Numerical solution of two dimensional scalar conservation laws using compact implicit numerical schemes on Cartesian meshes

This paper deals with the numerical solution of conservation laws in the two dimensional case using a novel compact implicit time discretization that enables applications of fast algebraic solvers. We present details for the second order accurate parametric scheme based on the finite volume method including simple variants of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) approximations. To avoid oscillatory numerical solutions for large time steps, we propose limiting in time which is provable non-oscillatory in the case of linear advection equation with variable velocity. We present numerical experiments for representative linear and nonlinear problems.