Advection-dispersion equation derivation

When pollution is established and its plume is mapped dynamic dispersivity may be derived by way of selecting advection-dispersion equation of mass transfer (see below) and value of 6, which produce modeled concentration distribution for tracer, adequate to a natural one. One difficulty in such determination is that the initial concentration and amoimt of tracer are often not known. 

Truncation error arises from approximating each of the various space and time derivatives in the transport equation. The error resulting from the derivative in the advection term is especially notable and has its own name. It is known as numerical dispersion because it manifests itself in the calculation results in the same way as hydrodynamic dispersion. 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [205]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. 

An effective approximation to Equation 1 is obtained by segmenting the water body of interest into n volume elements of volume Vj and representing the derivatives in Equation 1 by differences. Let V be the n X n diagonal matrix of volumes V, A, the n X n matrix of dispersive and advective transport terms SPy the n vector of source terms SPjy averaged over the volume Vjy- and P, the n vector of concentrations P which are the concentrations in the volumes. Then the finite difference equations can be expressed as a vector differential equation... 

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