求解气体动力学方程组的高效差分格式

给出了求解多维无粘可压Euler方程组的二阶半离散中心迎风格式。因考虑到了非线性波在Riemann扇内传播的局部速度，从而能更加准确地估计出局部Riemann扇的宽度，最终既回避了计算网格的交错，又降低了格式的数值粘性，建立了介于迎风格式和中心格式之间的高分辨率的半离散中心迎风格式。同时，该格式利用Tadmor等人的耗散型MinMod限制器和Harten等人的压缩型UNO限制器的凸组合来重构分片线性多项式，不仅能快速求解多维无粘可压Euler方程组，还可有效地防止数值解产生伪振荡。

An Efficient Difference Scheme for Gas Dynamics Equations

Existing difference schemes for gas dynamics equations such as WENO (Weighted Essentially Non-Oscillatory) scheme and CWENO (Central WENO) scheme, in our opinion, still leave much room for further improvement in efficiency. In this paper we present a difference scheme that is much more efficient than WENO and CWENO schemes at the cost of a little lowering in resolution (any difference scheme of 2nd-order or higher is usually considered to belong to the high-resolution class; our proposed second-order scheme is a little inferior to the 4th-order WENO and CWENO schemes in high resolution). We present a second-order semi-discrete central-upwind scheme for multidimensional inviscid compressible Euler equations in this paper. Taking into consideration the local speed of nonlinear wave propagation in Riemann fans, we calculate the widths of the local Riemann fans more accurately, obviate the necessity of staggering between two sets of grids, and obtain a scheme with much smaller numerical viscosity. The high-resolution method in this paper is just a semi-discrete central-upwind scheme connecting upwind scheme and central scheme. At the same time, the piecewise linear polynomial is reconstructed by a convex combination of the dissipative MinMod limiter by Tadmor et al and the compressive UNO limiter by Harten et al. We checked the feasibility of our method with three numerical simulation examples. Two numerical examples (four-shock-wave Riemann problem and single-shock, single-rarefaction, two-contact-discontinuity problem) were the same as those in Ref.12 by Kurganov and Tadmor, which gives high-resolution results with 3rd-order difference scheme. For the four shock-wave Riemann problem, results obtained with our efficient 2nd-order difference scheme agree with those in Ref.12 and even the precision obtained is about the same as that in Ref.12. For the single-shock, single-rarefaction, two-contact-discontinuity problem, our numerical results also agree with those in Ref.12. i-discretecentral-upwi paper.Takingintoc calculatethewidthsof ntwosetsofgrids,an thodinthispaperis