In this presentation, we present a fifth-order Hermite weighted essentially non-oscillatory (HWENO) scheme with artificial linear weights for one and two dimensional hyperbolic conser- vation laws, where the zeroth-order and the first-order moments are used in the spatial reconstruction. We construct the HWENO methodology using a nonlinear convex combination of a high degree polynomial with several low degree polynomials, and the associated linear weights can be any artificial positive numbers with only requirement that their summation equals one. The one advantage of the HWENO scheme is its simplicity and easy extension to multi-dimension in engineering applications for we can use any artificial linear weights which are independent on geometry of mesh. The another advantage is its higher order numerical accuracy using less candidate stencils for two dimensional problems. In addition, the HWENO scheme still keeps thecompactness as only immediate neighbor information is needed in the reconstruction and has high efficiency for directly using linear approximation in the smooth regions. In order to avoid nonphysical oscillations nearby strong shocks or contact discontinuities, we adopt the thought of limiter for discontinuous Galerkin method to control the spurious oscillations. Some benchmark numerical tests are performed to demonstrate the capability of the proposed scheme.
邱建贤，男，博士，厦门大学数学科学学院教授，国际著名刊物 “J. Comp. Phys.” (计算物理) 编委。从事计算流体力学及微分方程数值解法的研究工作，在间断Galerkin（DG）、加权本质无振荡（WENO）数值方法的研究及其应用方面取得了一些重要成果，已发表论文一百多篇。主持国家自然科学基金重点项目和联合基金重点支持项目各一项, 参与欧盟第六框架特别研究项目, 是项目组中唯一非欧盟的成员。多次被邀请在国际会议上作大会报告。