Collocation method on modelling of groundwater flow

Engineering systems or problems can be expressed usually with partial differential equations in one, two or three dimensions. But because of complexity of problem comes from nature itself and considerable amount of limit conditions, differential equations cannot be solved precisely by mathematical methods. At this point approximate solutions with specific error assumption are presented to differential equations which cannot be solved absolutely. Beside classical methods like finite differences, finite elements and finite volumes, in last years collocation methods are used frequently in solving engineering problems. Collocation method that provides more accurate by using less tie points, can degrade given differential equation like finite elements and finite differences, to linear equation system that is bounded to unknown function values on tie point selected before for solution area.Until today many methods developed for acquiring approximate solutions and some of them cannot be applied due to technological insufficiency. Technological advances make many applicable methods in theory. Each method has their own advantages and disadvantages in respect to needed computation capacity, time or precision.In this study, by examining two-dimensional groundwater flow, a numerical model developed with collocation method. For reference, the problem is solved with finite differences method in open and close forms. First of all, 2 dimensional partial differential equation which has exact solution is solved and then that solution is used to compare to finite difference method and collocation method. This comparison shows us, that the collocation method is better than the finite differences method. For the second example a groundwater flow problem is solved with finite differences and collocation methods. MATLAB is used for programming.