Constraint-preserving upwind methods for multidimensional advection equations

A general framework for constructing constraint-preserving numerical methods is presented and applied to a multidimensional divergence-constrained advection equation. This equation is part of a set of hyperbolic equations that evolve a vector field while locally preserving either its divergence or its curl. We discuss the properties of these equations and their relation to ordinary advection. Due to the constraint, such equations form model equations for general evolution equations with intrinsic constraints which appear frequently in physics. The general framework allows the construction of numerical methods that preserve exactly the discretized constraint by special flux distribution. Assuming a rectangular, two-dimensional grid as a first approach, application of this framework leads to a locally constraint-preserving multidimensional upwind method. We prove consistency and stability of the new method and present several numerical experiments. Finally, extensions of the method to the three-dimensional case are described.