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The first term of the equation reflects the unsteadiness of the flow and is absent in case of steady flows. The finite volume integration of the governing equation is carried out over a control volume and also over a finite time step ∆t. The control volume integration of the steady part of the equation is similar to the steady state governing ...
Finite volume method for two dimensional diffusion problem. The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. The general equation for steady diffusion can be easily derived from the general transport equation for property Φ by deleting transient and convective terms [1] where,
Heat equation. Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.
The convection–diffusion equation is a combination of the diffusion and convection ( advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the ...
This article describes how to use a computer to calculate an approximate numerical solution of the discretized equation, in a time-dependent situation. In order to be concrete, this article focuses on heat flow, an important example where the convection–diffusion equation applies. However, the same mathematical analysis works equally well to ...
Alternating-direction implicit method. In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, [ 1] and can be formulated to construct solutions in a memory ...
v. t. e. In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. [ 1] It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable.
Discretization in the space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations.