Problem solving environments for the numerical solution of partial differential equations

Abstract

The complexity and sophistication of numerical codes for the simulation of complex problems modelled by partial differential equations (PDEs) has increased greatly over the last decade. This makes it difficult for those without direct knowledge of the PDE software to employ it efficiently. Problem Solving Environments (PSEs) are seen as a way of making it possible to provide an easy-to-use layer surrounding the numerical software. The users can then concentrate on gaining an understanding of the physical problem through the results the code is providing. PSEs aim to aid novice and expert users in the problem specification process and to provide a natural way to solve the problem. They also decrease the time spent on the problem solving process. This study is concerned with the construction of a PSE for the numerical solution of PDEs. This is one area where PSEs can be used to particularly good effect because the solution process is complicated and error prone. The driving of numerical software and the construction of mathematical models used by the software pose problems for users of the software. The interpretation of results provided by the numerical code may also be difficult. It will be shown how PSEs can remedy these issues by allowing the user to easily specify and solve the problem. The construction of a prototype PSE is achieved through the utilisation and integration of existing scientific software tools and systems. An examination of the solution process of PDEs is used to identify the various components required in a PSE for such problems. The PSE makes use of an open design environment and incorporates the knowledge of the users and developers of the numerical code together with a set of generic software tools based on emerging standards. This combination of tools allows the PSE to automate the solution procedure for a number of PDE problems. Finally, the success of this approach to building PSEs is examined by reference to an engineering PDE problem