We extend to multi-dimensions the work of [1], where new fully explicit kinetic methods were built for the approximation of linear and non-linear convection-diffusion problems. The fundamental principles from the earlier work are retained: (1) rather than aiming for the desired equations in the strict limit of a vanishing relaxation parameter, as is commonly done in the diffusion limit of kinetic methods, diffusion terms are sought as a first-order correction of this limit in a Chapman-Enskog expansion, (2) introducing a coupling between the conserved variables within the relaxation process by a specifically designed collision matrix makes it possible to systematically match a desired diffusion. Extending this strategy to multi-dimensions cannot, however, be achieved through simple directional splitting, as diffusion is likely to couple space directions with each other, such as with shear viscosity in the Navier-Stokes equations. In this work, we show how rewriting the collision matrix in terms of moments can address this issue, regardless of the number of kinetic waves, while ensuring conservation systematically. This rewriting allows for introducing a new class of kinetic models called \emph{regularized} models, simplifying the numerical methods and establishing connections with Jin-Xin models. Subsequently, new explicit arbitrary high-order kinetic schemes are formulated and validated on standard two-dimensional cases from the literature. Excellent results are obtained in the simulation of a shock-boundary layer interaction, validating their ability to approximate the Navier-Stokes equations with kinetic speeds obeying nothing but a subcharacteristic condition along with a hyperbolic constraint on the time step.

翻译：我们将文献[1]的工作推广到多维情形，该工作针对线性和非线性对流扩散问题的逼近，构建了全新的完全显式动力学方法。我们保留了先前工作的基本原理：(1) 并非像动力学方法的扩散极限中常见的那样，在松弛参数趋近于零的严格极限下追求目标方程，而是在Chapman-Enskog展开中，将扩散项视为该极限的一阶修正项进行求解；(2) 通过引入一个专门设计的碰撞矩阵，在松弛过程中实现守恒变量之间的耦合，从而能够系统地匹配所需的扩散系数。然而，将这一策略推广至多维情形无法通过简单的方向分裂实现，因为扩散可能耦合不同的空间方向（例如纳维-斯托克斯方程中的剪切粘性）。在本文中，我们展示了如何通过将碰撞矩阵重写为矩的形式来解决这一问题——无论动力学波的数量如何，都能保证系统的守恒性。这一重写引入了称之为“正则化”模型的新一类动力学模型，简化了数值方法，并建立了与Jin-Xin模型的联系。随后，我们构建了新的显式任意高阶动力学格式，并在文献中标准二维算例上进行了验证。在激波-边界层相互作用的模拟中获得了极优结果，验证了该方法在仅满足次特征条件以及时间步长双曲约束的条件下，利用动力学速度逼近纳维-斯托克斯方程的能力。