Simulating Partial Differential equations in the World-Wide Web

This paper describes the extensions we have added to our special-purpose objectoriented continous simulation language (OOCSMP) to solve partial differential equations. We also describe the procedure we have used to generate semiautomatically a web description of the numerical resolution of partial differential equations. INTRODUCTION The currently most successful hypermedia system is the World Wide Web (WWW), wich has many advantages on traditional hypertext applications. This has brought around the current proliferation of educational courses in the WWW, which run from a simple transposition of lecture notes, to pages including more sophisticated elements, such as animated graphics, simulations and so forth. We have been working for some time on the developement of advanced simulation tools, that simplify the generation of educational courses on the WWW. The language we are using is an extension of the old CSMP (Continuous System Modelling Program) language, sponsored by IBM [1]. We call the new language OOCSMP [2], for its main difference with CSMP is the addition of object-oriented constructs which makes it much easier the simulation of complex system based on the mutual interaction of many similar agents. We have used this language to build a course on Newton’s gravitation and the solar system [3], a course on ecology [4], and a basic course on electronics [5] Many physical phenomena encountered in engineering, fluid mechanics, chemistry and other fields are modelled by means of differential equations [6]. Usually the problem addressed is too complicated to be solved by classical analytical methods, and a numerical approach is necessary. We have extended OOCSMP to handle the numerical resolution of partial differential equations (PDEs). PDEs can be solved with OOCSMP using combinations of well known methods such as the finite element method [7] and several schemes of the finite differences method [8] ( implicit and explicit). Domains can be discretized by mixing several techniques such as Delaunay triangulation [9] , isoparametric elements or interpolation meshing [10] and grid generation solving two elliptic equations[11]. In this paper, we are presenting some examples of the use of partial differential equations organized as a set of web pages containing examples of elliptic, hyperbolic and parabolic equations in 1D and 2D. A short example of grid generation is also included. The pages have been generated automatically with OOCSMP. 1. SOLVING PDE’s WITH OOCSMP We have extended OOCSMP to solve systems of PDE’s of the second order in one or two spatial dimensions, plus (optionally) time. The model equation to be solved is: Fig. 1 : Model equation 1.1 Spatial Domains The equations have to be solved in some spatial domain. Domains can be declared in OOCSMP by means of primitives. The domain primitives in 2D (and its shapes once discretized using an interpolation mesh or grid ) are : Fig.2:Circular sectors Fig,3:Quadrilaterals Fig.4:Quadrilaterals whose Fig.5:Triangles edges are splines In 1D the only primitive is the line. The syntax for all these primitives is very similar : DOMAIN := ( , , ) where the boundary conditions can be specified in one or more edges and can be of Dirichlet or Newmann type. 1.2 Discretizing Domains The next step to solve the PDE is to discretize the domains previously declared. Meshing a domain with OOCSMP can be made by using several techniques such as : • Delaunay triangulation. C x y t d dt c x y t du dt C x y t du dt A x y t d dx a x y t du dx A x y t d dy a x y t du dy A x y t d dx a x y t du dy B x y t du dx B tt tt t