We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO) reconstructions, and in time with high-order explicit Runge-Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of non-differentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton's method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for non-conservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley--Leverett equations. For many test problems, the solver converges in just a handful of iterations with convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.

翻译：本文研究一维空间中标量非线性守恒方程的时间并行求解方法。方程在空间上采用守恒型有限体积法结合加权本质无振荡（WENO）重构进行离散，时间上采用高阶显式Runge-Kutta方法。通过非线性迭代求解全局离散时空问题，对于不可微方程情形采用新型线性化策略。在特定离散与算法参数选择下，该非线性迭代退化为牛顿法，一般情况下则为预条件残差校正格式。每次非线性迭代中，线性化问题表现为所涉时空域上线性守恒方程特定离散格式的形式。通过单次多网格时间约化（MGRIT）迭代计算线性化问题的近似时间并行解。MGRIT迭代采用新型粗网格算子，该算子为修正的守恒型半拉格朗日离散格式，是我们先前针对非守恒标量线性双曲问题所开发方法的推广。对无粘Burgers方程与Buckley-Leverett方程进行数值测试。在多数测试问题中，包括具有（相互作用的）激波与稀疏波的问题，求解器仅需少量迭代即可收敛，且收敛速度与网格分辨率无关。