The method of lines is a widely used algorithm for solving parabolic partial differential equations that could benefit greatly from implementation on Grid computing environments. This paper outlines the issues involved in executing method-of-lines codes on a Grid and in developing model-driven adaptive control strategies for these codes. We have developed a parameterizable benchmark called MOL that captures a wide range of realistic method-of-lines codes. We are using this benchmark to develop performance models that can be used to achieve specific optimality criteria under the available (and dynamically varying) resources of a Grid environment, and under user-specified goals for solution error and computational rate-of-progress. We are developing a componentization strategy that can enable effective adaptive control of MOL, as well as language and compiler support that can simplify the development of adaptive distributed applications. If successful, this work should yield a much better understanding than we have at present of how an important class of parallel numerical applications can be executed effectively in a dynamic Grid environment.