Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier--Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the '90s and have several interesting features, such as being grid free, automatically adapting to the solution by concentrating elements where the gradient is large and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

翻译：基于解的空间导数演化的粒子方法最初是为模拟反应扩散过程而引入的，其灵感来源于纳维-斯托克斯方程中的涡旋方法。这类方法被称为梯度随机游走方法，在20世纪90年代得到了广泛研究，并具有若干有趣特性，例如无网格化、通过在高梯度区域集中粒子自动适应解、以及显著降低标准随机游走方法的方差。在本工作中，我们重新探讨了这些思想，展示了如何将该方法推广到更大类别的偏微分方程中，包括双曲守恒律方程组。为实现这一目标，我们首先将经典蒙特卡洛方法扩展至守恒律方程组的松弛近似，随后基于解的空间导数引入一种新颖的粒子动力学。该方法结合渐近保持型分裂离散化，为双曲守恒律方程组构建了一类新的基于梯度的蒙特卡洛方法。一维空间中标量方程及守恒律方程组的多个数值结果表明，新方法前景广阔，在方差降低以及激波结构描述方面均相较于标准蒙特卡洛方法取得了显著改进。