| Abstract [eng] |
Differential equations often arise in real problems, however, are usually solved only using numerical methods. Boundary value problem is even more difficult than initial value problem, as in boundary problems it is neccessary to solve system of equations. In educational literature usually only several simpler difference schemes are analysed, however, in full spectrum – from theory of stability to efficiency. But we know that it is possible to implement finite difference schemes using higher order approximations. In this paper we implement them seeking not efficiency but generality – we analyse broad accuracy spectrum of finite difference schemes. Folowing theory, higher order approximations would lead to more accurate solutions, however, in real simulations, computer’s errors put limits to this. Besides, full theoretical analysis is also not possible when analyzing higher order schemes. In this paper there are analysed several boundary value problems: ordinary linear differential equations, partial differential equations of two types: eliptic and parabolic, with two variables. In this paper there are implemented some general methods for solving these problems, where one can choose order of approximation. For analyzing stability of implemented schemes simple numerical analysis is chosen. For inicializing parabolic problem (initial-boundary problem) extrapolation idea is applied. For stability analysis some known properties of the solution are used to draw converging domains. Also, solution with higher order residual terms is found using only extrapolation, and comparative analysis is provided. |
