We study the numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous and to have at most polynomial growth. The numerical scheme we employ discretizes the stochastic partial differential equation (SPDE) according to a finite-volume method in space and a split-step backward Euler method in time. As a first result, we prove the well posedness as well as the existence and uniqueness of an invariant measure for both the semidiscrete and the split-step scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. We investigate rates of convergence theoretically, in the case where the flux function is globally Lipschitz continuous with a small Lipschitz constant, and numerically for the Burgers equation.

中文翻译：

---

粘性随机标量守恒定律不变测度的有限体积逼近

我们研究了粘性标量守恒定律的不变测度的数值逼近，它是空间变量中的一维和周期性，并且随机地被时间中的白色但空间相关的噪声强制。假设通量函数是局部 Lipschitz 连续的，并且最多具有多项式增长。我们采用的数值方案根据空间中的有限体积方法和时间上的分步后向欧拉方法对随机偏微分方程 (SPDE) 进行离散化。作为第一个结果，我们证明了半离散和分步方案的不变测度的适定性以及存在性和唯一性。我们的主要结果是离散近似的不变度量的收敛，因为空间和时间步长变为零，相对于二阶 Wasserstein 距离，走向 SPDE 的不变测度。我们从理论上研究收敛率，在通量函数是全局 Lipschitz 连续且具有小的 Lipschitz 常数的情况下，以及 Burgers 方程的数值。