A Direct Finite - Di erence Scheme for SolvingPDEs over

Classical nite-diierence methods apply only over regions that are rectangular or decomposed in several rectangular subdomains. We call a nite-diierence scheme direct if it can be applied over any region decomposed by means of a logically-rectangular grid. We introduce a second-order and easy-to-use direct scheme to solve boundary-value problems. Deduction of the method is elementary and its results compare well with other methods. The design of numerical methods to solve partial diierential equations (PDEs) must take in account the geometrical complexity of the physical region where the problem is deened. The classical nite-diierence schemes are well-suited for simple regions and, in that case, their implementation is straightforward. Over diicult regions, the nite-element method is commonly used. It has a strong mathematical-background supporting it and tools for subdividing the region into simpler cells are used; however, the complexity of data structure and the amount of memory required to its implementation has suggested the search for simpler methods. The use of the Mapping Method ((8]) and nite-diierence schemes is an alternate way, it requires the knowledge of a diieomorphism {a change of coordinates{ between the unit cell (logical space) and the region. The problem is then transformed and solved, not on the original region, but on the unit cell. Changes of coordinates are known in an analytical form only for few regions; in general, they have to be calculated in a discrete way, this is done using numerical grid generation methods. A disadvantage of the mapping method is that calculations involve the use of the jacobian; if the mapping is nearly singular in a point of the region, the jacobian is close to zero with the corresponding lack of precision. Another approach is to design schemes for discretizing partial derivatives directly on the physical region. As with the mapping method, it requires to have a grid over the region. Some such schemes have been reported by Steinberg, Shashkov, Hyman and Castillo ((10], 6]). In the present paper we introduce a simple bidimensional nite-diierence scheme applied directly on a grid over the physical region. We present some examples over irregular regions in order to show its performance. In section 1, we introduce the kind of problems to be solved. Section 2 contains a brief review of grid generations. Section 3 is devoted to explain our scheme. Finally, in section 4, some test problems are solved and we discuss the results.