Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups and discontinuities. Here we present an explicit numerical scheme for solving non-linear advection-diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for non-linear advection-diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.

中文翻译：

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通过离散时间随机游走方案的对流-扩散方程的数值解

众所周知，偏微分方程的显式数值有限差分格式易于实现，但对于求解其解允许冲击、爆炸和不连续性的方程，它们尤其成问题。在这里，我们提出了一个显式的数值方案，用于求解非线性对流扩散方程，该方程既易于实现又稳定。该数值方案是通过考虑非线性对流扩散的离散时间和空间随机过程的连续极限而获得的。随机过程是适定的，这保证了方案的稳定性。提供了几个例子来强调随机过程的公式化在获得稳定和准确的数值方案中的重要性。