标题： 关于非线性双曲守恒律系统的一般和完全的多维Riemann解算器 显示英文标题

标题： On general and complete multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws

摘要： 在这项工作中，我们引入了一个框架来设计非线性双曲守恒律系统在一般非结构化多边形Voronoi-like镶嵌上的多维Riemann解算器。 在这个框架下，我们提出了两种简单但完整的解算器。 第一种方法是Osher-Solomon Riemann解算器直接扩展到多个空间维度。 在这里，通过在原始多边形网格每个节点周围的对偶三角网格上积分通量雅可比行列式的绝对值，得到了多维数值阻尼。 然后，在涉及每个角处相遇的d+1个邻居的局部计算单纯形上评估所需的节点梯度。 第二种方法是一种真正的多维上风通量。 通过引入带有角通量的有限体积方法的波动形式，我们展示了它与残差分布方案（RD）的等价性。 这自然允许从现有的和已知的RD波动构造真正的多维上风角通量。 特别是，我们探讨了从所谓的N格式（即Roe的原始最优多维上风对流格式）获得的角通量的使用。 这两种方法都利用了基础双曲系统的完整特征结构，因此在构造上是完整的。 然后，在空间上使用CWENO重构，在时间上使用ADER方法，引入了一种简单的高阶扩展，直至四阶空间和时间精度，从而形成一组基于真正多维Riemann解算器的高阶精确全离散单步格式。 我们应用这些新的数值格式解决了一些可压缩Euler方程组的气体动力学测试问题。 特别是，我们展示了所提出的格式同时避免了网格状不稳定现象，并且能够精确保持某些静止剪切波。 显示英文摘要

摘要： In this work, we introduce a framework to design multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws on general unstructured polygonal Voronoi-like tessellations. In this framework we propose two simple but complete solvers. The first method is a direct extension of the Osher-Solomon Riemann solver to multiple space dimensions. Here, the multidimensional numerical dissipation is obtained by integrating the absolute value of the flux Jacobians over a dual triangular mesh around each node of the primal polygonal grid. The required nodal gradient is then evaluated on a local computational simplex involving the d+1 neighbors meeting at each corner. The second method is a genuinely multidimensional upwind flux. By introducing a fluctuation form of finite volume methods with corner fluxes, we show an equivalence with residual distribution schemes (RD). This naturally allows to construct genuinely multidimensional upwind corner fluxes starting from existing and well-known RD fluctuations. In particular, we explore the use of corner fluxes obtained from the so-called N scheme, i.e. the Roe's original optimal multidimensional upwind advection scheme. Both methods use the full eigenstructure of the underlying hyperbolic system and are therefore complete by construction. A simple higher order extension up to fourth order in space and time is then introduced at the aid of a CWENO reconstruction in space and an ADER approach in time, leading to a family of high order accurate fully-discrete one-step schemes based on genuinely multidimensional Riemann solvers. We present applications of our new numerical schemes to several test problems for the compressible Euler equations of gas-dyanamics. In particular, we show that the proposed schemes are at the same time carbuncle-free and preserve certain stationary shear waves exactly.