem is finding the minimum order μ(d, c, v) of the (d, c, v) graph, which is a c-vertex connected v-regular graph with diameter d. Many authors contributed finding μ(d, c, v) and they also enumerated and classified the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, we deal with an equivalent problem, finding the maximum diameter of digraphs with given order, focused on 2-regular case. We show that the maximum diameter of strongly connected 2-regular digraphs with order n is n − 3, and classify the digraphs which have diameter n− 3. All 15 nonisomorphic extremal digraphs are listed.

The exponent of Cartesian product of cycles

Abstract A digraph D is primitive if for each pair of vertices v , w of D , there is a positive integer k such that there is a directed walk of length k from v to w . The minimum of such k is the exponent of D . In this paper, we show that for a primitive graph G and a strongly connected bipartite digraph D , the exponent of the Cartesian product G × D is equal to the addition of the exponent of G and the diameter of D . Finally, we find the exponents of Cartesian products of cycles.

Two dimensional Riemann problem for a 2 × 2 system of hyperbolic conservation laws involving three constant states

Abstract Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions.

Radio number for the product of a path and a complete graph

A multilevel distance labeling of a graph

Long-time simulations of the Kelvin-Helmholtz instability using an adaptive vortex method.

G=(V,E)