Implicit Finite Difference Methods for Hyperbolic Conservation Laws

Hyperbolic conservation laws (HCLs) are a class of partial differential equations that model transport processes. Many important phenomena in natural sciences are described by them. In this paper we consider finite difference methods for the approximation of HCLs. As HCLs describe an evolution in time, one may distinguish explicit and implicit schemes by the corresponding time integration mechanism employed by them. Explicit numerical schemes are well-analysed. In the explicit setting, the monotonicity property of a method is the key to approximate the physically relevant entropy solution of a HCL. However, there does not exist a rigorous general approach to implicit monotone methods in the literature up to now. In the current work, this open issue is addressed. We propose monotonicity conditions for fully implicit schemes, and we prove that they are meaningful. The relation between an implicit monotone scheme and a discrete entropy inequality is constructed in a similar fashion as in the classic explicit approach of Crandall and Majda. The convergence of implicit monotone schemes is verified in a framework that does not rely on a compactness property of the underlying function space. All proofs are given for the case of a scalar HCL in multiple space dimensions. They can easily be extended to HCLs with source terms or specialised to the 1-D case. We apply the developed notion of monotonicity by investigating implicit variants of some well known explicit monotone schemes. As a surprising result, a stability restriction on the time step size may arise. This contradicts the usual intuition that implicit schemes give unconditional stability. Such a restriction is established for the implicit Lax-Friedrichs scheme, and it is illustrated by numerical tests.