Finite Difference Methods for Linear Transport Equations with Sobolev Velocity Fields

DiPerna and Lions (Invent Math 98(3):511–547, 1989) established the existence and uniqueness results for weak solutions to linear transport equations with Sobolev velocity fields. Motivated by fluid mechanics, this paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is \(L^p\) $L^p$ -strongly convergent in the class of DiPerna–Lions weak solutions. The second method is based on an implicit scheme with \(L^2\) $L^2$ -estimates, where the discrete Helmholtz–Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and \(L^2\) $L^2$ -strongly convergent in the class of DiPerna–Lions weak solutions. The key point for both of the methods is to obtain fine \(L^2\) $L^2$ -bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna–Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method for sharp interfaces involving transport equations, where rigorous discrete approximation of level-sets and their geometric quantities is discussed.