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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 ...
Solution of equation: 1. For solving the one- dimensional convection- diffusion problem we have to express equation (8) at all the grid nodes. 2. Now obtained set of algebraic equations is then solved to obtain the distribution of the transported property .
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]
The hybrid difference scheme [1] [2] is a method used in the numerical solution for convection–diffusion problems. It was introduced by Spalding (1970). It is a combination of central difference scheme and upwind difference scheme as it exploits the favorable properties of both of these schemes. [3] [4]
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
Over the past 20 years many numerical techniques have been developed to solve convection-diffusion equations and none are problem-free, but false diffusion is one of the most serious problems and a major topic of controversy and confusion among numerical analysts.
Upwind differencing scheme for convection — first-order scheme for convection–diffusion problems; Lax–Wendroff theorem — conservative scheme for hyperbolic system of conservation laws converges to the weak solution; Alternating direction implicit method (ADI) — update using the flow in x-direction and then using flow in y-direction
Gradient of parabola is used to evaluate diffusion terms. If F w > 0 and F e > 0 and if we use above equations for the convective terms and central differencing for the diffusion terms, the discretized form of the one-dimensional convection–diffusion transport equation will be written as: