Discretization can be understood as replacement of an exact solution of a PDE or a system of PDEs in a continuum domain by an approximate numerical solution in a discrete domain. Instead of continuous distributions of solution variables we find a finite set of numerical values that represent an approximation of the solution.
Computational Fluid Dynamics
The elliptic equations do not have real characteristics at all. Effect of any perturbation is felt immediately and to full degree in the entire domain of solution. There are no limited domains of influence or dependence
Physically, the elliptic systems describe equilibrium distributions of properties in spatial domains with boundary conditions
The elliptic PDE problems are always of equilibrium type. The solution has to be found at once in the entire domain space.
The quasilinear PDE can be classified into three types according to the existence and form of their characteristics, the special lines in the solution domain. . One aspect is, however, very important for us: The information in the solutions tends to propagate along the characteristics if they exist. This has deep implications not only for the mathematical properties of the solution but also for the choice of numerical methods. To put it simply, different numerical methods must be used for equations of different types.
In this case, we disregard temperature variations across the rod and assume that the temperature T is a function of the coordinate x and time t . The solution domain consists of the space interval [0, x] and the time interval, which can be finite [t(0), t(end)] or extended to infinity [t(0),∞). Different kinds of boundary conditions are possible. The situation when the ends of the rod are kept at constant temperature corresponds to the Dirichlet boundary conditions
The main entity of the PDE analysis is not a separate equation but a complete PDE problem consisting of an equation, domain of solution, boundary and initial conditions. The problem has to be solved in a spatial domain and, in the case of time-dependency, in a time interval between t(0) and t(end) . tend can be a finite number or infinity. Similarly, the domain may have a finite size or extend to infinity in one or several directions. In numerical simulations, the infinite limits of the spatial or time domain are replaced by sufficiently large finite numbers.Boundary conditions have to be imposed at the boundaries of the spatial domain Ω,
In principle, one can say that all parts of the universe are connected to each other by fluxes of heat and mass and, thus, must be included into a good CFD solution. Since such an enterprise is hardly feasible, we have to compromise and formulate CFD problems for finite domains limited by boundaries. Such boundaries …