Implicit fast sweeping method for hyperbolic systems of conservation laws

Abstract

Implicit time-accurate methods are often used to integrate stiff problems where explicit schemes impose severe time step restrictions. This paper presents an efficient numerical framework based on the fast sweeping method (FSM) for solving linear and nonlinear hyperbolic systems of conservation laws. The solution at each discrete location is computed by sweeping the numerical domain in several predetermined directions that follow the causality of the characteristic families. The use of a fractional step strategy eliminates the need for a solution selection criterion while one-sided stencils limit the number of sweeps to at most 2d for d space dimensions. This work focuses on the first-order implicit upwind method since it constitutes the building block for high-order conservative schemes. For problems where the degree of stiffness evolves over time, implicit-explicit hybridization can be accomplished with the same algorithm by simply switching the stencil at each time level. As opposed to traditional implicit solvers, the sweeping method does not require a local time linearization of the fluxes thereby preserving the nonlinear stability properties of the original implicit scheme. It also avoids the large computational and memory requirements associated with solving large block-diagonal systems of equations. A series of one- and two-dimensional test cases are presented for the inviscid Burgers' equation and the reactive Euler equations. The results indicate that the implicit FSM can allow for a major reduction in the number of time steps even in the presence of discontinuous solution profiles.