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