This paper deals with the development of efficient incomplete multidimensional Riemann solvers for hyperbolic systems. Departing from a four-waves model for the speeds of propagation arising at each vertex of the computational structured mesh, we present a general strategy for constructing genuinely multidimensional Riemann solvers, that can be applied for solving systems including source and coupling terms.

In particular, a simple version of a well-balanced 2d HLL scheme is presented, which is later taken as a basis to build a general class of incomplete Riemann solvers, the so-called Approximate Viscosity Matrix (AVM) schemes. The great advantage of the AVM strategy is the possibility to control the amount of numerical diffusion considered for each hyperbolic system at an affordable computational cost.

The presented numerical schemes are shown to be linearly $L^{\infty }$-stable for a CFL number up to unity. Our schemes can be used as building blocks for constructing high-order schemes. In this work, a second-order scheme is constructed by using a predictor-corrector MUSCL-Hancock procedure.

To test the performances of the proposed schemes, a number of challenging numerical experiments in one-layer and two-layer shallow water systems have been run. The presence of the bottom topography and the coupling terms represent an additional difficulty, that has been solved by reformulating the problem within the path-conservative framework. Finally, the 2d schemes have been shown to be more efficient than their projected 1d×1d counterparts.

本文涉及双曲系统的高效不完全多维黎曼求解器的开发。从计算结构化网格每个顶点产生的传播速度的四波模型出发，我们提出了构建真正多维黎曼求解器的一般策略，该策略可应用于求解包括源和耦合项在内的系统。

特别是，提出了一个简单版本的良好平衡的 2d HLL 方案，后来将其作为构建一类不完全黎曼求解器的基础，即所谓的近似粘度矩阵 (AVM) 方案。AVM 策略的巨大优势是以可承受的计算成本控制为每个双曲线系统考虑的数值扩散量的可能性。

所提出的数值方案显示为线性 ${升}^{\infty }$- 稳定的 CFL 数高达统一。我们的方案可以用作构建高阶方案的构建块。在这项工作中，通过使用预测器-校正器 MUSCL-Hancock 程序构建了二阶方案。

为了测试所提出方案的性能，在一层和两层浅水系统中进行了许多具有挑战性的数值实验。底部地形和耦合项的存在代表了一个额外的困难，通过在路径保守框架内重新表述问题已经解决了这个问题。最后，2d 方案已被证明比他们预计的 1d×1d 方案更有效。