High Order Adaptive Implicit Method for Conservation Laws

Numerical solution of conservation laws with large variations in the local CFL number, requires impractically small time steps for stable, explicit time integration. Implicit time stepping with large time steps is therefore necessary but is computationally expensive and consequently limited to first-order discretization schemes. A numerical scheme that is accurate and at the same time allows large time steps is clearly desirable. We develop a numerical formulation based on the Adaptive Implicit Method (AIM) that achieves both objectives by treating implicitly only a few grid cells that restrict the time step size, and by using high-order spatial discretization in both the implicit and explicit cells. Our novel approach solves the fully coupled transport and flow equations with high order-accuracy within the AIM framework to reduce the computational cost and improve the solution accuracy. We show that high-order discretization is computationally efficient when combined with AIM and leads to significant improvements in accuracy compared to the first-order solution.