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In this article, a concept of implicit methods for scalar conservation lawsin one or more spatial dimensions allowing also for source terms of varioustypes is presented. This material is a significant extension of previous workof the first author [3]. Implicit notions are developed that are centeredaround a monotonicity criterion. We demonstrate a connection between anumerical scheme and a discrete entropy inequality, which is based on aclassical approach by Crandall and Majda. Additionally, three implicit methodsare investigated using the developed notions. Next, we conduct a convergenceproof which is not based on a classical compactness argument. Finally, thetheoretical results are confirmed by various numerical tests.

Implicit monotone difference methods for scalar conservation laws with source terms