求解双曲型守恒律的一类高阶加权紧致中心格式
A class of high-order weighted compact central schemes for solving hyperbolic conservation laws
摘要：
提出了一类求解双曲型守恒律方程的加权紧致中心（WCC）格式。线性格式可以看作是中心Lax-Friedrichs（LxF）格式和中心守恒元和解元（CESE）格式的高阶推广。在每个单元上，通过P阶多项式来近似解，其中所有DOF被单独地存储和更新。单元平均值由经典的有限体积格式更新，该有限体积格式基于时空交错网格构造，使得通量在相邻控制体积的界面上是连续的，因此，绕过了局部黎曼问题。k阶空间导数通过单元顶点处的（k-1）阶空间导数的中心差来更新。所有的时空信息都是通过Cauchy-Kovalewski过程计算的。通过这样做，该方案能够实现任意一致的时空高阶超紧模板上只有一个显式的时间步长。为了捕捉不连续性而不产生虚假振荡，为格式量身定制了加权本质不振荡（WENO）型限制器。限制器保持了格式的紧致性和高阶精度。通过标量守恒律和可压缩Euler方程的数值算例验证了格式的精度、鲁棒性和有效性。
英文摘要：
We propose a class of weighted compact central (WCC) schemes for solving hyperbolic conservation laws. The linear version can be considered as a high-order extension of the central Lax-Friedrichs (LxF) scheme and the central conservation element and solution element (CESE) scheme. On every cell, the solution is approximated by a Pth order polynomial of which all the DOFs are stored and updated separately. The cell average is updated by a classical finite volume scheme which is constructed based on space-time staggered meshes such that the fluxes are continuous across the interfaces of the adjacent control volumes and, therefore, the local Riemann problem is bypassed. The kth order spatial derivatives are updated by a central difference of (k-1)th order spatial derivatives at cell vertices. All the space-time information is calculated by the Cauchy-Kovalewski procedure. By doing so, the schemes are able to achieve arbitrarily uniform spacetime high order on a super-compact stencil with only one explicit time step. In order to capture discontinuities without spurious oscillations, a weighted essentially non-oscillatory (WENO) type limiter is tailor-made for the schemes. The limiter preserves the compactness and high order accuracy of the schemes. The accuracy, robustness, and efficiency of the schemes are verified by several numerical examples of scalar conservation laws and the compressible Euler equations.
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