论文标题
一种原始的二重方法，用于求解与时间近似隐含的保护法律
A primal-dual approach for solving conservation laws with implicit in time approximations
论文作者
论文摘要
在这项工作中，我们提出了一个新颖的框架，用于通过原始的二重混合梯度方法使用隐式方案的时间依赖性保护定律解决方案。我们通过将其作为最小 - 最大问题的鞍点并使用迭代优化方法来找到鞍点来解决部分微分方程（PDE）的初始值问题（IVP）。我们的方法是灵活的，可以选择时间和空间离散方案。它受益于隐式结构并获得巨大的稳定性区域，并通过courant-friedrichs-lewwy（cfl）条件（实际上是通过von Neumann稳定性分析）来克服对网格大小的限制。然而，它是高度可行的且易于实现的。特别是，不需要非线性反转！具体而言，我们使用有限的差异方案和空间方案的不连续的Galerkin方法说明了我们的方法。向后的Euler和向后分化公式，用于随时间的隐式离散化。数值实验说明了方法的有效性和鲁棒性。在未来的工作中，我们将证明，我们在许多其他情况下，用这种原始的偶发梯度方法代替初始值演化方程的想法具有很大的优势。
In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential equation (PDE) by casting it as a saddle point of a min-max problem and using iterative optimization methods to find the saddle point. Our approach is flexible with the choice of both time and spatial discretization schemes. It benefits from the implicit structure and gains large regions of stability, and overcomes the restriction on the mesh size in time by explicit schemes from Courant--Friedrichs--Lewy (CFL) conditions (really via von Neumann stability analysis). Nevertheless, it is highly parallelizable and easy-to-implement. In particular, no nonlinear inversions are required! Specifically, we illustrate our approach using the finite difference scheme and discontinuous Galerkin method for the spatial scheme; backward Euler and backward differentiation formulas for implicit discretization in time. Numerical experiments illustrate the effectiveness and robustness of the approach. In future work, we will demonstrate that our idea of replacing an initial-value evolution equation with this primal-dual hybrid gradient approach has great advantages in many other situations.
